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newnumal5p1.txt. May 13, 1992
The set of plaintext ASCII files numalinx.txt newnumal5p1.txt newnumal5p2.txt newnumal5p3.txt together contains an update of the index and manual of the library NUMAL of Algol 60 procedures in numerical mathematics as published in the Mathematical Centre publication: P.W. Hemker (ed.)[1981]: NUMAL. Numerical Procedures in ALGOL 60. 7 volumes. MC Syllabus 47, Mathematical Centre, Amsterdam.
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1SECTION : 1.1.1 (DECEMBER 1979) PAGE 1
AUTHOR: P.A.BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FIVE PROCEDURES.
INIVEC INITIALIZES A ( PART OF A ) VECTOR WITH A CONSTANT.
INIMAT INITIALIZES A ( PART OF A ) MATRIX WITH A CONSTANT.
INIMATD INITIALIZES ELEMENTS A[I, I+SHIFT], I= LR(1)UR OF A MATRIX.
INISYMD INITIALIZES A (PART OF A) CODIAGONAL OF A SYMMETRIC MATRIX,
WHOSE UPPERTRIANGLE IS STORED COLUMNWISE IN A ONE-DIMENSIONAL
ARRAY.
INISYMROW INITIALIZES A (PART OF A) ROW OF A SYMMETRIC MATRIX,WHOSE
UPPERTRIANGLE IS STORED COLUMNWISE IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
INITIALIZATION.
SUBSECTION: INIVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" INIVEC(L, U, A, X); "VALUE" L,U,X;
"INTEGER" L,U; "REAL" X; "ARRAY" A;
"CODE" 31010;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER INDEX OF THE VECTOR A, RESPECTIVELY;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], THE ARRAY TO BE INITIALIZED;
X: <ARITHMETIC EXPRESSION>;
INITIALIZATION CONSTANT.
LANGUAGE: COMPASS.
1SECTION : 1.1.1 (DECEMBER 1979) PAGE 2
SUBSECTION: INIMAT.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" INIMAT(LR, UR, LC, UC, A, X); "VALUE" LR,UR,LC,UC,X;
"INTEGER" LR,UR,LC,UC; "REAL" X; "ARRAY" A;
"CODE" 31011;
FORMAL PARAMETERS:
LR,UR,LC,UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX, AND LOWER AND UPPER COLUMN-INDEX
OF THE MATRIX A, RESPECTIVELY;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR : UR, LC : UC], THE ARRAY TO BE INITIALIZED;
X: <ARITHMETIC EXPRESSION>;
INITIALIZATION CONSTANT.
LANGUAGE: COMPASS.
SUBSECTION: INIMATD.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" INIMATD(LR, UR, SHIFT, A, X); "VALUE" LR,UR,SHIFT,X;
"INTEGER" LR,UR,SHIFT; "REAL" X; "ARRAY" A;
"CODE" 31012;
FORMAL PARAMETERS:
LR,UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX OF THE CODIAGONAL TO BE
INITIALIZED;
SHIFT: <ARITHMETIC EXPRESSION>;
DISTANCE BETWEEN DIAGONAL AND CODIAGONAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR : UR, LR + SHIFT : UR + SHIFT], THE ARRAY TO
BE INITIALIZED;
X: <ARITHMETIC EXPRESSION>;
INITIALIZATION CONSTANT.
LANGUAGE: COMPASS.
1SECTION : 1.1.1 (DECEMBER 1979) PAGE 3
SUBSECTION: INISYMD.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" INISYMD(LR, UR, SHIFT, A, X); "VALUE" LR,UR,SHIFT,X;
"INTEGER" LR,UR,SHIFT; "REAL" X; "ARRAY" A;
"CODE" 31013;
FORMAL PARAMETERS:
LR,UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX OF AN UPPER CODIAGONAL ( OF A
SYMMETRIC MATRIX OF ORDER N ) TO BE INITIALIZED;
LR AND UR SHOULD SATISFY : LR >= 1, UR <= N;
SHIFT: <ARITHMETIC EXPRESSION>;
DISTANCE BETWEEN DIAGONAL AND CODIAGONAL, (0 =< SHIFT < N);
A: <ARRAY IDENTIFIER>;
A ONE-DIMENSIONAL ARRAY A[1 : N * (N+1)//2] CONTAINING THE
COLUMNWISE STORED UPPERTRIANGLE OF A SYMMETRIC MATRIX,
SUCH THAT THE (I,J) - TH ELEMENT OF THE MATRIX IS
A[ (J-1) * J//2 + I ]; J = 1,...,N; I = 1,...,J;
X: <ARITHMETIC EXPRESSION>;
INITIALIZATION CONSTANT.
LANGUAGE: ALGOL 60.
SUBSECTION: INISYMROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" INISYMROW(L, U, I, A, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A;
"CODE" 31014;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER INDEX OF ROW-ELEMENT TO BE INITIALIZED;
I: <ARITHMETIC EXPRESSION>;
ROW INDEX;
A: <ARRAY IDENTIFIER>;
A ONE-DIMENSIONAL ARRAY A[1 : N * (N+1)//2];
ARRAY A SHOULD CONTAIN A COLUMNWISE STORED UPPERTRIANGLE OF
A SYMMETRIC MATRIX OF ORDER N,
SUCH THAT THE (I,J) - TH ELEMENT OF THE MATRIX IS
A[ (J - 1) * J//2 + I ]; J = 1, ... ,N; I = 1, ... ,J.
FOR FIXED ORDER N, THE PARAMETERS L, U AND I SHOULD
SATISFY THE CONDITIONS : 1 <=L<= N, 1 <=U<= N, 1 <=I<= N ;
X: <ARITHMETIC EXPRESSION>;
INITIALIZATION CONSTANT.
LANGUAGE: ALGOL 60.
1SECTION : 1.1.1 (DECEMBER 1979) PAGE 4
SOURCE TEXT(S):
THE PROCEDURES INIVEC, INIMAT AND INIMATD ARE WRITTEN IN COMPASS, AN
EQUIVALENT ALGOL 60 TEXT OF THESE COMPASS ROUTINES IS GIVEN.
"CODE" 31010;
"PROCEDURE" INIVEC(L, U, A, X); "VALUE" L,U,X;
"INTEGER" L,U; "REAL" X; "ARRAY" A;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= X;
"EOP"
"CODE" 31011;
"PROCEDURE" INIMAT(LR, UR, LC, UC, A, X); "VALUE" LR,UR,LC,UC,X;
"INTEGER" LR,UR,LC,UC; "REAL" X; "ARRAY" A;
"BEGIN" "INTEGER" J;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"FOR" J:= LC "STEP" 1 "UNTIL" UC "DO" A[LR, J]:= X
"END" INIMAT;
"EOP"
"CODE" 31012;
"PROCEDURE" INIMATD(LR, UR, SHIFT, A, X); "VALUE" LR,UR,SHIFT,X;
"INTEGER" LR,UR,SHIFT; "REAL" X; "ARRAY" A;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO" A[LR, LR + SHIFT]:= X;
"EOP"
"CODE" 31013;
"PROCEDURE" INISYMD(LR, UR, SHIFT, A, X); "VALUE" LR,UR,SHIFT,X;
"INTEGER" LR,UR,SHIFT; "REAL" X; "ARRAY" A;
"BEGIN" SHIFT:= ABS(SHIFT); UR:= UR + SHIFT + 1; SHIFT:=LR + SHIFT;
LR := (SHIFT - 3) * SHIFT // 2 + LR;
"FOR" LR := SHIFT + LR "WHILE" SHIFT < UR "DO"
"BEGIN" A[LR]:= X; SHIFT:= SHIFT + 1 "END"
"END" INISYMD;
"EOP"
"CODE" 31014;
"PROCEDURE" INISYMROW(L, U, I, A, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A;
"BEGIN" "INTEGER" K;
"IF" L <= I "THEN"
"BEGIN" K:= (I - 1) * I//2; L := K + L;
K := ("IF" U < I "THEN" U "ELSE" I) + K;
"FOR" L:= L "STEP" 1 "UNTIL" K "DO" A[L]:= X;
L := I + 1
"END";
"IF" U>I "THEN""FOR" K:=(L-1)*L//2+I, K+L-1 "WHILE" L<= U "DO"
"BEGIN" A[K]:= X; L:= L + 1 "END"
"END" INISYMROW;
"EOP"
1SECTION : 1.1.2 (DECEMBER 1979) PAGE 1
AUTHOR: P.A.BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS SIX PROCEDURES.
DUPVEC COPIES THE VECTOR GIVEN IN ARRAY B[L+SHIFT : U+SHIFT] TO THE
VECTOR GIVEN IN ARRAY A[L:U].
DUPVECROW COPIES THE ROW VECTOR GIVEN IN ARRAY B[I:I, L:U] TO THE
VECTOR GIVEN IN ARRAY A[L:U].
DUPROWVEC COPIES THE VECTOR GIVEN IN ARRAY B[L:U] TO THE ROW VECTOR
GIVEN IN ARRAY A[I:I, L:U].
DUPVECCOL COPIES THE COLUMN VECTOR GIVEN IN ARRAY B[L:U, J:J] TO
THE VECTOR GIVEN IN ARRAY A[L:U].
DUPCOLVEC COPIES THE VECTOR GIVEN IN ARRAY B[L:U] TO THE COLUMN
VECTOR GIVEN IN ARRAY A[L:U, J:J].
DUPMAT COPIES THE MATRIX GIVEN IN ARRAY B[L:U, I:J] TO THE MATRIX
GIVEN IN ARRAY A[L:U, I:J].
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
DUPLICATION.
SUBSECTION: DUPVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPVEC(L, U, SHIFT, A, B); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A,B;
"CODE" 31030;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER VECTOR-INDEX, RESPECTIVELY;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L + SHIFT : U + SHIFT], B IS COPIED
INTO A.
LANGUAGE: COMPASS.
1SECTION : 1.1.2 (DECEMBER 1979) PAGE 2
SUBSECTION: DUPVECROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPVECROW(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"CODE" 31031;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER VECTOR ( COLUMN )-INDEX, RESPECTIVELY;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[I : I, L : U], B IS COPIED INTO A.
LANGUAGE: COMPASS.
SUBSECTION: DUPROWVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPROWVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"CODE" 31032;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER VECTOR ( COLUMN )-INDEX, RESPECTIVELY;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[L : U], B IS COPIED INTO A.
LANGUAGE: COMPASS.
1SECTION : 1.1.2 (DECEMBER 1979) PAGE 3
SUBSECTION: DUPVECCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPVECCOL(L, U, J, A, B); "VALUE" L,U,J;
"INTEGER" L,U,J; "ARRAY" A,B;
"CODE" 31033;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER VECTOR ( ROW )-INDEX, RESPECTIVELY;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L : U, I : I], B IS COPIED INTO A.
LANGUAGE: COMPASS.
SUBSECTION: DUPCOLVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPCOLVEC(L, U, J, A, B); "VALUE" L,U,J;
"INTEGER" L,U,J; "ARRAY" A,B;
"CODE" 31034;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER VECTOR ( ROW )-INDEX, RESPECTIVELY;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U], B IS COPIED INTO A.
LANGUAGE: COMPASS.
1SECTION : 1.1.2 (DECEMBER 1979) PAGE 4
SUBSECTION: DUPMAT.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" DUPMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"CODE" 31035;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX, RESPECTIVELY;
I,J: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER COLUMN-INDEX, RESPECTIVELY;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : J], B[L : U, I : J], B IS COPIED INTO
A.
LANGUAGE: COMPASS.
1SECTION : 1.1.2 (DECEMBER 1979) PAGE 5
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
"CODE" 31030;
"PROCEDURE" DUPVEC(L, U, SHIFT, A, B); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= B[L+SHIFT];
"EOP"
"CODE" 31031;
"PROCEDURE" DUPVECROW(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= B[I,L];
"EOP"
"CODE" 31032;
"PROCEDURE" DUPROWVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= B[L];
"EOP"
"CODE" 31033;
"PROCEDURE" DUPVECCOL(L, U, J, A, B); "VALUE" L,U,J;
"INTEGER" L,U,J; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= B[L,J];
"EOP"
"CODE" 31034;
"PROCEDURE" DUPCOLVEC(L, U, J, A, B); "VALUE" L,U,J;
"INTEGER" L,U,J; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,J]:= B[L];
"EOP"
"CODE" 31035;
"PROCEDURE" DUPMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"BEGIN" "INTEGER" K;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"FOR" K:= I "STEP" 1 "UNTIL" J "DO" A[L,K]:= B[L,K]
"END" DUPMAT;
"EOP"
1SECTION : 1.1.3 (DECEMBER 1979) PAGE 1
AUTHORS: P.A.BEENTJES, C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FIVE PROCEDURES.
MULVEC STORES X TIMES THE VECTOR GIVEN IN ARRAY B[L+SHIFT:U+SHIFT]
INTO THE VECTOR GIVEN IN ARRAY A[L:U].
MULROW STORES X TIMES THE ROW VECTOR GIVEN IN ARRAY B[J:J,L:U] INTO
THE ROW VECTOR GIVEN IN ARRAY A[I:I,L:U].
MULCOL STORES X TIMES THE COLUMN VECTOR GIVEN IN ARRAY B[L:U,J:J]
INTO THE COLUMN VECTOR GIVEN IN ARRAY A[L:U,I:I].
COLCST MULTIPLIES THE COLUMN VECTOR GIVEN IN ARRAY A[L:U,J:J] BY X.
ROWCST MULTIPLIES THE ROW VECTOR GIVEN IN ARRAY A[I:I,L:U] BY X.
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
MULTIPLICATIONS.
SUBSECTION: MULVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" MULVEC(L, U, SHIFT, A, B, X); "VALUE" L,U,SHIFT,X;
"INTEGER" L,U,SHIFT; "REAL" X; "ARRAY" A,B;
"CODE" 31020;
FORMAL PARAMETERS:
L,U: <ARITMETIC EXPRESSION>;
LOWER AND UPPER VECTOR-INDEX, RESPECTIVELY;
SHIFT: <ARITHMETIC EXPRESSION>;
SUBSCRIPT-SHIFTING PARAMETER;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L + SHIFT : U + SHIFT], THE PRODUCT OF
THE CONTENTS OF B ARE STORED IN A.
X: <ARITHMETIC EXPRESSION>;
MULTIPLICATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.3 (DECEMBER 1979) PAGE 2
SUBSECTION: MULROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" MULROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 31021;
FORMAL PARAMETERS:
L,U: <ARITMETIC EXPRESSION>;
LOWER AND UPPER COLUMN-INDEX, RESPECTIVELY;
I,J: <ARITHMETIC EXPRESSION>;
ROW-INDICES OF THE ROW VECTORS A AND B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[J : J, L : U], THE CONTENTS OF B
MULTIPLIED BY X ARE STORED INTO A.
X: <ARITHMETIC EXPRESSION>;
MULTIPLICATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: MULCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" MULCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 31022;
FORMAL PARAMETERS:
L,U: <ARITMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX, RESPECTIVELY;
I,J: <ARITHMETIC EXPRESSION>;
COLUMN-INDICES OF THE COLUMN VECTORS A AND B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U, J : J], THE CONTENTS OF B
MULTIPLIED BY X ARE STORED INTO A;
X: <ARITHMETIC EXPRESSION>;
MULTIPLICATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.3 (DECEMBER 1979) PAGE 3
SUBSECTION: COLCST.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" COLCST(L, U, J, A, X); "VALUE" L,U,J,X;
"INTEGER" L,U,J; "REAL" X; "ARRAY" A;
"CODE" 31131;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER ROW-INDEX, RESPECTIVELY;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, J : J];
X: <ARITHMETIC EXPRESSION>;
MULTIPLICATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: ROWCST.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ROWCST(L, U, I, A, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A;
"CODE" 31132;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER COLUMN-INDEX, RESPECTIVELY;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U];
X: <ARITHMETIC EXPRESSION>;
MULTIPLICATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.3 (DECEMBER 1979) PAGE 4
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
"CODE" 31020;
"PROCEDURE" MULVEC(L, U, SHIFT, A, B, X); "VALUE" L,U,SHIFT,X;
"INTEGER" L,U,SHIFT; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= B[L+SHIFT]*X;
"EOP"
"CODE" 31021;
"PROCEDURE" MULROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= B[J,L]*X;
"EOP"
"CODE" 31022;
"PROCEDURE" MULCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,I]:= B[L,J]*X;
"EOP"
"CODE" 31131;
"PROCEDURE" COLCST(L, U, J, A, X); "VALUE" L,U,J,X;
"INTEGER" L,U,J; "REAL" X; "ARRAY" A;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,J]:= A[L,J] * X;
"EOP"
"CODE" 31132;
"PROCEDURE" ROWCST(L, U, I, A, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= A[I,L] * X;
"EOP"
1SECTION : 1.1.4.1 (DECEMBER 1975) PAGE 1
AUTHORS: T.J.DEKKER, J.C.P.BUS, J.WOLLESWINKEL.
CONTRIBUTORS: P.A.BEENTJES, J.C.P.BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 741215.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS NINE PROCEDURES.
VECVEC:= SCALAR PRODUCT OF THE VECTOR GIVEN IN ARRAY A[L:U] AND
ARRAY B[SHIFT + L : SHIFT + U].
MATVEC:= SCALAR PRODUCT OF THE ROW VECTOR GIVEN IN ARRAY A[I:I,L:U]
AND THE VECTOR GIVEN IN ARRAY B[L:U].
TAMVEC:= SCALAR PRODUCT OF THE COLUMN VECTOR GIVEN IN ARRAY
A[L:U, I:I] AND THE VECTOR GIVEN IN ARRAY B[L:U].
MATMAT:= SCALAR PRODUCT OF THE ROW VECTOR GIVEN IN ARRAY A[I:I,L:U]
AND THE COLUMN VECTOR IN ARRAY B[L:U, J:J].
TAMMAT:= SCALAR PRODUCT OF THE COLUMN VECTORS GIVEN IN ARRAY
A[L:U, I:I] AND ARRAY B[L:U, J:J].
MATTAM := SCALAR PRODUCT OF THE ROW VECTORS GIVEN IN ARRAY
A[I:I,L:U] AND ARRAY B[J:J, L:U].
SEQVEC := SCALAR PRODUCT OF THE VECTORS GIVEN IN ARRAY
A[IL : IL + (U+L-1)*(U-L)//2] AND ARRAY B[SHIFT + L : SHIFT + U],
WHERE THE ELEMENTS OF THE FIRST VECTOR ARE A[IL+(J+L-1)*(J-L)//2]
FOR J = L, ..., U.
SCAPRD1:= SCALAR PRODUCT OF THE VECTORS GIVEN IN ARRAY
A[MIN(LA, LA + (N - 1) * SA) : MAX(LA,LA + (N - 1) * SA)] AND ARRAY
B[MIN(LB, LB + (N - 1) * SB) : MAX(LB,LB + (N - 1) * SB)] WHERE THE
ELEMENTS OF THE VECTORS ARE A[LA+(J-1)*SA] AND B[LB+(J-1)*SB]
FOR J = 1, ..., N.
SYMMATVEC := THE SCALARPRODUCT OF ( A PART OF ) A VECTOR AND
( A PART OF ) A ROW OF A SYMMETRIC MATRIX , WHOSE UPPERTRIANGLE IS
GIVEN COLUMNWISE IN AN ONE-DIMENSIONAL ARRAY.
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
SCALAR PRODUCTS.
1SECTION : 1.1.4.1 (DECEMBER 1979) PAGE 2
SUBSECTION: VECVEC.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" VECVEC(L, U, SHIFT, A, B); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A,B;
"CODE" 34010;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER OF THE VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L + SHIFT : U + SHIFT].
LANGUAGE: COMPASS.
SUBSECTION: MATVEC.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" MATVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"CODE" 34011;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[L : U].
LANGUAGE: COMPASS.
1SECTION : 1.1.4.1 (DECEMBER 1979) PAGE 3
SUBSECTION: TAMVEC.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" TAMVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"CODE" 34012;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U].
LANGUAGE: COMPASS.
SUBSECTION: MATMAT.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" MATMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"CODE" 34013;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A AND COLUMN-INDEX OF THE
COLUMN VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[L : U, J : J].
LANGUAGE: COMPASS.
1SECTION : 1.1.4.1 (DECEMBER 1979) PAGE 4
SUBSECTION: TAMMAT.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" TAMMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"CODE" 34014;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
COLUMN-INDICES OF THE COLUMN VECTORS A AND B, RESPECTIVELY;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U, J : J].
LANGUAGE: COMPASS.
SUBSECTION: MATTAM.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" MATTAM(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"CODE" 34015;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
ROW-INDICES OF THE ROW VECTORS A AND B, RESPECTIVELY;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[J : J, L : U].
LANGUAGE: COMPASS.
1SECTION : 1.1.4.1 (DECEMBER 1979) PAGE 5
SUBSECTION: SEQVEC.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" SEQVEC(L, U, IL, SHIFT, A, B);
"VALUE" L, U, IL, SHIFT; "INTEGER" L, U, IL, SHIFT; "ARRAY" A, B;
"CODE" 34016;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
IL: <ARITHMETIC EXPRESSION>;
LOWER BOUND OF THE VECTOR A;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER OF THE VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q], B[L + SHIFT, U + SHIFT];
THE VALUES OF P AND Q SHOULD SATISFY P <= IL AND Q >=
IL+(U+L-1)*(U-L)//2).
LANGUAGE: COMPASS.
SUBSECTION: SCAPRD1.
CALLING SEQUENCE:
HEADING:
"REAL" "PROCEDURE" SCAPRD1(LA, SA, LB, SB, N, A, B);
"VALUE" LA,SA,LB,SB,N; "INTEGER" LA,SA,LB,SB,N; "ARRAY" A,B;
"CODE" 34017;
FORMAL PARAMETERS:
N: <ARITHMETIC EXPRESSION>;
UPPER BOUND OF THE RUNNING SUBSCRIPT;
LA,LB: <ARITHMETIC EXPRESSION>;
LOWER BOUNDS OF THE VECTORS A AND B, RESPECTIVELY;
SA,SB: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETERS OF THE VECTORS A AND B,
RESPECTIVELY;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q], B[R : S];
THE SUBSCRIPTS ABOVE AND THE VALUES OF LA( +(J-1)*SA ) AND
LB( +(J-1)*SB ),J = 1(1)N SHOULD NOT CONTRADICT EACH OTHER.
LANGUAGE: COMPASS.
1SECTION : 1.1.4.1 (MARCH 1977) PAGE 6
SUBSECTION: SYMMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" SYMMATVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"CODE" 34018;
SYMMATVEC:= THE VALUE OF THE SCALAR PRODUCT OF THE VECTORS GIVEN
IN ARRAY A[P:Q] AND ARRAY B[L:U] , WHERE THE ELEMENTS
OF THE FIRST VECTOR ARE: IF L<I THEN A[(I-1)*I//2 + J],
J=L,..., MIN(U, I-1) AND A[(J-1)*J//2 + I], J=I,..., U,
RESPECTIVELY , OTHERWISE A[(J-1)*J//2 + I], J=L,..., U.
THE MEANING OF THE FORMAL PARAMETERS IS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE VECTOR B, RESPECTIVELY; L >=1;
I: <ARITHMETIC EXPRESSION>;
ROW INDEX OF THE MATRIX A; I >= 1;
A: <ARRAY IDENTIFIER>;
A ONE-DIMENSIONAL ARRAY A[P : Q] WITH: IF I > L THEN
P=(I-1)*I//2 + L ELSE P=(L-1)*L//2 + I AND IF I > U THEN
Q=(I-1)*I//2 + U ELSE Q=(U-1)*U//2 + I;
B: <ARRAY IDENTIFIER>;
A ONE-DIMENSIONAL ARRAY B[L:U];
PROCEDURES USED:
VECVEC = CP34010,
SEQVEC = CP34016.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SEE REFERENCE [2].
REFERENCES :
[1] T.J.DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA,PART 1,
MATHEMATICAL CENTRE TRACT 22,AMSTERDAM (1970)
[2] J.C.P.BUS.
MINIMALISERING VAN FUNCTIES VAN MEERDERE VARIABELEN,
MATHEMATICAL CENTRE, NR 29/72,AMSTERDAM (1972)
1SECTION : 1.1.4.1 (DECEMBER 1979) PAGE 7
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES, EXCEPT FOR SYMMATVEC ARE WRITTEN IN COMPASS,
AN EQUIVALENT ALGOL 60 TEXT OF THESE COMPASS ROUTINES IS GIVEN.
0"CODE" 34010;
"REAL" "PROCEDURE" VECVEC(L, U, SHIFT, A, B); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[K] * B[SHIFT + K] + S;
VECVEC:= S
"END" VECVEC;
"EOP"
0"CODE" 34011;
"REAL" "PROCEDURE" MATVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[I,K] * B[K] + S;
MATVEC:= S
"END" MATVEC;
"EOP"
0"CODE" 34012;
"REAL" "PROCEDURE" TAMVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[K,I] * B[K] + S;
TAMVEC:= S
"END" TAMVEC;
"EOP"
0"CODE" 34013;
"REAL" "PROCEDURE" MATMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[I,K] * B[K,J] + S;
MATMAT:= S
"END" MATMAT
1SECTION : 1.1.4.1 (DECEMBER 1975) PAGE 8
;
"EOP"
0"CODE" 34014;
"REAL" "PROCEDURE" TAMMAT(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[K,I] * B[K,J] + S;
TAMMAT:= S
"END" TAMMAT;
"EOP"
0"CODE" 34015;
"REAL" "PROCEDURE" MATTAM(L, U, I, J, A, B); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" S;
S:= 0;
"FOR" K:=L "STEP" 1 "UNTIL" U "DO" S:= A[I,K] * B[J,K] + S;
MATTAM:= S
"END" MATTAM;
"EOP"
0"CODE" 34016;
"REAL" "PROCEDURE" SEQVEC(L, U, IL, SHIFT, A, B);
"VALUE" L,U,IL,SHIFT; "INTEGER" L,U,IL,SHIFT; "ARRAY" A,B;
"BEGIN" "REAL" S;
S:= 0;
"FOR" L:=L "STEP" 1 "UNTIL" U "DO"
"BEGIN" S:= A[IL] * B[L + SHIFT] + S; IL:= IL + L "END";
SEQVEC:= S
"END" SEQVEC;
"EOP"
0"CODE" 34017;
"REAL" "PROCEDURE" SCAPRD1(LA, SA, LB, SB, N, A, B);
"VALUE" LA,SA,LB,SB,N; "INTEGER" LA,SA,LB,SB,N; "ARRAY" A,B;
"BEGIN" "REAL" S;"INTEGER" K;
S:= 0;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:=A[LA] * B[LB] + S; LA:= LA + SA; LB:= LB + SB "END";
SCAPRD1:= S
"END" SCAPRD1;
"EOP"
0"CODE" 34018;
"REAL" "PROCEDURE" SYMMATVEC(L, U, I, A, B); "VALUE" L,U,I;
"INTEGER" L,U,I; "ARRAY" A,B;
"BEGIN" "INTEGER" K, M;
M:= "IF" L > I "THEN" L "ELSE" I; K:= M * (M - 1) // 2;
SYMMATVEC:= VECVEC(L, "IF" I <= U "THEN" I-1 "ELSE" U, K, B, A)
+ SEQVEC(M, U, K + I, 0, A, B)
"END" SYMMATVEC;
"EOP"
1SECTION : 1.1.4.2 (DECEMBER 1979) PAGE 1
AUTHOR : D.WINTER.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 741215.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS FIVE PROCEDURES :
FULMATVEC CALCULATES THE VECTOR A * B, WHERE A IS A GIVEN MATRIX
AND B IS A VECTOR.
FULTAMVEC CALCULATES THE VECTOR A' * B, WHERE A' IS THE TRANSPOSED
OF THE MATRIX A AND B IS A VECTOR.
FULSYMMATVEC CALCULATES THE VECTOR A * B, WHERE A IS A SYMMETRIC
MATRIX WHOSE UPPERTRIANGLE IS STORED COLUMNWISE IN A
ONE-DIMENSIONAL ARRAY AND B IS A VECTOR.
RESVEC CALCULATES THE RESIDUAL VECTOR A * B + X * C, WHERE A IS A
GIVEN MATRIX, B AND C ARE VECTORS AND X IS A SCALAR.
SYMRESVEC CALCULATES THE RESIDUAL VECTOR A * B + X * C, WHERE A IS
A SYMMETRIC MATRIX WHOSE UPPERTRIANGLE IS STORED IN A
ONE-DIMENSIONAL ARRAY, B AND C ARE VECTORS AND X IS A SCALAR.
KEYWORDS :
ELEMENTARY PROCEDURE,
VECTOR OPERATION.
1SECTION : 1.1.4.2 (DECEMBER 1975) PAGE 2
SUBSECTION: FULMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" FULMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31500;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B IS DELIVERED IN C.
LANGUAGE: COMPASS 3.
(FOR THE COMPASS-VERSION, THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE: SEE REFERENCE [1].
SUBSECTION: FULTAMVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" FULTAMVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31501;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LR:UR]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LC:UC];
THE RESULT A' * B IS DELIVERED IN C; HERE A' DENOTES THE
TRANSPOSED OF THE MATRIX A.
1SECTION : 1.1.4.2 (DECEMBER 1975) PAGE 3
LANGUAGE: COMPASS 3.
(FOR THE COMPASS-VERSION, THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE: ELEMENTARY.
SUBSECTION: FULSYMMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" FULSYMMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC, B; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31502;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX; LR >= 1;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX; LC >= 1;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U], WHERE:
L = MIN(LR * (LR - 1) // 2 + LC, LC * (LC - 1) // 2 + LR),
U = MAX(UR * (UR - 1) // 2 + UC, UC * (UC - 1) // 2 + UR)
AND FOR I <= J THE (I,J)-TH ELEMENT, OTHERWISE THE
(J,I)-TH ELEMENT OF THE SYMMETRIC MATRIX SHOULD BE
GIVEN IN A[J * (J - 1) // 2 + I];
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B IS DELIVERED IN C.
1SECTION : 1.1.4.2 (DECEMBER 1975) PAGE 4
PROCEDURES USED:
SYMMATVEC = CP34018.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
ELEMENTARY.
SUBSECTION: RESVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" RESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"CODE" 31503;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE VALUE OF THE MULTIPLYING SCALAR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B + X * C IS OVERWRITTEN ON C.
LANGUAGE: COMPASS 3.
(FOR THE COMPASS-VERSION, THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.1.4.2 (DECEMBER 1975) PAGE 5
SUBSECTION: SYMRESVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" SYMRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, B, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"CODE" 31504;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX; LR >= 1;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX; LC >= 1;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U], WHERE:
L = MIN(LR * (LR - 1) // 2 + LC, LC * (LC - 1) // 2 + LR),
U = MAX(UR * (UR - 1) // 2 + UC, UC * (UC - 1) // 2 + UR)
AND THE (I,J)-TH ELEMENT OF THE SYMMETRIC MATRIX SHOULD BE
GIVEN IN A[J * (J - 1) // 2 + I];
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE VALUE OF THE MULTIPLYING SCALAR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B + X * C IS DELIVERED IN C.
PROCEDURES USED:
SYMMATVEC = CP34018.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
ELEMENTARY.
REFERENCES:
[1].T.J.DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART1,
MATHEMATICAL CENTRE TRACT 22, AMSTERDAM (1970).
[2].J.C.P.BUS.
MINIMALISERING VAN FUNKTIES VAN MEERDERE VARIABELEN,
MATHEMATICAL CENTRE, NR 29/72, AMSTERDAM (1972).
1SECTION : 1.1.4.2 (DECEMBER 1975) PAGE 6
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES, EXCEPT FOR FULSYMMATVEC AND SYMRESVEC ARE
WRITTEN IN COMPASS 3, AN EQUIVALENT ALGOL TEXT OF THESE COMPASS
ROUTINES IS GIVEN.
0"CODE" 31500;
"PROCEDURE" FULMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
C[LR]:= MATVEC(LC, UC, LR, A, B);
"END" FULMATVEC;
"EOP"
0"CODE" 31501;
"PROCEDURE" FULTAMVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN"
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
C[LC]:= TAMVEC(LR, UR, LC, A, B);
"END" FULTAMVEC;
"EOP"
0"CODE" 31502;
"PROCEDURE" FULSYMMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC, B; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
C[LR]:= SYMMATVEC(LC, UC, LR, A, B)
"END" FULSYMMATVEC;
"EOP"
0"CODE" 31503;
"PROCEDURE" RESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
C[LR]:= MATVEC(LC, UC, LR, A, B) + C[LR] * X
"END" RESVEC;
"EOP"
0"CODE" 31504;
"PROCEDURE" SYMRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, B, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
C[LR]:= SYMMATVEC(LC, UC, LR, A, B) + C[LR] * X
"END" SYMRESVEC;
"EOP"
1SECTION : 1.1.4.3 (DECEMBER 1979) PAGE 1
AUTHOR: D.T.WINTER.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 751208.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS PROCEDURES THAT MULTIPLY A GIVEN MATRIX BY A
(GENERALIZED) HOUSEHOLDER MATRIX, I.E. A MATRIX M = I + X * U * U',
WHERE I IS THE UNIT MATRIX, X A REAL CONSTANT AND U A VECTOR
(CALLED THE HOUSEHOLDER CONSTANT AND HOUSEHOLDER VECTOR,
RESPECTIVELY)
HSHVECMAT PREMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX, THE
HOUSEHOLDER VECTOR HAS BEEN STORED
IN A ONE-DIMENSIONAL ARRAY.
HSHCOLMAT PREMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX,
THE HOUSEHOLDER VECTOR HAS BEEN STORED AS A COLUMN
IN A TWO-DIMENSIONAL ARRAY.
HSHROWMAT PREMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX,
THE HOUSEHOLDER VECTOR HAS BEEN STORED AS A ROW
IN A TWO-DIMENSIONAL ARRAY.
HSHVECTAM POSTMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX,
THE HOUSEHOLDER VECTOR HAS BEEN STORED
IN A ONE-DIMENSIONAL ARRAY.
HSHCOLTAM POSTMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX,
THE HOUSEHOLDER VECTOR HAS BEEN STORED
AS A COLUMN IN A TWO-DIMENSIONAL ARRAY.
HSHROWTAM POSTMULTIPLIES A MATRIX BY A HOUSEHOLDER MATRIX,
THE HOUSEHOLDER VECTOR HAS BEEN STORED
AS A ROW IN A TWO-DIMENSIONAL ARRAY.
KEYWORDS:
HOUSEHOLDER MATRIX
ORTHOGONAL TRANSFORMATION
LANGUAGE: ALGOL 60
1SECTION : 1.1.4.3 (JANUARY 1976) PAGE 2
SUBSECTION: HSHVECMAT
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHVECMAT(LR, UR, LC, UC, X, U, A);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" U, A;
"CODE" 31070;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[LR:UR];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE PREMULTIPLIED BY THE HOUSEHOLDER MATRIX.
PROCEDURES USED:
TAMVEC = CP34012
ELMCOLVEC = CP34022
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
SUBSECTION: HSHCOLMAT
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHCOLMAT(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"CODE" 31071;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
I: <ARITHMETIC EXPRESSION>;
THE COLUMN INDEX OF THE HOUSEHOLDER VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[LR:UR,I:I];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE PREMULTIPLIED BY THE HOUSEHOLDER MATRIX.
PROCEDURES USED:
TAMMAT = CP34014
ELMCOL = CP34023
1SECTION : 1.1.4.3 (JANUARY 1976) PAGE 3
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
SUBSECTION: HSHROWMAT
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHROWMAT(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"CODE" 31072;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
I: <ARITHMETIC EXPRESSION>;
THE ROW INDEX OF THE HOUSEHOLDER VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[I:I,LR:UR];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE PREMULTIPLIED BY THE HOUSEHOLDER MATRIX.
PROCEDURES USED:
MATMAT = CP34013
ELMCOLROW = CP34027
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
SUBSECTION: HSHVECTAM
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHVECTAM(LR, UR, LC, UC, X, U, A);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" U, A;
"CODE" 31073;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[LC:UC];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE POSTMULTIPLIED BY THE HOUSEHOLDER MATRIX.
1SECTION : 1.1.4.3 (JANUARY 1976) PAGE 4
PROCEDURES USED:
MATVEC = CP34011
ELMROWVEC = CP34027
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
SUBSECTION: HSHCOLTAM
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHCOLTAM(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"CODE" 31074;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
I: <ARITHMETIC EXPRESSION>;
THE COLUMN INDEX OF THE HOUSEHOLDER VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[LC:UC,I:I];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE POSTMULTIPLIED BY THE HOUSEHOLDER MATRIX.
PROCEDURES USED:
MATMAT = CP34013
ELMROWCOL = CP34028
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
1SECTION : 1.1.4.3 (JANUARY 1976) PAGE 5
SUBSECTION: HSHROWTAM
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHROWTAM(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"CODE" 31075;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR,UR: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER ROW INDICES;
LC,UC: <ARITHMETIC EXPRESSIONS>;
THE LOWER AND UPPER COLUMN INDICES;
I: <ARITHMETIC EXPRESSION>;
THE ROW INDEX OF THE HOUSEHOLDER VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE HOUSEHOLDER CONSTANT;
U: <ARRAY IDENTIFIER>; "ARRAY" U[I:I,LC:UC];
THE HOUSEHOLDER VECTOR;
A: <ARRAY IDENTIFIER>; "ARRAY" A[LR:UR,LC:UC];
THE MATRIX TO BE POSTMULTIPLIED BY THE HOUSEHOLDER MATRIX.
PROCEDURES USED:
MATTAM = CP34015
ELMROW = CP34024
RUNNING TIME: PROPORTIONAL TO (UC-LC+1)*(UR-LR+1)
SOURCE TEXTS:
0"CODE" 31070;
"PROCEDURE" HSHVECMAT(LR, UR, LC, UC, X, U, A);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
ELMCOLVEC(LR, UR, LC, A, U, TAMVEC(LR, UR, LC, A, U) * X)
"END";
"EOP"
0"CODE" 31071;
"PROCEDURE" HSHCOLMAT(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
ELMCOL(LR, UR, LC, I, A, U, TAMMAT(LR, UR, LC, I, A, U) * X)
"END"
1SECTION : 1.1.4.3 (JANUARY 1976) PAGE 6
;
"EOP"
0"CODE" 31072;
"PROCEDURE" HSHROWMAT(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
ELMCOLROW(LR, UR, LC, I, A, U, MATMAT(LR, UR, I, LC, U, A) * X)
"END";
"EOP"
0"CODE" 31073;
"PROCEDURE" HSHVECTAM(LR, UR, LC, UC, X, U, A);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
ELMROWVEC(LC, UC, LR, A, U, MATVEC(LC, UC, LR, A, U) * X)
"END";
"EOP"
0"CODE" 31074;
"PROCEDURE" HSHCOLTAM(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
ELMROWCOL(LC, UC, LR, I, A, U, MATMAT(LC, UC, LR, I, A, U) * X)
"END";
"EOP"
0"CODE" 31075;
"PROCEDURE" HSHROWTAM(LR, UR, LC, UC, I, X, U, A);
"VALUE" LR, UR, LC, UC, I, X; "INTEGER" LR, UR, LC, UC, I;
"REAL" X; "ARRAY" U, A;
"BEGIN"
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
ELMROW(LC, UC, LR, I, A, U, MATTAM(LC, UC, LR, I, A, U) * X)
"END";
"EOP"
1SECTION : 1.1.5 (APRIL 1974) PAGE 1
AUTHORS: T.J.DEKKER, C.G. VAN DER LAAN.
CONTRIBUTOR: P.A.BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TEN PROCEDURES.
ELMVEC ADDS X TIMES THE VECTOR GIVEN IN ARRAY B[SHIFT+L : SHIFT+U]
TO THE VECTOR GIVEN IN ARRAY A[L:U].
ELMCOL ADDS X TIMES THE COLUMN VECTOR GIVEN IN ARRAY B[L:U, J:J] TO
THE COLUMN VECTOR GIVEN IN ARRAY A[L:U, I:I].
ELMROW ADDS X TIMES THE ROW VECTOR GIVEN IN ARRAY B[J:J, L:U] TO
THE ROW VECTOR GIVEN IN ARRAY A[I:I, L:U].
ELMVECCOL ADDS X TIMES THE COLUMN VECTOR GIVEN IN ARRAY B[L:U, I:I]
TO THE VECTOR GIVEN IN ARRAY A[L:U].
ELMCOLVEC ADDS X TIMES THE VECTOR GIVEN IN ARRAY B[L:U] TO THE
COLUMN VECTOR GIVEN IN ARRAY A[L:U, I:I].
ELMVECROW ADDS X TIMES THE ROW VECTOR GIVEN IN ARRAY B[I:I, L:U] TO
THE VECTOR GIVEN IN ARRAY A[L:U].
ELMROWVEC ADDS X TIMES THE VECTOR GIVEN IN ARRAY B[L:U] TO THE
ROW VECTOR GIVEN IN ARRAY A[I:I, L:U].
ELMCOLROW ADDS X TIMES THE ROW VECTOR GIVEN IN ARRAY B[J:J, L:U] TO
THE COLUMN VECTOR GIVEN IN ARRAY A[L:U, I:I].
ELMROWCOL ADDS X TIMES THE COLUMN VECTOR GIVEN IN ARRAY B[L:U, J:J]
TO THE ROW VECTOR GIVEN IN ARRAY A[I:I, L:U].
MAXELMROW ADDS X TIMES THE ROW VECTOR GIVEN IN ARRAY B[J:J, L:U] TO
THE ROW VECTOR GIVEN IN ARRAY A[I:I, L:U].
MOREOVER, MAXELMROW:= THE VALUE OF THE SECOND SUBSCRIPT OF AN
ELEMENT OF THE NEW ROW VECTOR IN ARRAY A WHICH IS OF MAXIMUM
ABSOLUTE VALUE.
IF, HOWEVER, L > U, THEN MAXELMROW:= L.
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
ELIMINATION.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 2
SUBSECTION: ELMVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMVEC(L, U, SHIFT, A, B, X); "VALUE" L,U,SHIFT,X;
"INTEGER" L,U,SHIFT; "REAL" X; "ARRAY" A,B;
"CODE" 34020;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER OF THE VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L + SHIFT : U + SHIFT];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: ELMCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 34023;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR A;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U, J : J];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 3
SUBSECTION: ELMROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 34024;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
J: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[J : J, L : U];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: ELMVECCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMVECCOL(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"CODE" 34021;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[L : U, I : I];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 4
SUBSECTION ELMCOLVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMCOLVEC(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"CODE" 34022;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[L : U];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: ELMVECROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMVECROW(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"CODE" 34026;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U], B[I : I, L : U];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 5
SUBSECTION: ELMROWVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMROWVEC(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"CODE" 34027;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[L : U];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
SUBSECTION: ELMCOLROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMCOLROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 34029;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR A;
J: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, I : I], B[J : J, L : U], WHEN A = B THEN
CORRECT ELIMINATION IS GUARANTEED ONLY WHEN THE ROW AND
COLUMN ARE DISJUNCT;
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 6
SUBSECTION: ELMROWCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ELMROWCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 34028;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[L : U, J : J], WHEN A = B THEN
CORRECT ELIMINATION IS GUARANTEED ONLY WHEN THE ROW AND
COLUMN ARE DISJUNCT;
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 7
SUBSECTION: MAXELMROW.
CALLING SEQUENCE:
HEADING:
"INTEGER" "PROCEDURE" MAXELMROW(L, U, I, J, A, B, X);
"VALUE" L,U,I,J,X; "INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"CODE" 34025;
MAXELMROW DELIVERS THE INDEX OF THE MAXIMAL ELEMENT AFTER ELIMINA-
TION STEP IN ARRAY A.
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR A;
J: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[I : I, L : U], B[I : I, L : U];
X: <ARITHMETIC EXPRESSION>;
ELIMINATION FACTOR.
LANGUAGE: COMPASS.
REFERENCES:
[1].T.J.DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MATHEMATICAL CENTRE TRACT 22, AMSTERDAM (1970).
1SECTION : 1.1.5 (DECEMBER 1979) PAGE 8
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
ELMVEC
ELMROW
ELMVECCOL
ELMCOLVEC
MAXELMROW
"CODE" 34020;
"PROCEDURE" ELMVEC(L, U, SHIFT, A, B, X); "VALUE" L,U,SHIFT,X;
"INTEGER" L,U,SHIFT; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= A[L] + B[L + SHIFT] * X;
"EOP"
"CODE" 34023;
"PROCEDURE" ELMCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,I]:= A[L,I] + B[L,J] * X;
"EOP"
"CODE" 34024;
"PROCEDURE" ELMROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= A[I,L] + B[J,L] * X;
"EOP"
"CODE" 34021;
"PROCEDURE" ELMVECCOL(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= A[L] + B[L,I] * X;
"EOP"
"CODE" 34022;
"PROCEDURE" ELMCOLVEC(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,I]:= A[L,I] + B[L] * X;
"EOP"
"CODE" 34026;
"PROCEDURE" ELMVECROW(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L]:= A[L] + B[I,L] * X;
"COMMENT"
1SECTION : 1.1.5 (APRIL 1974) PAGE 9
;
"EOP"
"CODE" 34027;
"PROCEDURE" ELMROWVEC(L, U, I, A, B, X); "VALUE" L,U,I,X;
"INTEGER" L,U,I; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= A[I,L] + B[L] * X;
"EOP"
"CODE" 34029;
"PROCEDURE" ELMCOLROW(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[L,I]:= A[L,I] + B[J,L] * X;
"EOP"
"CODE" 34028;
"PROCEDURE" ELMROWCOL(L, U, I, J, A, B, X); "VALUE" L,U,I,J,X;
"INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" A[I,L]:= A[I,L] + B[L,J] * X;
"EOP"
"CODE" 34025;
"INTEGER" "PROCEDURE" MAXELMROW(L, U, I, J, A, B, X);
"VALUE" L,U,I,J,X; "INTEGER" L,U,I,J; "REAL" X; "ARRAY" A,B;
"BEGIN" "INTEGER" K; "REAL" R, S;
S:= 0;
"FOR" K:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[I,K]:= A[I,K] + B[J,K] * X;"IF" ABS(R) > S "THEN"
"BEGIN" S:= ABS(R); L:= K "END"
"END";
MAXELMROW:= L
"END" MAXELMROW;
"EOP"
1SECTION : 1.1.6 (APRIL 1974) PAGE 1
AUTHOR: T.J.DEKKER.
CONTRIBUTOR: P.A.BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS SIX PROCEDURES.
ICHVEC INTERCHANGES THE ELEMENTS OF THE VECTOR GIVEN IN ARRAY
A[L:U] AND ARRAY A[SHIFT + L : SHIFT + U].
ICHCOL INTERCHANGES THE ELEMENTS OF THE COLUMN VECTORS GIVEN IN
ARRAY A[L:U, I:I] AND ARRAY A[L:U, J:J].
ICHROW INTERCHANGES THE ELEMENTS OF THE ROW VECTORS GIVEN IN ARRAY
A[I:I, L:U] AND ARRAY A[J:J, L:U].
ICHROWCOL INTERCHANGES THE ELEMENTS OF THE ROW VECTOR GIVEN IN
ARRAY A[I:I, L:U] AND THE COLUMN VECTOR GIVEN IN ARRAY A[L:U, J:J].
ICHSEQVEC INTERCHANGES THE ELEMENTS OF THE VECTORS GIVEN IN ARRAY
A[IL : IL + (U + L - 1)*(U - L)//2] AND ARRAY A[SHIFT+L : SHIFT+U],
WHERE THE ELEMENTS OF THE FIRST VECTOR ARE A[IL+(J+L-1)*(J-L)//2]
FOR J = L,..., U.
ICHSEQ INTERCHANGES THE ELEMENTS OF THE VECTORS GIVEN
IN ARRAY A[IL : IL + (U + L - 1) * (U - L) // 2] AND ARRAY
A[SHIFT + IL : SHIFT + IL + (U + L - 1) * (U - L) // 2] WHERE THE
ELEMENTS OF THE VECTORS ARE A[IL + (J + L - 1) * (J - L) // 2] AND
A[SHIFT + IL + (J + L - 1) * (J - L) // 2] FOR J = L ,..., U .
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
INTERCHANGING.
1SECTION : 1.1.6 (DECEMBER 1979) PAGE 2
SUBSECTION: ICHVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHVEC(L, U, SHIFT, A); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A;
"CODE" 34030;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q]; P AND Q SHOULD SATISFY: P <= L, Q >= U,
P <= L + SHIFT AND Q >= U + SHIFT.
LANGUAGE: COMPASS.
SUBSECTION: ICHCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHCOL(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"CODE" 34031;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
COLUMN-INDICES OF THE COLUMN VECTORS OF ARRAY A;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, P : Q]; P AND Q SHOULD SATISFY: P <= I,
P <= J, Q >= I AND Q >= J.
LANGUAGE: COMPASS.
1SECTION : 1.1.6 (DECEMBER 1979) PAGE 3
SUBSECTION: ICHROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHROW(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"CODE" 34032;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
ROW-INDICES OF THE ROW VECTORS OF ARRAY A;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q, L : U]; P AND Q SHOULD SATISFY: P <= I,
P <= J, Q >= I AND Q >= J.
LANGUAGE: COMPASS.
SUBSECTION: ICHROWCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHROWCOL(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"CODE" 34033;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR OF ARRAY A;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR OF ARRAY A;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q, R : S]; P, Q, R AND S SHOULD SATISFY:
P <= I, P <= L, Q >= I, Q >= U, R <= J, R <= L, S >= J AND
S >= U, FURTHERMORE THE ROW AND COLUMN TO BE INTERCHANGED
SHOULD BE DISJUNCT.
LANGUAGE: COMPASS.
1SECTION : 1.1.6 (DECEMBER 1979) PAGE 4
SUBSECTION: ICHSEQVEC.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHSEQVEC(L, U, IL, SHIFT, A); "VALUE" L,U,IL,SHIFT;
"INTEGER" L,U,IL,SHIFT; "ARRAY" A;
"CODE" 34034;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
IL: <ARITHMETIC EXPRESSION>;
LOWER BOUND OF THE VECTOR A;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q];
THE SUBSCRIPTS ABOVE AND THE VALUES OF L(+SHIFT), U(+SHIFT)
AND IL+(U+L-1)*(U-L)//2 SHOULD NOT CONTRADICT EACH OTHER.
LANGUAGE: COMPASS.
1SECTION : 1.1.6 (DECEMBER 1979) PAGE 5
SUBSECTION: ICHSEQ.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ICHSEQ(L, U, IL, SHIFT, A); "VALUE" L,U,IL,SHIFT;
"INTEGER" L,U,IL,SHIFT; "ARRAY" A;
"CODE" 34035;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
IL: <ARITHMETIC EXPRESSION>;
LOWER BOUND OF THE VECTOR A;
SHIFT: <ARITHMETIC EXPRESSION>;
INDEX-SHIFTING PARAMETER;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q];
THE SUBSCRIPTS ABOVE AND THE VALUES OF IL+(J+L-1)*(J-L)//2
( +SHIFT ),J = L(1)U, SHOULD NOT CONTRADICT EACH OTHER.
LANGUAGE: COMPASS.
REFERENCES:
[1].T.J.DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MATHEMATICAL CENTRE TRACT 22, AMSTERDAM (1970).
1SECTION : 1.1.6 (DECEMBER 1979) PAGE 6
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
0"CODE" 34030;
"PROCEDURE" ICHVEC(L, U, SHIFT, A); "VALUE" L,U,SHIFT;
"INTEGER" L,U,SHIFT; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[L]; A[L]:= A[L + SHIFT]; A[L + SHIFT]:= R "END"
"END" ICHVEC;
"EOP"
"CODE" 34031;
"PROCEDURE" ICHCOL(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[L,I]; A[L,I]:= A[L,J]; A[L,J]:= R "END"
"END" ICHCOL;
"EOP"
"CODE" 34032;
"PROCEDURE" ICHROW(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[I,L]; A[I,L]:= A[J,L]; A[J,L]:= R "END"
"END" ICHROW
1SECTION : 1.1.6 (APRIL 1974) PAGE 7
;
"EOP"
"CODE" 34033;
"PROCEDURE" ICHROWCOL(L, U, I, J, A); "VALUE" L,U,I,J;
"INTEGER" L,U,I,J; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[I,L]; A[I,L]:= A[L,J]; A[L,J]:= R "END"
"END" ICHROWCOL;
"EOP"
"CODE" 34034;
"PROCEDURE" ICHSEQVEC(L, U, IL, SHIFT, A); "VALUE" L,U,IL,SHIFT;
"INTEGER" L,U,IL,SHIFT; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[IL]; A[IL]:= A[L + SHIFT]; A[L + SHIFT]:= R;
IL:= IL + L
"END"
"END" ICHSEQVEC;
"EOP"
"CODE" 34035;
"PROCEDURE" ICHSEQ(L, U, IL, SHIFT, A); "VALUE" L,U,IL,SHIFT;
"INTEGER" L,U,IL,SHIFT; "ARRAY" A;
"BEGIN" "REAL" R;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= A[IL]; A[IL]:= A[IL + SHIFT]; A[IL + SHIFT]:= R;
IL:= IL + L
"END"
"END" ICHSEQ;
"EOP"
1SECTION : 1.1.7 (DECEMBER 1979) PAGE 1
AUTHOR: P.A.BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730715.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES.
ROTCOL REPLACES THE COLUMN VECTOR X GIVEN IN ARRAY A[L:U, I:I] AND
THE COLUMN VECTOR Y GIVEN IN ARRAY A[L:U, J:J] BY THE VECTORS
CX + SY AND CY - SX.
ROTROW REPLACES THE ROW VECTOR X GIVEN IN ARRAY A[I:I, L:U] AND THE
ROW VECTOR Y GIVEN IN ARRAY A[J:J, L:U] BY THE VECTORS CX + SY AND
CY - SX.
KEYWORDS:
ELEMENTARY PROCEDURE,
VECTOR OPERATIONS,
ROTATION.
SUBSECTION: ROTCOL.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ROTCOL(L, U, I, J, A, C, S); "VALUE" L,U,I,J,C,S;
"INTEGER" L,U,I,J; "REAL" C,S; "ARRAY" A;
"CODE" 34040;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
COLUMN-INDICES OF THE COLUMN VECTORS OF ARRAY A;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L : U, P : Q]; P AND Q SHOULD SATISFY: P <= I,
P <= J, Q >= I AND Q >= J;
C,S: <ARITHMETIC EXPRESSION>;
ROTATION FACTORS.
LANGUAGE: COMPASS.
1SECTION : 1.1.7 (DECEMBER 1979) PAGE 2
SUBSECTION: ROTROW.
CALLING SEQUENCE:
HEADING:
"PROCEDURE" ROTROW(L, U, I, J, A, C, S); "VALUE" L,U,I,J,C,S;
"INTEGER" L,U,I,J; "REAL" C,S; "ARRAY" A;
"CODE" 34041;
FORMAL PARAMETERS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE RUNNING SUBSCRIPT;
I,J: <ARITHMETIC EXPRESSION>;
ROW-INDICES OF THE ROW-VECTORS OF ARRAY A;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[P : Q, L : U]; P AND Q SHOULD SATISFY: P <= I,
P <= J, Q >= I AND Q >= J;
C,S: <ARITHMETIC EXPRESSION>;
ROTATION FACTORS.
LANGUAGE: COMPASS.
REFERENCES:
[1].T.J.DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MATHEMATICAL CENTRE TRACT 22, AMSTERDAM (1970).
1SECTION : 1.1.7 (DECEMBER 1979) PAGE 3
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
"CODE" 34040;
"PROCEDURE" ROTCOL(L, U, I, J, A, C, S); "VALUE" L,U,I,J,C,S;
"INTEGER" L,U,I,J; "REAL" C,S; "ARRAY" A;
"BEGIN" "REAL" X, Y;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" X:= A[L,I]; Y:= A[L,J]; A[L,I]:= X * C + Y * S;
A[L,J]:= Y * C - X * S
"END"
"END" ROTCOL;
"EOP"
"CODE" 34041;
"PROCEDURE" ROTROW(L, U, I, J, A, C, S); "VALUE" L,U,I,J,C,S;
"INTEGER" L,U,I,J; "REAL" C,S; "ARRAY" A;
"BEGIN" "REAL" X, Y;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" X:= A[I,L]; Y:= A[J,L]; A[I,L]:= X * C + Y * S;
A[J,L]:= Y * C - X * S
"END"
"END" ROTROW;
"EOP"
1SECTION : 1.1.8 (OCTOBER 1975) PAGE 1
AUTHORS: C.G. VAN DER LAAN AND J.C.P. BUS.
CONTRIBUTOR: J.C.P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 740921.
BRIEF DESCRIPTION:
INFNRMVEC CALCULATES THE INFINITY-NORM OF A VECTOR;
INFNRMROW CALCULATES THE INFINITY-NORM OF A ROW VECTOR;
INFNRMCOL CALCULATES THE INFINITY-NORM OF A COLUMN VECTOR;
INFNRMMAT CALCULATES THE INFINITY-NORM OF A MATRIX;
ONENRMVEC CALCULATES THE ONE-NORM OF A VECTOR;
ONENRMROW CALCULATES THE ONE-NORM OF A ROW VECTOR;
ONENRMCOL CALCULATES THE ONE-NORM OF A COLUMN VECTOR;
ONENRMMAT CALCULATES THE ONE-NORM OF A MATRIX;
ABSMAXMAT CALCULATES FOR A GIVEN MATRIX THE MODULUS OF AN ELEMENT
WHICH IS OF MAXIMUM ABSOLUTE VALUE;
KEYWORDS:
VECTOR NORMS,
MATRIX NORMS.
1SECTION : 1.1.8 (DECEMBER 1979) PAGE 2
SUBSECTION: INFNRMVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" INFNRMVEC(L, U, K, A);
"VALUE" L, U; "INTEGER" L, U, K; "ARRAY" A;
"CODE" 31061;
INFNRMVEC := MAX( ABS(A[I]), I= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE INDEX OF THE
VECTOR A, RESPECTIVELY;
K: <VARIABLE>;
EXIT:THE FIRST INDEX FOR WHICH ABS(A[I]), I = L, ..., U,
IS MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U].
PROCEDURES USED: NONE.
SUBSECTION: INFNRMROW.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" INFNRMROW(L, U, I, K, A);
"VALUE" L, U, I; "INTEGER" L, U, I, K; "ARRAY" A;
"CODE" 31062;
INFNRMROW := MAX( ABS(A[I,J]), J= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE COLUMN INDEX
OF THE ROW VECTOR A, RESPECTIVELY;
I: <ARITHMETIC EXPRESSION>;
ENTRY:THE ROW INDEX;
K: <VARIABLE>;
EXIT:THE FIRST INDEX FOR WHICH ABS(A[I,J]), J = L, ..., U,
IS MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[I:I,L:U].
PROCEDURES USED: NONE.
1SECTION : 1.1.8 (DECEMBER 1979) PAGE 3
SUBSECTION: INFNRMCOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" INFNRMCOL(L, U, J, K, A);
"VALUE" L, U, J; "INTEGER" L, U, J, K; "ARRAY" A;
"CODE" 31063;
INFNRMCOL := MAX( ABS(A[I,J]), I= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE ROW INDEX OF
THE COLUMN VECTOR A, RESPECTIVELY;
J: <ARITHMETIC EXPRESSION>;
ENTRY:THE COLUMN INDEX;
K: <VARIABLE>;
EXIT:THE FIRST INDEX FOR WHICH ABS(A[I,J]), I = L, ..., U,
IS MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U,J:J].
PROCEDURES USED: NONE.
SUBSECTION: INFNRMMAT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" INFNRMMAT(LR, UR, LC, UC, KR, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, KR; "ARRAY" A;
"CODE" 31064;
INFNRMMAT := MAX( ONENRMROW(LC, UC, I, A), I=LR, ..., UR );
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE ROW INDEX,
RESPECTIVELY;
LC, UC: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE COLUMN INDEX,
RESPECTIVELY;
KR: <VARIABLE>;
EXIT:THE FIRST ROW INDEX FOR WHICH THE ONE-NORM IS MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC].
PROCEDURES USED: ONENRMROW.
1SECTION : 1.1.8 (DECEMBER 1979) PAGE 4
SUBSECTION: ONENRMVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" ONENRMVEC(L, U, A);
"VALUE" L, U; "INTEGER" L, U; "ARRAY" A;
"CODE" 31065;
ONENRMVEC := SUM( ABS(A[I]), I= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE INDEX OF THE
VECTOR A, RESPECTIVELY;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U].
PROCEDURES USED: NONE.
SUBSECTION: ONENRMROW.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" ONENRMROW(L, U, I, A);
"VALUE" L, U, I; "INTEGER" L, U, I; "ARRAY" A;
"CODE" 31066;
ONENRMROW := SUM( ABS(A[I,J]), J= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE COLUMN INDEX
OF THE ROW VECTOR A, RESPECTIVELY;
I: <ARITHMETIC EXPRESSION>;
ENTRY: THE ROW INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[I:I,L:U].
PROCEDURES USED: NONE.
1SECTION : 1.1.8 (DECEMBER 1979) PAGE 5
SUBSECTION: ONENRMCOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" ONENRMCOL(L, U, J, A);
"VALUE" L, U, J; "INTEGER" L, U, J; "ARRAY" A;
"CODE" 31067;
ONENRMCOL := SUM( ABS(A[I,J]), I= L, ..., U );
THE MEANING OF THE FORMAL PARAMETERS IS:
L, U: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE ROW INDEX OF
THE COLUMN VECTOR A, RESPECTIVELY;
J: <ARITHMETIC EXPRESSION>;
ENTRY: THE COLUMN INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U,J:J].
PROCEDURES USED: NONE.
SUBSECTION: ONENRMMAT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" ONENRMMAT(LR, UR, LC, UC, KC, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, KC; "ARRAY" A;
"CODE" 31068;
ONENRMMAT := MAX(ONENRMCOL(LR, UR, J, A), J=LC, ..., UC);
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE ROW INDEX,
RESPECTIVELY;
LC, UC: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE COLUMN INDEX,
RESPECTIVELY;
KC: <VARIABLE>;
EXIT:THE FIRST COLUMN INDEX FOR WHICH THE ONE-NORM IS
MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC].
PROCEDURES USED: ONENRMCOL.
1SECTION : 1.1.8 (OCTOBER 1975) PAGE 6
SUBSECTION: ABSMAXMAT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" ABSMAXMAT(LR, UR, LC, UC, KR, KC, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, KR, KC;
"ARRAY" A;
"CODE" 31069;
ABSMAXMAT := MAX( ABS(A[I,J]), I= LR, ..., UR, J= LC, ..., UC );
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE ROW INDEX,
RESPECTIVELY;
LC, UC: <ARITHMETIC EXPRESSION>;
ENTRY:THE LOWER BOUND AND UPPER BOUND OF THE COLUMN INDEX,
RESPECTIVELY;
KR, KC: <VARIABLE>;
EXIT:THE ROW AND COLUMN INDEX OF AN ELEMENT FOR WHICH THE
MODULUS IS MAXIMAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC].
PROCEDURES USED: INFNRMCOL.
LANGUAGE: COMPASS.
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.1.8 (OCTOBER 1975) PAGE 7
SOURCE TEXT(S):
THE FOLLOWING PROCEDURES ARE WRITTEN IN COMPASS, AN EQUIVALENT ALGOL 60
TEXT OF THESE COMPASS ROUTINES IS GIVEN.
0"CODE" 31061;
"REAL" "PROCEDURE" INFNRMVEC(L, U, K, A); "VALUE" L, U;
"INTEGER" L, U, K; "ARRAY" A;
"BEGIN" "REAL" R, MAX;
MAX:= 0; K:= L;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= ABS(A[L]); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; K:= L "END"
"END";
INFNRMVEC:= MAX
"END" INFNRMVEC;
"EOP"
0"CODE" 31062;
"REAL" "PROCEDURE" INFNRMROW(L, U, I, K, A); "VALUE" L, U, I;
"INTEGER" L, U, I, K; "ARRAY" A;
"BEGIN" "REAL" R, MAX;
MAX:= 0; K:= L;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= ABS(A[I,L]); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; K:= L "END"
"END";
INFNRMROW:= MAX
"END" INFNRMROW;
"EOP"
0"CODE" 31063;
"REAL" "PROCEDURE" INFNRMCOL(L, U, J, K, A); "VALUE" L, U, J;
"INTEGER" L, U, J, K; "ARRAY" A;
"BEGIN" "REAL" R, MAX;
MAX:= 0; K:= L;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" R:= ABS(A[L,J]); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; K:= L "END"
"END";
INFNRMCOL:= MAX
"END" INFNRMCOL
1SECTION : 1.1.8 (OCTOBER 1975) PAGE 8
;
"EOP"
0"CODE" 31064;
"REAL" "PROCEDURE" INFNRMMAT(LR, UR, LC, UC, KR, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, KR; "ARRAY" A;
"BEGIN" "REAL" R, MAX;
MAX:= 0; KR:= LR;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"BEGIN" R:= ONENRMROW(LC, UC, LR, A); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; KR:= LR "END"
"END";
INFNRMMAT:= MAX
"END" INFNRMMAT;
"EOP"
0"CODE" 31065;
"REAL" "PROCEDURE" ONENRMVEC(L, U, A); "VALUE" L, U;
"INTEGER" L, U; "ARRAY" A;
"BEGIN" "REAL" SUM;
SUM:= 0; "FOR" L:= L "STEP" 1 "UNTIL" U "DO"
SUM:= SUM + ABS(A[L]);
ONENRMVEC:= SUM
"END" ONENRMVEC;
"EOP"
0"CODE" 31066;
"REAL" "PROCEDURE" ONENRMROW(L, U, I, A); "VALUE" L, U, I;
"INTEGER" L, U, I; "ARRAY" A;
"BEGIN" "REAL" SUM;
SUM:= 0; "FOR" L:= L "STEP" 1 "UNTIL" U "DO"
SUM:= SUM + ABS(A[I,L]);
ONENRMROW:= SUM
"END" ONENRMROW
1SECTION : 1.1.8 (OCTOBER 1975) PAGE 9
;
"EOP"
0"CODE" 31067;
"REAL" "PROCEDURE" ONENRMCOL(L, U, J, A); "VALUE" L, U, J;
"INTEGER" L, U, J; "ARRAY" A;
"BEGIN" "REAL" SUM;
SUM:= 0; "FOR" L:= L "STEP" 1 "UNTIL" U "DO"
SUM:= SUM + ABS(A[L,J]);
ONENRMCOL:= SUM
"END" ONENRMCOL;
"EOP"
0"CODE" 31068;
"REAL" "PROCEDURE" ONENRMMAT(LR, UR, LC, UC, KC, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, KC; "ARRAY" A;
"BEGIN" "REAL" MAX, R;
MAX:= 0; KC:= LC;
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
"BEGIN" R:= ONENRMCOL(LR, UR, LC, A); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; KC:= LC "END"
"END";
ONENRMMAT:= MAX
"END" ONENRMMAT;
"EOP"
0"CODE" 31069;
"REAL" "PROCEDURE" ABSMAXMAT(LR, UR, LC, UC, I, J, A);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC, I, J; "ARRAY" A;
"BEGIN" "INTEGER" II; "REAL" MAX, R;
MAX:= 0; I:= LR; J:= LC;
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
"BEGIN" R:= INFNRMCOL(LR, UR, LC, II, A); "IF" R > MAX "THEN"
"BEGIN" MAX:= R; I:= II; J:= LC "END"
"END";
ABSMAXMAT:= MAX
"END" ABSMAXMAT;
"EOP"
1SECTION : 1.1.9 (APRIL 1974) PAGE 1
AUTHORS : T.J. DEKKER, W. HOFFMANN.
CONTRIBUTORS: W. HOFFMANN, S.P.N. VAN KAMPEN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731030.
BRIEF DESCRIPTION:
THE PROCEDURE REASCL NORMALIZES THE (NON-NULL) COLUMNS OF A TWO-
DIMENSIONAL ARRAY IN SUCH A WAY THAT, IN EACH COLUMN, AN ELEMENT OF
MAXIMUM ABSOLUTE VALUE EQUALS 1. THE NORMALIZED VECTORS ARE
DELIVERED IN THE CORRESPONDING COLUMNS OF THE ARRAY.
KEYWORDS:
NORMALIZATION,
VECTOR SCALING.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" REASCL(A, N, N1, N2); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A;
"CODE" 34183;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
A TWO-DIMENSIONAL ARRAY A[1:N,N1:N2];
ENTRY: THE N2 - N1 + 1 COLUMN VECTORS MUST BE GIVEN IN A;
EXIT: THE NORMALIZED VECTORS (I.E. IN EACH VECTOR AN
ELEMENT OF MAXIMUM ABSOLUTE VALUE EQUALS 1) ARE
DELIVERED IN THE CORRESPONDING COLUMNS OF A;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF ARRAY A;
N1, N2: <ARITHMETIC EXPRESSION>;
THE LOWER AND UPPER BOUND OF THE COLUMN INDICES OF ARRAY A.
1SECTION : 1.1.9 (APRIL 1974) PAGE 2
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N * (N2 - N1 + 1).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE REF [1].
REFERENCES:
[1].T.J. DEKKER AND W. HOFFMANN.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2.
MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM.
EXAMPLE OF USE:
THE PROCEDURE REASCL IS USED IN REAEIG1, SECTION 3.3.1.2.2.
0SOURCE TEXT(S) :
"CODE" 34183;
"COMMENT" MCA 2413;
"PROCEDURE" REASCL(A, N, N1, N2); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A;
"BEGIN" "INTEGER" I, J; "REAL" S;
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" S:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"IF" ABS(A[I,J]) > ABS(S) "THEN" S:= A[I,J];
"IF" S ^= 0 "THEN"
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" A[I,J]:= A[I,J] / S
"END"
"END" REASCL;
"EOP"
1SECTION : 1.2.3 (MAY 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730928.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURES COMCOLCST AND COMROWCST.
COMCOLCST MULTIPLIES THE COMPLEX COLUMN-VECTOR GIVEN IN ARRAY
AR,AI[L:U,J:J] BY XR+I*XI.
COMROWCST MULTIPLIES THE COMPLEX ROW-VECTOR GIVEN IN ARRAY
AR,AI[I:I,L:U] BY XR+I*XI.
KEYWORDS :
COMPLEX VECTOR OPERATIONS,
MULTIPLICATION.
SUBSECTION: COMCOLCST.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" COMCOLCST(L,U,J,AR,AI,XR,XI);
"VALUE" L,U,J,XR,XI;"INTEGER" L,U,J;"REAL" XR,XI;
"ARRAY" AR,AI;
"CODE" 34352;
THE MEANING OF THE FORMAL PARAMETERS IS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN VECTOR;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTOR;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[L:U,J:J]
ENTRY:
AR : REAL PART,
AI : IMAGINARY PART OF THE COLUMN VECTOR
EXIT:
THE TRANSFORMED COMPLEX COLUMN;
XR,XI: <ARITHMETIC EXPRESSION>;
ENTRY:
XR: REAL PART OF THE MULTIPLICATION FACTOR;
XI: IMAGINARY PART OF THE MULTIPLICATION FACTOR.
PROCEDURES USED: COMMUL = CP34341.
1SECTION : 1.2.3 (DECEMBER 1975) PAGE 2
RUNNING TIME: ROUGHLY PROPORTIONAL TO (U-L+1).
LANGUAGE: ALGOL 60.
SUBSECTION: COMROWCST.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" COMROWCST(L, U, I, AR, AI, XR, XI);
"VALUE" L, U, I, XR, XI; "INTEGER" L, U, I; "REAL" XR, XI;
"ARRAY" AR, AI;
"CODE" 34353;
THE MEANING OF THE FORMAL PARAMETERS IS:
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW VECTOR;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTOR;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY"AR,AI[I:I,L:U];
ENTRY:
AR : REAL PART,
AI : IMAGINARY PART OF THE ROW VECTOR
EXIT:
THE TRANSFORMED COMPLEX ROW;
XR,XI: <ARITHMETIC EXPRESSION>;
XR: REAL PART OF THE MULTIPLICATION FACTOR;
XI: IMAGINARY PART OF THE MULTIPLICATION FACTOR.
PROCEDURES USED: COMMUL = CP34341.
RUNNING TIME: ROUGHLY PROPORTIONAL TO (U-L).
LANGUAGE: ALGOL 60.
1SECTION : 1.2.3 (MAY 1974) PAGE 3
SOURCE TEXT(S) :
0"CODE" 34352;
"PROCEDURE" COMCOLCST(L,U,J,AR,AI,XR,XI);
"VALUE" L,U,J,XR,XI;"INTEGER" L,U,J;"REAL" XR,XI;
"ARRAY" AR,AI;
"BEGIN"
"FOR" L:=L "STEP" 1 "UNTIL" U "DO"
COMMUL(AR[L,J],AI[L,J],XR,XI,AR[L,J],AI[L,J]);
"END" COMCOLCST;
"EOP"
0"CODE" 34353;
"PROCEDURE" COMROWCST(L, U, I, AR, AI, XR, XI);
"VALUE" L, U, I, XR, XI; "INTEGER" L, U, I; "REAL" XR, XI;
"ARRAY" AR, AI;
"BEGIN"
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" COMMUL(AR[I,L], AI[I,L], XR,
XI, AR[I,L], AI[I,L]);
"END" COMROWCST;
"EOP"
1SECTION : 1.2.4 (MAY 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED : 731016.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THREE PROCEDURES:
COMMATVEC CALCULATES THE SCALAR PRODUCT OF A COMPLEX ROWVECTOR
GIVEN IN ARRAY AR,AI[I:I,L:U] AND THE COMPLEX VECTOR GIVEN IN
ARRAY BR,BI[L:U].
HSHCOMCOL TRANSFORMS A COMPLEX VECTOR INTO A VECTOR
PROPORTIONAL TO A UNIT VECTOR;
HSHCOMPRD PREMULTIPLIES A COMPLEX MATRIX WITH A COMPLEX
HOUSEHOLDER MATRIX.
HSHCOMCOL AND HSHCOMPRD ARE AUXILIARY PROCEDURES FOR PREMULTIPLYING
A COMPLEX MATRIX OR VECTOR WITH A COMPLEX HOUSEHOLDER MATRIX;
KEYWORDS:
COMPLEX SCALAR PRODUCTS.
HOUSEHOLDER TRANSFORMATION
SUBSECTION: COMMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" COMMATVEC(L, U, I, AR, AI, BR, BI, RR, RI);
"VALUE" L, U, I; "INTEGER" L, U, I; "REAL" RR, RI;
"ARRAY" AR, AI, BR, BI;
"CODE" 34354;
THE MEANING OF THE FORMAL PARAMETERS IS:
L,U : <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE VECTORS;
I : <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTORS AR AND AI;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[I:I,L:U];
ENTRY:
AR: REAL PART AND
AI: IMAGINARY PART OF THE MATRIX;
1SECTION : 1.2.4 (MAY 1974) PAGE 2
BR,BI : <ARRAY IDENTIFIER>;
"ARRAY" BR,BI[L:U];
ENTRY:
BR: REAL PART AND
BI: IMAGINARY PART OF THE VECTOR;
RR,RI: <VARIABLE>;
EXIT:
RR: THE REAL PART AND
RI: THE IMAGINARY PART OF THE SCALAR PRODUCT.
PROCEDURES USED:MATVEC=CP34011.
RUNNING TIME: PROPORTIONAL TO U-L.
LANGUAGE: ALGOL 60.
SUBSECTION: HSHCOMCOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"BOOLEAN" "PROCEDURE" HSHCOMCOL(L, U, J, AR, AI, TOL, K, C, S, T);
"VALUE" L, U, J, TOL; "INTEGER" L, U, J; "REAL" TOL, K, C, S, T;
"ARRAY" AR, AI;
"CODE" 34355;
HSHCOMCOL DELIVERS THE FOLLOWING BOOLEAN VALUE:
IF AR[L+1,J]**2+AI[L+1,J]**2+...+AR[U,J]**2+AI[U,J]**2>TOL THEN
A TRANSFORMATION IS PERFORMED AND HSHCOMCOL:="TRUE",OTHERWISE
HSHCOMCOL:="FALSE" AND THE VECTOR TO BE TRANSFORMED IS CONSIDERED
TO BE PROPORTIONAL TO THE DESIRED UNIT VECTOR AND NO
TRANSFORMATION IS PERFORMED.
THE MEANING OF THE FORMAL PARAMETERS IS:
L,U,J: <ARITHMETIC EXPRESSION>;
THE COMPLEX VECTOR TO BE TRANSFORMED, MUST BE GIVEN IN
THE J-TH COLUMN FROM ROW L UNTIL ROW U OF A COMPLEX
MATRIX;
AR,AI: <ARRAY IDENTIIER>;
"ARRAY" AR,AI[L:U,J:J];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE VECTOR TO BE
TRANSFORMED MUST BE GIVEN IN THE ARRAYS AR AND AI,
RESPECTIVELY;
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE VECTOR U,
OF THE HOUSEHOLDER MATRIX I-UU"/T (WHERE " DENOTES
CONJUGATING AND TRANSPOSING)ARE DELIVERED IN THE ARRAYS
AR AND AI,RESPECTIVELY,PROVIDED A TRANSFORMATION IS
PERFORMED.IF NO TRANSFORMATION IS PERFORMED THE ARRAYS
AR AND AI ARE UNALTERED;
1SECTION : 1.2.4 (DECEMBER 1979) PAGE 3
TOL: <ARITHMETIC EXPRESSION>;
ENTRY: A TOLERANCE;
(E.G. THE SQUARE OF THE MACHINE PRECISION TIMES A NORM
OF THE MATRIX IN CONSIDERATION);
T: <ARITHMETIC EXPRESSION>;
EXIT:
INFORMATION CONCERNING THE TRANSFORMATION,I.E. THE SCALAR
T OF THE HOUSEHOLDER MATRIX ,PROVIDED A TRANSFORMATION IS
PERFORMED.OTHERWISE,T:=-1;
K,C,S: <VARIABLE>;
EXIT:
THE MODULUS , COSINE AND SINE OF THE ARGUMENT OF THE
FIRST ELEMENT OF THE TRANSFORMED VECTOR ARE DELIVERED IN
K,C AND S,RESPECTIVELY,PROVIDED A TRANSFORMATION IS
PERFORMED.OTHERWISE THE MODULUS,COSINE AND SINE OF THE
ARGUMENT OF THE COMPLEX NUMBER AR[L,J]+AI[L,J]*I ARE
DELIVERED.
PROCEDURES USED:
CARPOL=CP34344,
TAMMAT=CP34014.
RUNNING TIME: PROPORTIONAL TO U-L.
METHOD AND PERFORMANCE:
SEE WILKINSON(1965,P.49,50).
LANGUAGE: ALGOL 60.
1SECTION : 1.2.4 (DECEMBER 1975) PAGE 4
SUBSECTION: HSHCOMPRD.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHCOMPRD(I, II, L, U, J, AR, AI, BR, BI, T);
"VALUE" I, II, L, U, J, T; "INTEGER" I, II, L, U, J; "REAL" T;
"ARRAY" AR, AI, BR, BI;
"CODE" 34356;
THE MEANING OF THE FORMAL PARAMETERS IS:
I,II,L,U: <ARITHMETIC EXPRESSION>;
ENTRY:
THE COMPLEX MATRIX TO BE PREMULTIPLIED, MUST BE GIVEN
IN THE L-TH TO U-TH COLUMN FROM ROW I TO ROW II
OF A COMPLEX MATRIX;
J: <ARITHMETIC EXPRESSION>;
ENTRY:
THE COMPLEX VECTOR V OF THE HOUSEHOLDER MATRIX
I-VV"/T,WHERE " DENOTES TRANSPOSING AND CONJUGATING,
MUST BE GIVEN IN THE J-TH COLUMN FROM ROW I TO ROW
II OF A COMPLEX MATRIX GIVEN IN (BR,BI);
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[I:II,L:U];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE MATRIX TO
BE PREMULTIPLIED,MUST BE GIVEN IN THE ARRAYS AR AND
AI,RESPECTIVELY;
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE
RESULTING MATRIX ARE DELIVERED IN THE ARRAYS AR AND
AI,RESPECTIVELY;
BR,BI: <ARRAY IDENTIFIER>;
"ARRAY" BR,BI[I:II,J:J];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE COMPLEX
VECTOR V OF THE HOUSEHOLDER MATRIX MUST BE GIVEN IN
THE ARRAYS BR AND BI,RESPECTIVELY;
(E.G. AS DELIVERED BY HSHCOMCOL);
T: <ARITHMETIC EXPRESSION>;
ENTRY:
THE SCALAR T OF THE HOUSEHOLDER MATRIX;
(E.G. AS DELIVERED BY HSHCOMCOL);
1SECTION : 1.2.4 (MAY 1974) PAGE 5
PROCEDURES USED:
TAMMAT =CP34014,
ELMCOMCOL=CP34377.
RUNNING TIME: PROPORTIONAL TO (U-L)*(II-I).
LANGUAGE: ALGOL 60.
REFERENCE:
WILKINSON,J.H(1965):
THE ALGEBRAIC EIGENVALUE PROBLEM,
CLARENDON PRESS,OXFORD.
EXAMPLE OF USE:
AS A FORMAL TEST OF THE PROCEDURES HSHCOMCOL AND HSHCOMPRD THE
FOLLOWING MATRIX:
3 4*I
4*I 5
IS TRANSFORMED INTO UPPER TRIANGULAR FORM.
"BEGIN""INTEGER"I;"REAL"K,C,S,T;
"ARRAY"AR,AI[1:2,1:2];
AR[1,1]:=3;AR[1,2]:=AR[2,1]:=0;AR[2,2]:=5;
AI[1,1]:=0;AI[1,2]:=AI[2,1]:=4;AI[2,2]:=0;
"IF"HSHCOMCOL(1,2,1,AR,AI,("-14*5)**2,K,C,S,T)"THEN"
HSHCOMPRD(1,2,2,2,1,AR,AI,AR,AI,T);
OUTPUT(61,"(""("AFTER USE HSHCOMCOL,HSHCOMPRD:")",/,
2(2(-D.D,+D.D,"("*I")",BB),/)")",
AR[1,1],AI[1,1],AR[1,2],AI[1,2],AR[2,1],AI[2,1],AR[2,2],AI[2,2]);
OUTPUT(61,"(""("K, C, S, T,")",/,3(-D.DB),-DD.D,/,")",K,C,S,
T);
"END"
OUTPUT:
AFTER USE HSHCOMCOL,HSHCOMPRD:
8.0+0.0*I 0.0+1.6*I
0.0+4.0*I 6.2+0.0*I
K, C, S, T,
5.0 -1.0 0.0 40.0
1SECTION : 1.2.4 (MAY 1974) PAGE 6
SOURCE TEXT(S) :
0"CODE" 34354;
"PROCEDURE" COMMATVEC(L, U, I, AR, AI, BR, BI, RR, RI);
"VALUE" L, U, I; "INTEGER" L, U, I; "REAL" RR, RI;
"ARRAY" AR, AI, BR, BI;
"BEGIN"
"REAL" MV;
MV:= MATVEC(L, U, I, AR, BR) - MATVEC(L, U, I, AI, BI);
RI:= MATVEC(L, U, I, AI, BR) + MATVEC(L, U, I, AR, BI);
RR:=MV
"END" COMMATVEC;
"EOP"
0"CODE" 34355;
"BOOLEAN" "PROCEDURE" HSHCOMCOL(L, U, J, AR, AI, TOL, K, C, S, T);
"VALUE" L, U, J, TOL; "INTEGER" L, U, J; "REAL" TOL, K, C, S, T;
"ARRAY" AR, AI;
"BEGIN" "REAL" VR, DEL, MOD, H, ARLJ, AILJ;
VR:= TAMMAT(L + 1, U, J, J, AR, AR) + TAMMAT(L + 1, U,
J, J, AI, AI); ARLJ:= AR[L,J]; AILJ:= AI[L,J];
CARPOL(ARLJ, AILJ, MOD, C, S); "IF" VR > TOL "THEN"
"BEGIN" VR:= VR + ARLJ ** 2 + AILJ ** 2; H:= K:= SQRT(VR);
T:= VR + MOD * H;
"IF" ARLJ = 0 "AND" AILJ = 0 "THEN" AR[L,J]:= H "ELSE"
"BEGIN" AR[L,J]:= ARLJ + C * K; AI[L,J]:= AILJ + S * K;
S:= - S
"END";
C:= - C; HSHCOMCOL:= "TRUE"
"END"
"ELSE"
"BEGIN" HSHCOMCOL:= "FALSE"; K:= MOD; T:= - 1 "END"
"END" HSHCOMCOL;
"EOP"
0"CODE" 34356;
"PROCEDURE" HSHCOMPRD(I, II, L, U, J, AR, AI, BR, BI, T);
"VALUE" I, II, L, U, J, T; "INTEGER" I, II, L, U, J; "REAL" T;
"ARRAY" AR, AI, BR, BI;
"BEGIN"
"FOR" L:= L "STEP" 1 "UNTIL" U "DO" ELMCOMCOL(I, II, L, J, AR, AI,
BR, BI, ( - TAMMAT(I, II, J, L, BR, AR) - TAMMAT(I, II, J,
L, BI, AI)) / T, (TAMMAT(I, II, J, L, BI, AR) - TAMMAT(I,
II, J, L, BR, AI)) / T);
"END" HSHCOMPRD;
"EOP"
1SECTION : 1.2.5 (MAY 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET , C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED : 730813.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURES ELMCOMVECCOL, ELMCOMCOL AND
ELMCOMROWVEC.
ELMCOMVECCOL ADDS XR+I*XI TIMES THE COMPLEX COLUMN VECTOR GIVEN
IN ARRAY BR,BI[L:U,J:J] TO THE COMPLEX VECTOR GIVEN IN ARRAY
AR,AI[L:U].
ELMCOMCOL ADDS XR+I*XI TIMES THE COMPLEX COLUMN VECTOR GIVEN IN
ARRAY BR,BI[L:U,J:J] TO THE COMPLEX COLUMN VECTOR GIVEN IN ARRAY
AR,AI[L:U,I:I].
ELMCOMROWVEC ADDS XR+I*XI TIMES THE COMPLEX VECTOR GIVEN IN ARRAY
BR,BI[L:U] TO THE COMPLEX ROW VECTOR GIVEN IN ARRAY AR,AI[I:I,L:U].
KEYWORDS :
COMPLEX VECTOR OPERATIONS ,
ELIMINATION.
SUBSECTION : ELMCOMVECCOL.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" ELMCOMVECCOL(L,U,J,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,J,XR,XI;
"INTEGER" L,U,J;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"CODE" 34376;
THE MEANING OF THE FORMAL PARAMETERS IS :
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE VECTORS;
J: <ARITHMETIC EXPRESSION>;
COLUMN-INDEX OF THE COLUMN VECTORS BR AND BI;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[L:U]
ENTRY:
AR : REAL PART OF THE VECTOR,
AI : IMAGINARY PART OF THE VECTOR.
EXIT:
THE RESULTING VECTOR (SEE ALSO BRIEF DESCRIPTION);
1SECTION : 1.2.5 (DECEMBER 1979) PAGE 2
BR,BI: <ARRAY IDENTIFIER>;
"ARRAY" BR,BI[L:U,J:J];
ENTRY:
BR : REAL PART OF THE COLUMN VECTOR,
BI : IMAGINARY PART OF THE COLUMN VECTOR.
XR,XI: <ARITHMETIC EXPRESSION>;
ENTRY:
XR: REAL PART OF THE ELIMINATION FACTOR;
XI: IMAGINARY PART OF THE ELIMINATION FACTOR .
PROCEDURES USED : ELMVECCOL = CP34021 .
RUNNING TIME : ROUGHLY PROPORTIONAL TO (U-L) .
LANGUAGE: ALGOL 60.
SUBSECTION : ELMCOMCOL.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" ELMCOMCOL(L,U,I,J,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,I,J,XR,XI;
"INTEGER" L,U,I,J;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"CODE" 34377;
THE MEANING OF THE FORMAL PARAMETERS IS :
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE VECTORS;
I,J: <ARITHMETIC EXPRESSION>;
I: COLUMN-INDEX OF THE COLUMN VECTORS AR AND AI;
J: COLUMN-INDEX OF THE COLUMN VECTORS BR AND BI;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[L:U,I:I]
ENTRY:
AR : REAL PART OF THE COLUMN VECTOR,
AI : IMAGINARY PART OF THE COLUMN VECTOR.
EXIT:
THE RESULTING VECTOR (SEE ALSO BRIEF DESCRIPTION);
BR,BI: <ARRAY IDENTIFIER>;
"ARRAY" BR,BI[L:U,J:J]
ENTRY:
BR : REAL PART OF THE COLUMN VECTOR,
BI : IMAGINARY PART OF THE COLUMN VECTOR.
XR,XI: <ARITHMETIC EXPRESSION>;
ENTRY:
XR: REAL PART OF THE ELIMINATION FACTOR;
XI: IMAGINARY PART OF THE ELIMINATION FACTOR .
1SECTION : 1.2.5 (MAY 1974) PAGE 3
PROCEDURES USED : ELMCOL = CP34023 .
RUNNING TIME : ROUGHLY PROPORTIONAL TO (U-L) .
LANGUAGE: ALGOL 60.
SUBSECTION : ELMCOMROWVEC .
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" ELMCOMROWVEC(L,U,I,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,I,XR,XI;
"INTEGER" L,U,I;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"CODE" 34378;
THE MEANING OF THE FORMAL PARAMETERS IS :
L,U: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE VECTORS;
I: <ARITHMETIC EXPRESSION>;
ROW-INDEX OF THE ROW VECTORS AR AND AI;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[I:I,L:U]
ENTRY:
AR : REAL PART OF THE ROW VECTOR,
AI : IMAGINARY PART OF THE ROW VECTOR.
EXIT:
THE RESULTING VECTOR (SEE ALSO BRIEF DESCRIPTION);
BR,BI: <ARRAY IDENTIFIER>;
"ARRAY" BR,BI[L:U]
ENTRY:
BR : REAL PART OF THE VECTOR,
BI : IMAGINARY PART OF THE VECTOR
XR,XI: <ARITHMETIC EXPRESSION>;
ENTRY:
XR: REAL PART OF THE ELIMINATION FACTOR;
XI: IMAGINARY PART OF THE ELIMINATION FACTOR .
PROCEDURES USED : ELMROWVEC = CP34027 .
RUNNING TIME : ROUGHLY PROPORTIONAL TO (U-L) .
LANGUAGE: ALGOL 60.
1SECTION : 1.2.5 (MAY 1974) PAGE 4
EXAMPLE OF USE :
"BEGIN"
"COMMENT" EXAMPLE OF USE ELMCOMCOL;
"REAL" "ARRAY" AR,AI[1:2,1:2];
"INTEGER" I,J;
"PROCEDURE" OUT(K);"INTEGER" K;
OUTPUT(61,"("2(-D,+D,"("*I ")"),/")",
AR[K,1],AI[K,1],AR[K,2],AI[K,2]);
AR[1,1]:=+1;AR[1,2]:=-9;AR[2,1]:=-1;AR[2,2]:=-1;
AI[1,1]:=+2;AI[1,2]:=+2;AI[2,1]:=+2;AI[2,2]:=-2;
OUTPUT(61,"(""("INPUT MATRIX:")",/")");
"FOR" I:=1,2 "DO" OUT(I);
ELMCOMCOL(1,2,2,1,AR,AI,AR,AI,1,-4);
OUTPUT(61,"("/,"("MATRIX AFTER ELIMINATION:")",/")");
OUTPUT(61,"("-D,+D,"("*I")",4B,Z,D/")",
AR[1,1],AI[1,1],AR[1,2],AI[1,2]);
OUT(2)
"END"
OUTPUT:
INPUT MATRIX:
1+2*I -9+2*I
-1+2*I -1-2*I
MATRIX AFTER ELIMINATION:
1+2*I 0
-1+2*I 6+4*I .
1SECTION : 1.2.5 (MAY 1974) PAGE 5
SOURCE TEXT(S) :
0"CODE" 34376;
"PROCEDURE" ELMCOMVECCOL(L,U,J,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,J,XR,XI;
"INTEGER" L,U,J;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"BEGIN"
ELMVECCOL(L,U,J,AR,BR,XR);
ELMVECCOL(L,U,J,AR,BI,-XI);
ELMVECCOL(L,U,J,AI,BR,XI);
ELMVECCOL(L,U,J,AI,BI,XR)
"END" ELMCOMVECCOL;
"EOP"
0"CODE" 34377;
"PROCEDURE" ELMCOMCOL(L,U,I,J,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,I,J,XR,XI;
"INTEGER" L,U,I,J;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"BEGIN"
ELMCOL(L,U,I,J,AR,BR,XR);
ELMCOL(L,U,I,J,AR,BI,-XI);
ELMCOL(L,U,I,J,AI,BR,XI);
ELMCOL(L,U,I,J,AI,BI,XR)
"END" ELMCOMCOL;
"EOP"
0"CODE" 34378;
"PROCEDURE" ELMCOMROWVEC(L,U,I,AR,AI,BR,BI,XR,XI);
"VALUE" L,U,I,XR,XI;
"INTEGER" L,U,I;"REAL" XR,XI;"ARRAY" AR,AI,BR,BI;
"BEGIN"
ELMROWVEC(L,U,I,AR,BR,XR);
ELMROWVEC(L,U,I,AR,BI,-XI);
ELMROWVEC(L,U,I,AI,BR,XI);
ELMROWVEC(L,U,I,AI,BI,XR)
"END" ELMCOMROWVEC;
"EOP"
1SECTION : 1.2.7 (DECEMBER 1979) PAGE 1
AUTHORS : J.J.G. ADMIRAAL, C.G. VAN DER LAAN.
CONTRIBUTORS : J.J.G. ADMIRAAL, H. FIOLET, C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE, UNIVERSITY OF AMSTERDAM.
RECEIVED : 730817.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURES ROTCOMCOL, ROTCOMROW AND CHSH2
ROTCOMCOL REPLACES THE COLUMN VECTOR VR+I*VI GIVEN IN THE ARRAYS
AR,AI[L:U,I:I] AND THE COLUMN VECTOR YR+I*YI GIVEN IN THE ARRAYS
AR,AI[L:U,J:J] BY THE VECTORS (VR+I*VI)*(CR-I*CI)-(YR+I*YI)*S AND
(YR+I*YI)*(CR+I*CI)+(VR+I*VI)*S, RESPECTIVELY.
ROTCOMROW REPLACES THE ROW VECTOR VR+I*VI GIVEN IN THE ARRAYS
AR,AI[I:I,L:U] AND THE ROW VECTOR YR+I*YI GIVEN IN THE ARRAYS
AR,AI[J:J,L:U] BY THE VECTORS (VR+I*VI)*(CR-I*CI)+(YR+I*YI)*S AND
(YR+I*YI)*(CR+I*CI)-(VR+I*VI)*S, RESPECTIVELY.
CHSH2 COMPUTES THE COMPLEX HOUSEHOLDERMATRIX THAT
MAPS THE COMPLEX VECTOR (A1,A2) INTO THE DIRECTION (1,0).
WARNING : IN ROTCOMCOL AND ROTCOMROW THE COSINE IS COMPLEX AND THE
SINE IS REAL, IN CONTRAST TO THIS, IN CHSH2 THE SINE IS COMPLEX
AND THE COSINE IS REAL.
KEYWORDS :
COMPLEX VECTOR OPERATIONS,
ROTATION,
HOUSEHOLDER MATRIX.
1SECTION : 1.2.7 (JANUARY 1976) PAGE 2
SUBSECTION : ROTCOMCOL .
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" ROTCOMCOL(L, U, I, J, AR, AI, CR, CI, S);
"VALUE" L, U, I, J, CR, CI, S; "INTEGER" L, U, I, J;
"REAL" CR, CI, S; "ARRAY" AR, AI;
"CODE" 34357;
THE MEANING OF THE FORMAL PARAMETERS IS :
L,U,I,J: <ARITHMETIC EXPRESSION>;
THE ROTATION IS PERFORMED ON THE COLUMN VECTORS
AR,AI[L:U,I:I] AND AR,AI[L:U,J:J];
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[L:U,I:J];
ENTRY:
AR:THE REAL PARTS OF THE COLUMN VECTORS
AI:THE IMAGINARY PARTS OF THE COLUMN VECTORS
EXIT:
THE RESULTING VECTORS (SEE ALSO BRIEF DESCRIPTION);
CR,CI,S: <ARITHMETIC EXPRESSION>;
ENTRY:
ROTATION FACTORS; SEE ALSO BRIEF DESCRIPTION.
RUNNING TIME : ROUGHLY PROPORTIONAL TO (U-L) .
LANGUAGE: ALGOL 60.
1SECTION : 1.2.7 (JANUARY 1976) PAGE 3
SUBSECTION : ROTCOMROW .
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" ROTCOMROW(L, U, I, J, AR, AI, CR, CI, S);
"VALUE" L, U, I, J, CR, CI, S; "INTEGER" L, U, I, J;
"REAL" CR, CI, S; "ARRAY" AR, AI;
"CODE" 34358;
THE MEANING OF THE FORMAL PARAMETERS IS :
L,U,I,J: <ARITHMETIC EXPRESSION>;
THE ROTATION IS PERFORMED ON THE ROW VECTORS
AR,AI[I:I,L:U] AND AR,AI[J:J,L:U];
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[I:J,L:U];
ENTRY:
AR:THE REAL PARTS OF THE ROW VECTORS
AI:THE IMAGINARY PARTS OF THE ROW VECTORS
EXIT:
THE RESULTING VECTORS (SEE ALSO BRIEF DESCRIPTION);
CR,CI,S: <ARITHMETIC EXPRESSION>;
ENTRY:
ROTATION FACTORS; SEE ALSO BRIEF DESCRIPTION.
PROCEDURES USED : NONE .
RUNNING TIME : ROUGHLY PROPORTIONAL TO (U-L) .
LANGUAGE: ALGOL 60.
1SECTION : 1.2.7 (JANUARY 1976) PAGE 4
EXAMPLE OF USE :
"BEGIN"
"COMMENT" EXAMPLE OF USE ROTCOMCOL;
"REAL" "ARRAY" AR,AI[1:2,1:2];
"INTEGER" I,J;
AR[1,1]:=+4;AR[1,2]:=+5;AR[2,1]:=-5;AR[2,2]:=+4;
AI[1,1]:=+3;AI[1,2]:= 0;AI[2,1]:= 0;AI[2,2]:=-3;
OUTPUT(61,"(""("INPUT MATRIX:")",/")");
OUTPUT(61,"("-D,+D,"("*I")",4B,-D,Z/,BB-D,Z,3B,-D,+D,"("*I")",/")",
AR[1,1],AI[1,1],AR[1,2],AI[1,2],AR[2,1],AI[2,1],AR[2,2],AI[2,2]);
OUTPUT(61,"("//,"("AFTER POSTMULTIPLICATION WITH:")",/")");
OUTPUT(61,"(""(".08-.06*I -.1")",/,
"(" .1 .08+.06*I")",//")");
ROTCOMCOL(1,2,1,2,AR,AI,.08,.06,-.1);
OUTPUT(61,"(""("DELIVERS:")",/")");
OUTPUT(61,"("-D,Z,2BD,Z/,BD,Z,B-D,Z")",
AR[1,1],AI[1,1],AR[1,2],AI[1,2],AR[2,1],AI[2,1],AR[2,2],AI[2,2]);
"END"
OUTPUT:
INPUT MATRIX:
4+3*I 5
-5 4-3*I
AFTER POSTMULTIPLICATION WITH:
.08-.06*I -.1
.1 .08+.06*I
DELIVERS:
1 0
0 1 .
1SECTION : 1.2.7 (DECEMBER 1979) PAGE 5
SUBSECTION: CHSH2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHSH2(A1R,A1I,A2R,A2I,C,SR,SI);
"VALUE" A1R,A1I,A2R,A2I;"REAL" A1R,A1I,A2R,A2I,C,SR,SI;
"CODE" 34611;
THE MEANING OF THE FORMAL PARAMETERS IS:
A1R: <ARITHMETIC EXPRESSION>;
ENTRY: THE REAL PART OF THE FIRST VECTORCOMPONENT;
A1I: <ARITHMETIC EXPRESSION>;
ENTRY: THE IMAGINARY PART OF THE FIRST VECTORCOMPONENT;
A2R: <ARITHMETIC EXPRESSION>;
ENTRY: THE REAL PART OF THE SECOND VECTORCOMPONENT;
A2I: <ARITHMETIC EXPRESSION>;
ENTRY: THE IMAGINARY PART OF THE SECOND VECTORCOMPONENT;
C,SR,SI: <VARIABLE>;
EXIT: THE FACTORS THAT DETERMINE THE HOUSEHOLDER MATRIX.
THE HOUSEHOLDERMATRIX, DEFINED BY:
HA = B
A = (A1,A2)'
B = (-SIGN(A1R)*SQRT(A1*A1+A2*A2),O)',
IS DETERMINED BY:
( -C SR+I*SI)
(SR+I*SI C )
PROCEDURES USED: NONE;
LANGUAGE: ALGOL 60;
METHOD AND PERFORMANCE:
AFTER A CALL OF CHSH2 YOU ARE ABLE TO ROTATE A COMPLEX VECTOR OF
DIMENSION TWO BY MEANS OF THE FACTORS C,SR AND SI.
EXAMPLE OF USE: CHSH2 IS USED IN QZI AND QZIVAL,SECTION 3.4
1SECTION : 1.2.7 (JANUARY 1976) PAGE 6
SOURCE TEXT(S) :
0"CODE" 34357;
"PROCEDURE" ROTCOMCOL(L, U, I, J, AR, AI, CR, CI, S);
"VALUE" L, U, I, J, CR, CI, S; "INTEGER" L, U, I, J;
"REAL" CR, CI, S; "ARRAY" AR, AI;
"BEGIN" "REAL" ARLI, AILI, ARLJ, AILJ;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" ARLI:= AR[L,I]; AILI:= AI[L,I]; ARLJ:= AR[L,J];
AILJ:= AI[L,J];
AR[L,I]:= CR * ARLI + CI * AILI - S * ARLJ;
AI[L,I]:= CR * AILI - CI * ARLI - S * AILJ;
AR[L,J]:= CR * ARLJ - CI * AILJ + S * ARLI;
AI[L,J]:= CR * AILJ + CI * ARLJ + S * AILI;
"END"
"END" ROTCOMCOL;
"EOP"
0"CODE" 34358;
"PROCEDURE" ROTCOMROW(L, U, I, J, AR, AI, CR, CI, S);
"VALUE" L, U, I, J, CR, CI, S; "INTEGER" L, U, I, J;
"REAL" CR, CI, S; "ARRAY" AR, AI;
"BEGIN" "REAL" ARIL, AIIL, ARJL, AIJL;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" ARIL:= AR[I,L]; AIIL:= AI[I,L]; ARJL:= AR[J,L];
AIJL:= AI[J,L];
AR[I,L]:= CR * ARIL + CI * AIIL + S * ARJL;
AI[I,L]:= CR * AIIL - CI * ARIL + S * AIJL;
AR[J,L]:= CR * ARJL - CI * AIJL - S * ARIL;
AI[J,L]:= CR * AIJL + CI * ARJL - S * AIIL;
"END"
"END" ROTCOMROW;
"EOP"
0"CODE" 34611;
"PROCEDURE"CHSH2(A1R,A1I,A2R,A2I,C,SR,SI);
"VALUE"A1R,A1I,A2R,A2I;"REAL"A1R,A1I,A2R,A2I,C,SR,SI;
"BEGIN" "REAL" R;
"IF" A2R^=0 "OR" A2I^=0 "THEN"
"BEGIN" "IF" A1R^=0 "OR" A1I^=0 "THEN"
"BEGIN" R:=SQRT(A1R*A1R+A1I*A1I);C:=R;
SR:=(A1R*A2R+A1I*A2I)/R;SI:=(A1R*A2I-A1I*A2R)/R;
R:=SQRT(C*C+SR*SR+SI*SI);C:=C/R;SR:=SR/R;SI:=SI/R
"END" "ELSE"
"BEGIN" SI:=C:=0;SR:=1 "END"
"END" "ELSE" "BEGIN" C:=1;SR:=SI:=0 "END"
"END" CHSH2;
"EOP"
1SECTION : 1.2.8 (DECEMBER 1975) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731016.
BRIEF DESCRIPTION:
COMEUCNRM CALCULATES THE EUCLIDEAN NORM OF A COMPLEX MATRIX
WITH LW LOWER CODIAGONALS.
KEYWORDS:
EUCLIDEAN NORM,
COMPLEX MATRIX.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" COMEUCNRM(AR, AI, LW, N); "VALUE" N, LW;
"INTEGER" N, LW; "ARRAY" AR, AI;
"CODE" 34359;
COMEUCNRM DELIVERS THE EUCLIDEAN NORM OF A COMPLEX MATRIX WITH LW
LOWER CODIAGONALS;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
LW: <ARITHMETIC EXPRESSION>;
THE NUMBER OF LOWER CODIAGONALS;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[1:N,1:N];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE COMPLEX
MATRIX,WITH LW LOWER CODIAGONALS,MUST BE GIVEN IN THE
ARRAYS AR AND AI,RESPECTIVELY.
1SECTION : 1.2.8 (DECEMBER 1975) PAGE 2
PROCEDURES USED: MATTAM = CP34015.
RUNNING TIME: PROPORTIONAL TO N**2.
LANGUAGE: ALGOL 60.
EXAMPLE OF USE:SEE EIGVALCOM OR EIGCOM (SECTION 3.3.2.2.2).
SOURCE TEXT(S) :
0"CODE" 34359;
"REAL" "PROCEDURE" COMEUCNRM(AR, AI, LW, N); "VALUE" N, LW;
"INTEGER" N, LW; "ARRAY" AR, AI;
"BEGIN" "INTEGER" I, L;
"REAL" R;
R:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" L:= "IF" I > LW "THEN" I - LW "ELSE" 1;
R:= MATTAM(L, N, I, I, AR, AR) + MATTAM(L, N, I,
I, AI, AI) + R;
"END";
COMEUCNRM:= SQRT(R)
"END" COMEUCNRM;
"EOP"
1SECTION : 1.2.9 (DECEMBER 1975) PAGE 1
AUTHORS :
T.J. DEKKER, W. HOFFMANN (COMSCL),
C.G. VAN DER LAAN (SCLCOM).
CONTRIBUTORS:
W. HOFFMANN, S.P.N. VAN KAMPEN (COMSCL),
H. FIOLET, C.G. VAN DER LAAN (SCLCOM).
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731030.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES :
COMSCL NORMALIZES THE REAL AND COMPLEX EIGENVECTORS
GIVEN COLUMNWISE IN A TWO-DIMENSIONAL ARRAY; THE IMAGINARY PARTS OF
THE CORRESPONDING EIGENVALUES MUST BE GIVEN IN A ONE-DIMENSIONAL
ARRAY;
THE EIGENVECTORS ARE NORMALIZED IN SUCH A WAY THAT, IN EACH EIGEN-
VECTOR, AN ELEMENT OF MAXIMUM MODULUS EQUALS 1;
THE NORMALIZED EIGENVECTORS ARE DELIVERED IN THE GIVEN ARRAY.
SCLCOM NORMALIZES THE (NON-NULL) COLUMNS OF A COMPLEX MATRIX
IN SUCH A WAY THAT IN EACH COLUMN AN ELEMENT OF MAXIMUM ABSOLUTE
VALUE BECOMES EQUAL TO ONE.
KEYWORDS:
NORMALIZATION,
SCALING OF COMPLEX EIGENVECTORS,
COMPLEX SCALING.
1SECTION : 1.2.9 (DECEMBER 1975) PAGE 2
SUBSECTION : COMSCL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" COMSCL(A, N, N1, N2, IM); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, IM;
"CODE" 34193;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,N1:N2];
ENTRY: EACH REAL EIGENVECTOR MUST BE GIVEN IN A COLUMN OF
ARRAY A, WHOSE CORRESPONDING ELEMENT OF ARRAY IM
EQUALS 0;
THE REAL AND IMAGINARY PART OF EACH COMPLEX EIGEN-
VECTOR MUST BE GIVEN IN CONSECUTIVE COLUMNS OF ARRAY
A, WHOSE CORRESPONDING ELEMENTS OF ARRAY IM ARE NOT
EQUAL TO 0;
EXIT: THE NORMALIZED EIGENVECTORS (I.E. IN EACH EIGEN-
VECTOR AN ELEMENT OF MAXIMUM MODULUS EQUALS 1) ARE
DELIVERED IN THE CORRESPONDING COLUMNS OF A;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF ARRAY A;
N1, N2: <ARITHMETIC EXPRESSION>;
THE LOWER AND UPPER BOUND OF THE COLUMN INDICES OF ARRAY A;
IM: <ARRAY IDENTIFIER>;
"ARRAY" IM[N1:N2];
THE IMAGINARY PARTS OF THE EIGENVALUES, OF WHICH THE EIGEN-
VECTORS ARE GIVEN IN THE CORRESPONDING COLUMNS OF ARRAY A,
MUST BE GIVEN IN ARRAY IM.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N * (N2 - N1 + 1).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE REF [1].
REFERENCES:
[1].T.J. DEKKER AND W. HOFFMANN.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2.
MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM.
EXAMPLE OF USE:
THE PROCEDURE COMSCL IS USED IN COMEIG1, SECTION 3.3.1.2.2.
1SECTION : 1.2.9 (DECEMBER 1975) PAGE 3
SUBSECTION : SCLCOM.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"SCLCOM(AR,AI,N,N1,N2);
"VALUE"N,N1,N2;"INTEGER"N,N1,N2;"ARRAY"AR,AI;
"CODE" 34360;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[1:N,N1:N2];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE MATRIX OF
WHICH THE COLUMNS ARE TO BE SCALED MUST BE GIVEN IN THE
ARRAYS AR AND AI,RESPECTIVELY;
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE MATRIX WITH
SCALED COLUMNS ARE DELIVERED IN THE ARRAYS AR AND AI,
RESPECTIVELY;
N,N1,N2:<ARITHMETIC EXPRESSION>;
N : ORDER OF THE MATRIX;
N1,N2: THE N1-TH TO N2-TH COLUMN VECTORS ARE TO BE
SCALED.
PROCEDURES USED: COMCOLCST = CP34352.
RUNNING TIME: PROPORTIONAL TO N*(N2-N1).
LANGUAGE: ALGOL 60.
EXAMPLE OF USE: SEE EIGCOM (SECTION 3.3.2.2.2).
1SECTION : 1.2.9 (DECEMBER 1975) PAGE 4
SOURCE TEXT(S) :
"CODE" 34193;
"COMMENT" MCA 2423;
"PROCEDURE" COMSCL(A, N, N1, N2, IM); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, IM;
"BEGIN" "INTEGER" I, J, K;
"REAL" S, U, V, W;
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" S:= 0; "IF" IM[J] ^= 0 "THEN"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" U:= A[I,J] ** 2 + A[I,J + 1] ** 2;
"IF" U > S "THEN" "BEGIN" S:= U; K:= I "END"
"END";
"IF" S ^= 0 "THEN"
"BEGIN" V:= A[K,J] / S; W:= - A[K,J + 1] / S;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" U:= A[I,J]; S:= A[I,J + 1];
A[I,J]:= U * V - S * W;
A[I,J + 1]:= U * W + S * V
"END"
"END";
J:= J + 1
"END"
"ELSE"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"IF" ABS(A[I,J]) > ABS(S) "THEN" S:= A[I,J];
"IF" S ^= 0 "THEN"
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" A[I,J]:= A[I,J] / S
"END"
"END"
"END" COMSCL;
"EOP"
0"CODE" 34360;
"PROCEDURE" SCLCOM(AR, AI, N, N1, N2); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" AR, AI;
"BEGIN" "INTEGER" I, J, K;
"REAL" S, R;
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" S:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= AR[I,J] ** 2 + AI[I,J] ** 2; "IF" R > S "THEN"
"BEGIN" S:= R; K:= I "END"
"END";
"IF" S ^= 0 "THEN" COMCOLCST(1, N, J, AR, AI, AR[K,J] /
S, - AI[K,J] / S)
"END"
"END" SCLCOM;
"EOP"
1SECTION : 1.3.1 (MAY 1974) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730815.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THREE PROECEDURES:
COMABS CALCULATES THE MODULUS OF A COMPLEX NUMBER.
COMSQRT CALCULATES THE SQUARE ROOT OF A COMPLEX NUMBER
CARPOL TRANSFORMS A COMPLEX NUMBER GIVEN IN CARTESIAN COORDINATES
INTO POLAR COORDINATES
KEYWORDS:
COMPLEX NUMBER.
MODULUS.
SQUARE ROOT.
TRANSFORMATION.
CARTESIAN COORDINATES.
POLAR COORDINATES.
SUBSECTION: COMABS.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"COMABS(XR,XI);
"VALUE"XR,XI;"REAL"XR,XI;
"CODE" 34340;
COMABS DELIVERS THE MODULUS OF THE COMPLEX NUMBER XR + I * XI;
THE MEANING OF THE FORMAL PARAMETERS IS:
XR,XI:<ARITHMETIC EXPRESSION>;
ENTRY:XR,XI ARE THE REAL PART AND THE IMAGINARY PART
OF THE COMPLEX NUMBER,RESPECTIVELY.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
1SECTION : 1.3.1 (MAY 1974) PAGE 2
EXAMPLE OF USE:
"BEGIN"
OUTPUT(61,"(""("THE MODULUS OF .3+.4*I EQUALS")",-D.DD")",
COMABS(.3,.4))
"END"
THE MODULUS OF .3+.4*I EQUALS 0.50
SUBSECTION : COMSQRT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMSQRT(AR,AI,PR,PI);
"VALUE"AR,AI;"REAL"AR,AI,PR,PI;
"CODE" 34343;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI:<ARITHMETIC EXPRESSION>;
ENTRY:AR,AI ARE THE REAL PART AND THE IMAGINARY PART
OF THE COMPLEX NUMBER,RESPECTIVELY;
PR,PI;<VARIABLE>;
EXIT:THE REAL PART AND THE IMAGINARY PART OF THE SQUARE ROOT
ARE DELIVERED IN PR AND PI,RESPECTIVELY.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE REPRESENTATION OF THE RESULTING COMPLEX NUMBER IS CHOSEN SUCH
THAT ITS REAL PART IS NONNEGATIVE;THE PROCEDURE IS PROTECTED
AGAINST INTERMEDIATE OVERFLOW.
EXAMPLE OF USE:
"BEGIN""REAL"R,I;
COMSQRT(-3,4,R,I);
OUTPUT(61,"(""("THE SQUARE ROOT OF -3+4*I IS")",-D.DD,+D.DD,"("*I")"
")",R,I);
"END"
THE SQUARE ROOT OF -3+4*I IS 1.00+2.00*I
1SECTION : 1.3.1 (MAY 1974) PAGE 3
SUBSECTION : CARPOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"CARPOL(AR,AI,R,C,S);
"VALUE"AR,AI;"REAL"AR,AI,R,C,S;
"CODE" 34344;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI:<ARITHMETIC EXPRESSION>;
ENTRY:AR,AI ARE THE REAL PART AND THE IMAGINARY PART OF THE
COMPLEX NUMBER ,RESPECTIVELY;
R,C,S:<VARIABLE>;
EXIT:THE MODULUS OF THE COMPLEX NUMBER IS DELIVERED IN R
AND THE COSINE AND THE SINE OF THE ARGUMENT ARE
DELIVERED IN C AND S,RESPECTIVELY;
WHEN AR=AI=0 THEN C:=1 AND R:=S:=0.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
EXAMPLE OF USE:
"BEGIN""REAL"R,C,S;
CARPOL(.3,.4,R,C,S);
OUTPUT(61,"(""("THE POLAR COORDINATES OF .3+.4*I ARE:")",/,
"("MODULUS:")",-D.DD,/,
"("COSINE OF ARGUMENT:")",-D.DD,/,
"("SINE OF ARGUMENT:")",-D.DD")",R,C,S)
"END"
THE POLAR COORDINATES OF .3+.4*I ARE:
MODULUS: 0.50
COSINE OF ARGUMENT: 0.60
SINE OF ARGUMENT: 0.80
1SECTION : 1.3.1 (MAY 1974) PAGE 4
SOURCE TEXT(S):
0"CODE"34340;
"REAL" "PROCEDURE" COMABS(XR,XI);"VALUE" XR,XI;"REAL" XR,XI;
"BEGIN" XR:= ABS(XR); XI:= ABS(XI);
COMABS:= "IF" XI > XR "THEN" SQRT((XR/XI)**2+1)*XI
"ELSE" "IF" XI= 0 "THEN" XR "ELSE" SQRT((XI/XR)**2+1)*XR
"END" COMABS;
"EOP"
0"CODE"34343;
"PROCEDURE" COMSQRT(AR,AI,PR,PI);
"VALUE" AR,AI; "REAL" AR,AI,PR,PI;
"IF" AR=0 & AI= 0 "THEN" PR:= PI:=0 "ELSE"
"BEGIN" "REAL" BR,BI,H;
BR:= ABS(AR); BI:= ABS(AI);
H:= "IF" BI < BR "THEN"
("IF" BR<1 "THEN" SQRT((SQRT((BI/BR)**2+1)*.5+.5)*BR)
"ELSE" SQRT((SQRT((BI/BR)**2+1)*.125+.125)*BR)*2)
"ELSE" "IF" BI<1 "THEN" SQRT((SQRT((BR/BI)**2+1)*BI+BR)*2)*.5
"ELSE" "IF" BR+1= 1 "THEN" SQRT(BI*.5)
"ELSE" SQRT(SQRT((BR/BI)**2+1)*BI*.125+BR*.125)*2;
"IF" AR >= 0 "THEN"
"BEGIN" PR:= H; PI:= AI/H*.5 "END"
"ELSE" "BEGIN" PI:= "IF" AI >= 0 "THEN" H "ELSE" -H;
PR:= BI/H*.5
"END"
"END" COMSQRT;
"EOP"
"CODE"34344;
"PROCEDURE"CARPOL(AR,AI,R,C,S);
"VALUE" AR,AI; "REAL" AR,AI,R,C,S;
"IF" AR=0&AI=0 "THEN"
"BEGIN" C:=1;R:=S:=0 "END"
"ELSE" "BEGIN"
R:="IF" ABS(AR)>ABS(AI) "THEN"
ABS(AR)*SQRT(1+(AI/AR)**2)
"ELSE" ABS(AI)* SQRT(1+(AR/AI)**2);
C:=AR/R;S:=AI/R
"END"CARPOL;
"EOP"
1SECTION : 1.3.2 (MAY 1974) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730815.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES :
COMMUL CALCULATES THE PRODUCT OF TWO COMPLEX NUMBERS.
COMDIV CALCULATES THE QUOTIENT OF TWO COMPLEX NUMBERS.
KEYWORDS:
COMPLEX MULTIPLICATION.
COMPLEX DIVISION.
SUBSECTION COMMUL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMMUL(AR,AI,BR,BI,RR,RI);
"VALUE"AR,AI,BR,BI;"REAL"AR,AI,BR,BI,RR,RI;
"CODE" 34341;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI,BR,BI:<ARITHMETIC EXPRESSION>;
ENTRY:AR,BR ARE THE REAL PARTS OF THE COMPLEX
NUMBERS AND AI,BI ARE THE IMAGINARY PARTS OF
THE COMPLEX NUMBERS;
RR,RI: <VARIABLE>;
EXIT:THE REAL PART AND THE IMAGINARY PART OF THE
RESULTING COMPLEX NUMBER ARE DELIVERED IN RR AND
RI,RESPECTIVELY.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
1SECTION : 1.3.2 (DECEMBER 1975) PAGE 2
EXAMPLE OF USE:
"BEGIN""REAL"R,I;
COMMUL(.1,.2,.3,.4,R,I);
OUTPUT(61,"(""("(.1+.2*I)*(.3+.4*I)=")",-D.DD,+D.DD,"("*I")"")",R,I
)
"END"
(.1+.2*I)*(.3+.4*I)=-0.05+0.10*I
SUBSECTION : COMDIV.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMDIV(XR,XI,YR,YI,ZR,ZI);
"VALUE"XR,XI,YR,YI;"REAL"XR,XI,YR,YI,ZR,ZI;
"CODE" 34342;
THE MEANING OF THE FORMAL PARAMETERS IS:
XR,XI,YR,YI:<ARITHMETIC EXPRESSION>;
ENTRY:XR,YR ARE THE REAL PARTS OF THE NUMERATOR
AND THE DENOMINATOR,RESPECTIVELY AND XI,YI ARE
THE CORRESPONDING IMAGINARY PARTS;
ZR,ZI: <VARIABLE>;
EXIT:THE REAL PART AND THE IMAGINARY PART OF THE
RESULTING COMPLEX NUMBER ARE DELIVERED IN RR
AND RI,RESPECTIVELY.
RUNNING TIME:
AT MOST SIX MULTIPLICATIONS AND/OR DIVISIONS ARE USED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE IS NOT PROTECTED AGAINST DIVISION BY ZERO.
1SECTION : 1.3.2 (MAY 1974) PAGE 3
EXAMPLE OF USE:
"BEGIN""REAL"R,I;
COMDIV(-.05,.1,.1,.2,R,I);
OUTPUT(61,"(""("(-.05+.1*I)/(.1+.2*I)=")",-D.DD,+D.DD,"("*I")"")"
,R,I)
"END"
(-.05+.1*I)/(.1+.2*I)= 0.30+0.40*I
SOURCE TEXT(S):
0"CODE"34341;
"PROCEDURE" COMMUL(AR,AI,BR,BI,RR,RI);
"VALUE" AR,AI,BR,BI; "REAL" AR,AI,BR,BI,RR,RI;
"BEGIN" RR:= AR * BR - AI * BI;
RI:= AR * BI + AI * BR
"END" COMMUL;
"EOP"
0"CODE"34342;
"PROCEDURE" COMDIV(XR,XI,YR,YI,ZR,ZI);
"VALUE" XR,XI,YR,YI;"REAL" XR,XI,YR,YI,ZR,ZI;
"BEGIN" "REAL" H,D;
"IF" ABS(YI) < ABS(YR) "THEN"
"BEGIN" "IF" YI= 0 "THEN"
"BEGIN" ZR:= XR/YR; ZI:= XI/YR "END" "ELSE"
"BEGIN" H:= YI/YR; D:= H*YI + YR;
ZR:= (XR + H * XI)/D; ZI:= (XI-H*XR)/D
"END"
"END" "ELSE"
"BEGIN" H:= YR/YI; D:= H*YR + YI;
ZR:= (XR*H + XI)/D; ZI:= (XI*H - XR)/D
"END"
"END" COMDIV;
"EOP"
1SECTION : 1.4 (OCTOBER 1974) PAGE 1
AUTHOR: H.J.J. TE RIELE.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 740125; REVISED: 740514;
BRIEF DESCRIPTION:
THIS SECTION CONTAINS A SET OF FIVE PROCEDURES FOR
THE BASIC ARITHMETIC OPERATIONS WITH LONG INTEGERS:
LNG INT ADD EXACTLY COMPUTES THE SUM OF TWO NONNEGATIVE INTEGERS.
LNG INT SUBTRACT EXACTLY COMPUTES THE DIFFERENCE OF TWO NONNEGATIVE
INTEGERS.
LNG INT MULT EXACTLY COMPUTES THE PRODUCT OF TWO NONNEGATIVE
INTEGERS.
LNG INT DIVIDE EXACTLY COMPUTES THE QUOTIENT WITH REMAINDER OF TWO
NONNEGATIVE INTEGERS.
LNG INT POWER EXACTLY COMPUTES U**POWER,
WHERE U IS A NONNEGATIVE LONG INTEGER AND POWER IS THE
POSITIVE (SINGLE-LENGTH) EXPONENT.
KEYWORDS:
LONG INTEGER ARITHMETIC,
ADDITION,
SUBTRACTION,
MULTIPLICATION,
DIVISION WITH REMAINDER,
EXPONENTIATION.
1SECTION : 1.4 (OCTOBER 1974) PAGE 2
SUBSECTION : LNG INT ADD.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNG INT ADD(U,V,SUM);
"INTEGER" "ARRAY" U,V,SUM;
"CODE" 31200;
THE MEANING OF THE FORMAL PARAMETERS IS:
U,V,SUM: <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" U[0:U[0]], V[0:V[0]],
SUM[0:MAX(U[0],V[0])+1];
BEFORE THE CALL OF LNG INT ADD, U AND V MUST
CONTAIN THE LONG INTEGERS TO BE ADDED;
AFTER THE CALL, SUM CONTAINS THE MULTI-LENGTH
SUM OF U AND V, WHILE U AND V REMAIN UNCHANGED.
PROCEDURES USED : NONE.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : 7.
RUNNING TIME :
WE GIVE A FORMULA FOR THE RUNNING TIME IN MILLISECONDS ON THE
CD CYBER 73-28 COMPUTER; THE RELATIVE PRECISION OF THE
COEFFICIENTS IS AT MOST ONE OR TWO DIGITS:
.10*MAX(U[0],V[0]) + .06*MIN(U[0],V[0]) + .56.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE : SEE LNG INT POWER (THIS SECTION).
EXAMPLE OF USE : SEE LNG INT POWER (THIS SECTION).
1SECTION : 1.4 (OCTOBER 1974) PAGE 3
SUBSECTION : LNG INT SUBTRACT.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" LNG INT SUBTRACT (U,V,DIFFERENCE);
"INTEGER" "ARRAY" U,V,DIFFERENCE;
"CODE" 31201;
THE MEANING OF THE FORMAL PARAMETERS IS:
U,V,DIFFERENCE: <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" U[0:U[0]],V[0:V[0]],DIFFERENCE[0:U[0]];
BEFORE THE CALL OF LNG INT SUBTRACT, U AND V MUST
CONTAIN THE LONG INTEGERS TO BE SUBTRACTED(V FROM U);
AFTER THE CALL, DIFFERENCE CONTAINS THE MULTI-LENGTH
DIFFERENCE U-V; IF U<V THEN DIFFERENCE[0]=0
IS DELIVERED; U AND V REMAIN UNCHANGED.
PROCEDURES USED : NONE.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH : 7.
RUNNING TIME :
WE GIVE A FORMULA FOR THE RUNNING TIME IN MILLISECONDS ON THE
CD CYBER 73-28 COMPUTER; THE RELATIVE PRECISION OF THE
COEFFICIENTS IS AT MOST ONE OR TWO DIGITS:
.10*U[0] + .06*V[0] + .64.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE : SEE LNG INT POWER (THIS SECTION).
EXAMPLE OF USE : SEE LNG INT POWER (THIS SECTION).
1SECTION : 1.4 (OCTOBER 1974) PAGE 4
SUBSECTION : LNG INT MULT.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNG INT MULT(U,V,PRODUCT);
"INTEGER" "ARRAY" U,V,PRODUCT;
"CODE" 31202;
THE MEANING OF THE FORMAL PARAMETERS IS:
U,V,PRODUCT: <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" U[0:U[0]], V[0:V[0]],
PRODUCT[0:U[0]+V[0]];
BEFORE THE CALL OF LNG INT MULT, U AND V MUST
CONTAIN THE LONG INTEGERS TO BE MULTIPLIED;
AFTER THE CALL, PRODUCT CONTAINS THE MULTI-LENGTH
PRODUCT OF U AND V, WHILE U AND V REMAIN UNCHANGED.
PROCEDURES USED : NONE.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : 7.
RUNNING TIME :
WE GIVE A FORMULA FOR THE RUNNING TIME IN MILLISECONDS ON THE
CD CYBER 73-28 COMPUTER; THE RELATIVE PRECISION OF THE
COEFFICIENTS IS AT MOST ONE OR TWO DIGITS:
.18*U[0]*V[0] + .15*U[0] + .06*V[0] + .46.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE : SEE LNG INT POWER (THIS SECTION).
EXAMPLE OF USE : SEE LNG INT POWER (THIS SECTION).
1SECTION : 1.4 (MARCH 1977) PAGE 5
SUBSECTION : LNG INT DIVIDE.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNG INT DIVIDE(U,V,QUOTIENT,REMAINDER); "VALUE" U;
"INTEGER" "ARRAY" U,V,QUOTIENT,REMAINDER;
"CODE" 31203;
THE MEANING OF THE FORMAL PARAMETERS IS:
U,V,QUOTIENT,REMAINDER: <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" U[0:U[0]], V[0:V[0]],
QUOTIENT[0:U[0]-V[0]+1], REMAINDER[0:V[0]];
BEFORE THE CALL OF LNG INT DIVIDE, U MUST CONTAIN THE
DIVIDEND, V THE DIVISOR(V ^= 0);
AFTER THE CALL, THE RESULTS OF THE LONG DIVISION
OF U BY V (I.E. U//V AND U-U//V) ARE STORED INTO
QUOTIENT AND REMAINDER; U AND V REMAIN UNCHANGED.
PROCEDURES USED : NONE.
REQUIRED CENTRAL MEMORY :
11 + U[0] + (IF V[0]=1 OR U[0]<V[0] THEN 0 ELSE V[0]+1),
RUNNING TIME :
WE GIVE A FORMULA FOR THE RUNNING TIME IN MILLISECONDS ON THE
CD CYBER 73-28 COMPUTER; THE RELATIVE PRECISION OF THE
COEFFICIENTS IS AT MOST ONE OR TWO DIGITS:
IF V[0]=1 THEN (.34*U[0] + .67) ELSE IF V[1] >= 5 000 000 THEN
(.26*DIFF*V[0] + .57*DIFF + .10*V[0] + 1.8)
ELSE (.27*DIFF*V[0] + .66*DIFF + .66*V[0] + 2.0)
(HERE DIFF=U[0]-V[0]+1, I.E. THE NUMBER OF EXECUTIONS
OF THE STATEMENT, IN WHICH DIVISION OF A (V[0]+1)-PLACE
NUMBER BY A V[0]-PLACE NUMBER IS PERFORMED).
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE : SEE LNG INT POWER (THIS SECTION).
EXAMPLE OF USE : SEE LNG INT POWER (THIS SECTION).
1SECTION : 1.4 (OCTOBER 1974) PAGE 6
SUBSECTION : LNG INT POWER.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNG INT POWER(U,EXPONENT,RESULT);
"VALUE" EXPONENT; "INTEGER" EXPONENT; "INTEGER" "ARRAY" U,RESULT;
"CODE" 31204;
THE MEANING OF THE FORMAL PARAMETERS IS:
EXPONENT: <ARITHMETIC EXPRESSION>;
THE (POSITIVE) POWER TO WHICH THE LONG INTEGER U
WILL BE RAISED;
U,RESULT: <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" U[0:U[0]], RESULT[0:U[0]*EXPONENT];
BEFORE THE CALL OF LNG INT POWER, U MUST CONTAIN
THE LONG INTEGER WHICH HAS TO BE RAISED TO THE
POWER EXPONENT;
AFTER THE CALL, RESULT CONTAINS THE VALUE OF THE
LONG INTEGER U**EXPONENT; U REMAINS UNCHANGED.
PROCEDURES USED :
LNG INT MULT = CP31202.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : 4 + 3 * (U[0] * EXPONENT + 1).
RUNNING TIME :
FOR THIS PROCEDURE THE TIME FORMULA IS A COMPLICATED FUNCTION OF
U[0], EXPONENT AND THE NUMBER OF ONES IN THE BINARY REPRESENTATION
OF EXPONENT, BUT ROUGHLY THE TIME IS OF THE ORDER :
(U(0)*EXPONENT)**2).
TWO TESTCASES :
EXPONENT TIME(IN SEC.) FOR:
U[0]=1 U[0]=2
20 .04 .10
40 .13 .34
100 .68 1.94
300 5.48 16.6
500 16.8 51.0
1SECTION : 1.4 (OCTOBER 1974) PAGE 7
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE:
DEFINITION:
A LONG INTEGER OF LENGTH N, OR AN N-PLACE INTEGER
(N>0) IS ANY NONNEGATIVE INTEGER LESS THAN BASE**N,
AND GREATER THAN OR EQUAL TO BASE**(N-1), WHERE BASE
IS THE (POSITIVE) RADIX OF THE POSITIONAL NOTATION, IN
WHICH THE INTEGERS ARE EXPRESSED.
ALL FIVE PROCEDURES USE THE BASE 10 000 000; THIS IS THE LARGEST
POWER OF 10, THE SQUARE OF WHICH CAN BE REPRESENTED EXACTLY
ON THE CD CYBER 73-28 COMPUTER. IF ONE WANTS TO USE THE
PROCEDURES WITH ANOTHER VALUE OF THE BASE, SAY R (NOT
NECESSARILY A POWER OF 10), THEN IN THE SOURCE TEXTS OF THE
PROCEDURES THE NUMBER 10 000 000 HAS TO BE REPLACED BY R
(8 TIMES IN LNG INT ADD,
2 TIMES IN LNG INT SUBTRACT,
2 TIMES IN LNG INT MULT AND
16 TIMES IN LNG INT DIVIDE).
MOREOVER, IN LNG INT DIVIDE THE NUMBER 9 999 999 HAS TO BE
REPLACED BY THE NUMBER R - 1.
IF A[1], A[2], ..., A[N] ARE THE N "DIGITS" OF THE LONG
INTEGER M OF LENGTH N (A[1] ^= 0), THEN
M=((...(A[1]*BASE+A[2])*BASE+...+A[N-2])*BASE+A[N-1])*BASE+A[N].
ACCORDINGLY, A LONG INTEGER M OF LENGTH N ALWAYS WILL
BE STORED INTO A CORRESPONDING "INTEGER" "ARRAY" A,
THE LENGTH N WILL BE STORED INTO THE ARRAY ELEMENT A[0].
FOR THE METHOD OF THE PROCEDURES LNG INT ADD, LNG INT SUBTRACT,
LNG INT MULT AND LNG INT DIVIDE, SEE [1; PP.229-248];
PROCEDURE LNG INT POWER USES THE BINARY METHOD FOR
EXPONENTIATION (SEE [1; PP.398-401]).
REFERENCES:
[1]. DONALD E. KNUTH.
THE ART OF COMPUTER PROGRAMMING, VOLUME 2/
SEMINUMERICAL ALGORITHMS.
ADDISON-WESLEY PUBLISHING COMPANY, 1969.
1SECTION : 1.4 (MARCH 1977) PAGE 8
EXAMPLE OF USE:
"BEGIN"
"PROCEDURE" OUT(A); "INTEGER" "ARRAY" A;
"BEGIN" "INTEGER" I,L; L:= A[0];
OUTPUT(61,"("B6ZD")",A[1]);
"FOR" I:= 2 "STEP" 1 "UNTIL" L "DO" OUTPUT(61,"("B7D")",
A[I]); OUTPUT(61,"("/")")
"END" OUT;
"INTEGER" "ARRAY" U,V,R1,R2[0:100];
U[0]:=5; U[1]:=333; U[2]:=U[3]:=U[4]:=U[5]:=7 000 000; OUT(U);
V[0]:=2; V[1]:=4 444; V[2]:=4 444 444; OUT(V);
LNG INT ADD(U,V,R1); OUT(R1);
LNG INT SUBTRACT(U,V,R1); OUT(R1);
LNG INT MULT(U,V,R1); OUT(R1);
LNG INT DIVIDE(U,V,R1,R2); OUT(R1); OUT(R2);
LNG INT POWER(V,5,R1); OUT(R1)
"END"
DELIVERS:
333 7000000 7000000 7000000 7000000
4444 4444444
333 7000000 7000000 7004445 1444444
333 7000000 7000000 6995556 2555556
1483111 1114073 9114221 9114221 9111110 8000000
750825 0001650 0826575
734 0700700
17341 5299149 6553709 6327185 8964586 9972395 8069589 6628224
1SECTION : 1.4 (OCTOBER 1974) PAGE 9
SOURCE TEXT(S):
"CODE" 31200;
"PROCEDURE" LNG INT ADD(U,V,SUM); "INTEGER""ARRAY" U,V,SUM;
"BEGIN""INTEGER" LU,LV,DIFF,CARRY,I,T,MAX;
LU:=U[0]; LV:=V[0];
"IF" LU >= LV "THEN"
"BEGIN" MAX:=LU; DIFF:=LU - LV + 1; CARRY:=0;
"FOR" I:=LU "STEP" -1 "UNTIL" DIFF "DO"
"BEGIN" T:=U[I] + V[I-DIFF+1] + CARRY;
CARRY:="IF" T<10 000 000 "THEN" 0 "ELSE" 1;
SUM[I]:=T - CARRY * 10 000 000
"END";
"FOR" I:=DIFF - 1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=U[I] + CARRY;
CARRY:="IF" T<10 000 000 "THEN" 0 "ELSE" 1;
SUM[I]:=T - CARRY * 10 000 000
"END"
"END" "ELSE"
"BEGIN" MAX:=LV; DIFF:=LV - LU + 1; CARRY:=0;
"FOR" I:=LV "STEP" -1 "UNTIL" DIFF "DO"
"BEGIN" T:=V[I] + U[I-DIFF+1] + CARRY;
CARRY:="IF" T<10 000 000 "THEN" 0 "ELSE" 1;
SUM[I]:=T - CARRY * 10 000 000
"END";
"FOR" I:=DIFF - 1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=V[I] + CARRY;
CARRY:="IF" T<10 000 000 "THEN" 0 "ELSE" 1;
SUM[I]:=T - CARRY * 10 000 000
"END"
"END";
"IF" CARRY=1 "THEN"
"BEGIN" "FOR" I:=MAX "STEP" -1 "UNTIL" 1 "DO"
SUM[I+1]:=SUM[I]; SUM[1]:=1; MAX:=MAX + 1
"END";
SUM[0]:=MAX
"END" LNG INT ADD
1SECTION : 1.4 (OCTOBER 1974) PAGE 10
;
"EOP"
"CODE" 31201;
"PROCEDURE" LNG INT SUBTRACT(U,V,DIFFERENCE);
"INTEGER""ARRAY" U,V,DIFFERENCE;
"BEGIN""INTEGER" LU,LV,DIFF,I,T,J,CARRY;
LU:=U[0]; LV:=V[0];
"IF" LU<LV "OR" LU=LV "AND" U[1]<V[1] "THEN" DIFFERENCE[0]:=0 "ELSE"
"BEGIN" DIFF:=LU - LV + 1; CARRY:=0;
"FOR" I:=LU "STEP" -1 "UNTIL" DIFF "DO"
"BEGIN" T:=U[I] - V[I-DIFF+1] + CARRY;
CARRY:="IF" T<0 "THEN" -1 "ELSE" 0;
DIFFERENCE[I]:=T - CARRY * 10 000 000
"END";
"FOR" I:=DIFF - 1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=U[I] + CARRY; CARRY:="IF" T<0 "THEN" -1 "ELSE" 0;
DIFFERENCE[I]:=T - CARRY * 10 000 000
"END";
"IF" CARRY=-1 "THEN"
"BEGIN" DIFFERENCE[0]:=0; "GOTO" READY "END";
I:=0; J:=LU;
"FOR" I:=I+1 "WHILE" DIFFERENCE[I]=0 "AND" J>1 "DO" J:=J-1;
DIFFERENCE[0]:=J;
"IF" J<LU "THEN"
"FOR" I:=1 "STEP" 1 "UNTIL" J "DO"
DIFFERENCE[I]:=DIFFERENCE[LU+I-J]
"END";
READY:
"END" LNG INT SUBTRACT;
"EOP"
"CODE" 31202;
"PROCEDURE" LNG INT MULT(U,V,PRODUCT);
"INTEGER""ARRAY" U,V,PRODUCT;
"BEGIN""INTEGER" LU,LV,LUV,I,J,CARRY,T;
LU:=U[0]; LV:=V[0]; LUV:=LU + LV;
"FOR" I:=LU + 1 "STEP" 1 "UNTIL" LUV "DO" PRODUCT[I]:=0;
"FOR" J:=LU "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" CARRY:=0;
"FOR" I:=LV "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=U[J] * V[I] + PRODUCT[J+I] + CARRY;
CARRY:=T//10 000 000; PRODUCT[J+I]:=T - CARRY * 10 000 000
"END"; PRODUCT[J]:=CARRY
"END";
"IF" PRODUCT[1]=0 "THEN"
"BEGIN" "FOR" I:=2 "STEP" 1 "UNTIL" LUV "DO"
PRODUCT[I-1]:=PRODUCT[I]; LUV:=LUV - 1
"END"; PRODUCT[0]:=LUV
"END" LNG INT MULT
1SECTION : 1.4 (OCTOBER 1974) PAGE 11
;
"EOP"
"CODE" 31203;
"PROCEDURE" LNG INT DIVIDE(U,V,QUOTIENT,REMAINDER); "VALUE" U;
"INTEGER""ARRAY" U,V,QUOTIENT,REMAINDER;
"BEGIN""INTEGER" LU,LV,V1,DIFF,I,T,SCALE,D,Q1,J,CARRY;
LU:=U[0]; LV:=V[0]; V1:=V[1]; DIFF:=LU - LV;
"IF" LV=1 "THEN"
"BEGIN" CARRY:=0;
"FOR" I:=1 "STEP" 1 "UNTIL" LU "DO"
"BEGIN" T:=CARRY * 10 000 000 + U[I]; QUOTIENT[I]:=T//V1;
CARRY:=T - QUOTIENT[I] * V1
"END"; REMAINDER[0]:=1; REMAINDER[1]:=CARRY;
"IF" QUOTIENT[1]=0 "THEN"
"BEGIN" "FOR" I:=2 "STEP" 1 "UNTIL" LU "DO"
QUOTIENT[I-1]:=QUOTIENT[I];
QUOTIENT[0]:=LU - ("IF" LU=1 "THEN" 0 "ELSE" 1)
"END" "ELSE" QUOTIENT[0]:=LU
"END" LV=1
"ELSE"
"IF" LU<LV "THEN"
"BEGIN" QUOTIENT[0]:=1; QUOTIENT[1]:=0;
"FOR" I:=0 "STEP" 1 "UNTIL" LU "DO" REMAINDER[I]:=U[I]
"END" LU<LV
"ELSE"
"BEGIN""INTEGER""ARRAY" A[0:LV];
SCALE:=10 000 000//(V1+1);
"IF" SCALE>1 "THEN"
"BEGIN""COMMENT" NORMALIZE U; CARRY:=0;
"FOR" I:=LU "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=SCALE * U[I] + CARRY; CARRY:=T//10 000 000;
U[I]:=T - CARRY * 10 000 000
"END"; U[0]:=CARRY;
"COMMENT" NORMALIZE V; CARRY:=0;
"FOR" I:=LV "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=SCALE * V[I] + CARRY; CARRY:=T//10 000 000;
V[I]:=T - CARRY * 10 000 000
"END"; V1:=V[1]
"END" NORMALIZATION
"ELSE" U[0]:=0; "COMMENT"
1SECTION : 1.4 (OCTOBER 1974) PAGE 12
;
"COMMENT" COMPUTE QUOTIENT AND REMAINDER;
"FOR" I:=0 "STEP" 1 "UNTIL" DIFF "DO"
"BEGIN" D:=U[I] * 10 000 000 + U[I+1];
Q1:="IF" U[I]=V1 "THEN" 9 999 999 "ELSE" D//V1;
"IF" V[2] * Q1 > (D-Q1*V1) * 10 000 000 + U[I+2] "THEN"
"BEGIN" Q1:=Q1 - 1;
"IF" V[2]*Q1>(D-Q1*V1)*10 000 000+U[I+2] "THEN" Q1:=Q1-1
"END";
"COMMENT" U[I:I+LV]:=U[I:I+LV] - Q1 * V[1:LV];
CARRY:=0;
"FOR" J:=LV "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=Q1 * V[J] + CARRY; CARRY:=T//10 000 000;
A[J]:=T - CARRY * 10 000 000
"END"; A[0]:=CARRY;
CARRY:=0;
"FOR" J:=LV "STEP" -1 "UNTIL" 0 "DO"
"BEGIN" T:=U[I+J] - A[J] + CARRY; CARRY:="IF" T<0 "THEN" -1
"ELSE" 0; U[I+J]:=T - CARRY * 10 000 000
"END";
"COMMENT" IF CARRY=-1 THEN Q1 IS ONE TOO LARGE,
AND V MUST BE ADDED BACK TO U[I:I+LV];
"IF" CARRY=-1 "THEN"
"BEGIN" Q1:=Q1 - 1; CARRY:=0;
"FOR" J:=LV "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" T:=U[I+J] + V[J] + CARRY; CARRY:="IF" T<10 000 000
"THEN" 0 "ELSE" 1; U[I+J]:=T - CARRY * 10 000 000
"END"
"END"; QUOTIENT[I]:=Q1
"END" I;
"COMMENT" CORRECT STORAGE OF QUOTIENT;
"IF" QUOTIENT[0] ^= 0 "THEN"
"BEGIN" "FOR" I:=DIFF "STEP" -1 "UNTIL" 0 "DO"
QUOTIENT[I+1]:=QUOTIENT[I]; QUOTIENT[0]:=DIFF + 1
"END" "ELSE"
"IF" DIFF < 2 "THEN"
"BEGIN" QUOTIENT[0]:= 1; "IF" DIFF = 0 "THEN" QUOTIENT[1]:= 0
"END" "ELSE"
"IF" QUOTIENT[1] ^= 0 "THEN" QUOTIENT[0]:=DIFF "ELSE"
"BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" DIFF - 1 "DO"
QUOTIENT[I]:=QUOTIENT[I+1]; QUOTIENT[0]:=DIFF - 1
"END"; "COMMENT"
1SECTION : 1.4 (OCTOBER 1974) PAGE 13
;
"COMMENT" REMAINDER := U[DIFF+1:LU]//SCALE;
"IF" SCALE>1 "THEN"
"BEGIN" CARRY:=0;
"FOR" I:=1 "STEP" 1 "UNTIL" LV "DO"
"BEGIN" T:=CARRY * 10 000 000 + U[DIFF+I];
REMAINDER[I]:=T//SCALE; CARRY:=T - REMAINDER[I] * SCALE
"END"
"END" "ELSE"
"FOR" I:=1 "STEP" 1 "UNTIL" LV "DO" REMAINDER[I]:=U[DIFF+I];
"COMMENT" CORRECT STORAGE OF REMAINDER;
I:=0; J:=LV;
"FOR" I:=I+1 "WHILE" REMAINDER[I]=0 "AND" J>1 "DO" J:=J-1;
REMAINDER[0]:=J;
"IF" J<LV "THEN"
"FOR" I:=1 "STEP" 1 "UNTIL" J "DO"
REMAINDER[I]:=REMAINDER[LV + I - J];
"COMMENT" UNNORMALIZE THE DIVISOR V;
"IF" SCALE>1 "THEN"
"BEGIN" CARRY:=0;
"FOR" I:=1 "STEP" 1 "UNTIL" LV "DO"
"BEGIN" T:=CARRY * 10 000 000 + V[I];
V[I]:=T//SCALE; CARRY:=T - V[I] * SCALE
"END"
"END"
"END"
"END" LNG INT DIVIDE;
"EOP"
"CODE" 31204;
"PROCEDURE" LNG INT POWER(U,EXPONENT,RESULT);
"VALUE" EXPONENT; "INTEGER" EXPONENT; "INTEGER""ARRAY" U,RESULT;
"BEGIN""INTEGER" MAX,I,N;
MAX:=U[0] * EXPONENT;
"BEGIN""INTEGER""ARRAY" Y,Z,H[0:MAX];
"COMMENT" Y:=1, Z:=U;
Y[0]:=Y[1]:=1;
"FOR" I:=U[0] "STEP" -1 "UNTIL" 0 "DO" Z[I]:=U[I];
HALVE: N:=EXPONENT//2; "IF" N+N=EXPONENT "THEN" "GOTO" SQUARE Z;
LNG INT MULT(Y,Z,H);
"FOR" I:=H[0] "STEP" -1 "UNTIL" 0 "DO" Y[I]:=H[I];
"IF" N=0 "THEN" "GOTO" READY;
SQUARE Z: LNG INT MULT(Z,Z,H);
"FOR" I:=H[0] "STEP" -1 "UNTIL" 0 "DO" Z[I]:=H[I];
EXPONENT:=N; "GOTO"HALVE;
READY: "FOR" I:=Y[0] "STEP" -1 "UNTIL" 0 "DO" RESULT[I]:=Y[I]
"END"
"END" LNG INT POWER;
"EOP"
1SECTION : 1.5.1 (MARCH 1977) PAGE 1
AUTHORS: D.T.WINTER(A-D,F-I), T.J.DEKKER(E,J)
CONTRIBUTOR: J.KOOPMAN(E,J)
INSTITUTES: MATHEMATICAL CENTRE,UNIVERSITY OF AMSTERDAM.
RECEIVED: 770328
BRIEF DESCRIPTION:
THIS SECTION CONTAINS PROCEDURES FOR THE ELEMENTARY OPERATIONS IN
DOUBLE PRECISION ARITHMETIC.
A. DPADD ADDS TWO SINGLE PRECISION NUMBERS TO A DOUBLE PRECISION
SUM.
B. DPSUB SUBTRACTS TWO SINGLE PRECISION NUMBERS TO A DOUBLE
PRECISION DIFFERENCE.
C. DPMUL MULTIPLIES TWO SINGLE PRECISION NUMBERS TO A
DOUBLE PRECISION PRODUCT.
D. DPDIV DIVIDES TWO SINGLE PRECISION NUMBERS TO A DOUBLE
PRECISION QUOTIENT.
E. DPPOW COMPUTES A**EXPON IN DOUBLE PRECISION,WHERE
A IS A SINGLE PRECISION REAL NUMBER AND EXPON THE
INTEGER EXPONENT.
F. LNGADD ADDS TWO DOUBLE PRECISION NUMBERS.
G. LNGSUB SUBTRACTS TWO DOUBLE PRECISION NUMBERS.
H. LNGMUL MULTIPLIES TWO DOUBLE PRECISION NUMBERS.
I. LNGDIV DIVIDES TWO DOUBLE PRECISION NUMBERS.
J. LNGPOW COMPUTES (A,AA)**EXPON IN DOUBLE PRECISION,WHERE
(A,AA) IS A DOUBLE PRECISION REAL NUMBER AND EXPON
THE INTEGER EXPONENT.
KEYWORDS:
DOUBLE PRECISION ARITHMETIC
EXPONENTIATION.
LANGUAGE: COMPASS(A-D,F-I),ALGOL 60(E,J)
1SECTION : 1.5.1 (DECEMBER 1979) PAGE 2
METHOD AND PERFORMANCE:
THE PROCEDURES A-D,F-I USE THE HARDWARE FUNCTIONS FOR DOUBLE
PRECISION THAT ARE AVAILIBLE ON THE CYBER.
THE PROCEDURES LNG ADD, LNG SUB, LNG MUL AND LNG DIV CHECK THE
INPUT PARAMETERS (A,AA) AND (B,BB) FOR CORRECTNESS. A HEAD/TAIL
PAIR IS A CORRECT DOUBLE PRECISION PARAMETER IN THE FOLLOWING
CASES:
A) THE TAIL IS ZERO;
B) THE EXPONENT IN THE BINARY REPRESENTATION OF THE TAIL IS 48
LESS THAN THE EXPONENT OF THE HEAD.
AN OUTPUT PARAMETER OF THESE PROCEDURES ALWAYS IS A CORRECT DOUBLE
PRECISION NUMBER. IF AN INPUT PARAMETER IS NOT CORRECT, THE ERROR
MESSAGE "DP PARAMETER TAIL ERROR" WILL BE ISSUED.
BOTH PROCEDURES E AND J MAKE USE OF THE BINARY REPRESENTATION OF
THE INTEGER EXPONENT. IF X DENOTES THE NUMBER THAT IS TO BE
EXPONENTIATED, THE PROCEDURES E AND J RUN AS FOLLOWS:
THE SEQUENCE X,X**2,X**4,X**8,... IS FORMED WHILE
SIMULTANEOUSLY THE BINARY REPRESENTATION OF THE EXPONENT
IS CHECKED; WHEN THE I-TH DIGIT EQUALS ONE,THE FACTOR
X**(2**(I-1)) IS TAKEN INTO ACCOUNT.
EXAMPLE OF USE:
SEE THE PROCEDURE LNGREATODECI (SECTION 1.5.3).
1SECTION : 1.5.1 (MARCH 1977) PAGE 3
SUBSECTION: DP ADD
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" DP ADD(A, B, C, CC);
"VALUE" A, B; "REAL" A, B, C, CC;
"CODE" 31101;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,B: <ARITHMETIC EXPRESSIONS>;
THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE DOUBLE PRECISION RESULT OF A+B.
SUBSECTION: DP SUB
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" DP SUB(A, B, C, CC);
"VALUE" A, B; "REAL" A, B, C, CC;
"CODE" 31102;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,B: <ARITHMETIC EXPRESSIONS>;
THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE DOUBLE PRECISION RESULT OF A-B.
SUBSECTION: DP MUL
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" DP MUL(A, B, C, CC);
"VALUE" A, B; "REAL" A, B, C, CC;
"CODE" 31103;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,B: <ARITHMETIC EXPRESSIONS>;
THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE DOUBLE PRECISION RESULT OF A*B.
1SECTION : 1.5.1 (MARCH 1977) PAGE 4
SUBSECTION: DP DIV
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" DP DIV(A, B, C, CC);
"VALUE" A, B; "REAL" A, B, C, CC;
"CODE" 31104;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,B: <ARITHMETIC EXPRESSIONS>;
THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE DOUBLE PRECISION RESULT OF A/B.
SUBSECTION: DP POW.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE"DP POW(A ,EXPON ,C ,CC );
"VALUE"A,EXPON;"INTEGER"EXPON;"REAL"A,C,CC;
"CODE"31109;
THE MEANING OF THE FORMAL PARAMETERS IS:
A : <ARITHMETIC EXPRESSION>;
THE NUMBER THAT IS TO BE EXPONENTIATED;
EXPON : <ARITHMETIC EXPRESSION>;
THE (INTEGER) POWER TO WHICH A WILL BE RAISED;
C , CC : <REAL VARIABLES>;
EXIT: THE HEAD (C) AND TAIL (CC) OF THE DOUBLE
PRECISION RESULT A**EXPON.
PROCEDURES USED:
LNG POW = CP31110.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO LN(EXPON).
1SECTION : 1.5.1 (MARCH 1977) PAGE 5
SUBSECTION: LNG ADD
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LNG ADD(A, AA, B, BB, C, CC);
"VALUE" A, AA, B, BB; "REAL" A, AA, B, BB, C, CC;
"CODE" 31105;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,AA,B,BB: <ARITHMETIC EXPRESSIONS>;
THE HEADS (A AND B) AND THE TAILS (AA AND BB) OF THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE RESULT (A,AA)+(B,BB).
SUBSECTION: LNG SUB
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LNG SUB(A, AA, B, BB, C, CC);
"VALUE" A, AA, B, BB; "REAL" A, AA, B, BB, C, CC;
"CODE" 31106;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,AA,B,BB: <ARITHMETIC EXPRESSIONS>;
THE HEADS (A AND B) AND THE TAILS (AA AND BB) OF THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE RESULT (A,AA)-(B,BB).
SUBSECTION: LNG MUL
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LNG MUL(A, AA, B, BB, C, CC);
"VALUE" A, AA, B, BB; "REAL" A, AA, B, BB, C, CC;
"CODE" 31107;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,AA,B,BB: <ARITHMETIC EXPRESSIONS>;
THE HEADS (A AND B) AND THE TAILS (AA AND BB) OF THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE RESULT (A,AA)*(B,BB).
1SECTION : 1.5.1 (MARCH 1977) PAGE 6
SUBSECTION: LNG DIV
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LNG DIV(A, AA, B, BB, C, CC);
"VALUE" A, AA, B, BB; "REAL" A, AA, B, BB, C, CC;
"CODE" 31108;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,AA,B,BB: <ARITHMETIC EXPRESSIONS>;
THE HEADS (A AND B) AND THE TAILS (AA AND BB) OF THE OPERANDS;
C,CC: <REAL VARIABLES>;
THE HEAD AND TAIL OF THE RESULT (A,AA)/(B,BB).
SUBSECTION: LNG POW.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE"LNG POW(A ,AA ,EXPON ,C ,CC );
"VALUE"A,AA,EXPON;"INTEGER"EXPON;"REAL"A,AA,C,CC;
"CODE"31110;
THE MEANING OF THE FORMAL PARAMETERS IS:
A,AA : <ARITHMETIC EXPRESSIONS>;
THE HEAD (A) AND TAIL (AA) OF THE NUMBER THAT
IS TO BE EXPONENTIATED;
EXPON : <ARITHMETIC EXPRESSION>;
THE (INTEGER) POWER TO WHICH (A,AA) WILL BE RAISED;
C,CC : <REAL VARIABLES>;
EXIT: THE HEAD (C) AND TAIL (CC) OF THE DOUBLE
PRECISION RESULT (A,AA)**EXPON.
PROCEDURES USED:
LNG MUL = CP31107.
LNG DIV = CP31108.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO LN(EXPON).
1SECTION : 1.5.1 (DECEMBER 1979) PAGE 7
SOURCE TEXT(S):
ALL PROCEDURES, EXCEPT POW AND LNG POW, ARE WRITTEN IN COMPASS, IT IS
NOT POSSIBLE TO SIMULATE THESE PROCEDURES IN ALGOL 60, SO ONLY THE TEXT
IS GIVEN FOR POW AND LNG POW.
"CODE"31109;
"PROCEDURE"DP POW(A,EXPON,C,CC);
"VALUE"A,EXPON;"INTEGER"EXPON;"REAL"A,C,CC;
"BEGIN"
LNG POW(A,0,EXPON,C,CC)
"END" DP POW;
"EOP"
"CODE"31110;
"PROCEDURE"LNG POW(A,AA,EXPON,C,CC);
"VALUE"A,AA,EXPON;"INTEGER"EXPON;"REAL"A,AA,C,CC;
"BEGIN""INTEGER"OLDEX,NEWEX;"REAL"D,DD;
D:=A;DD:=AA;C:=1;CC:=0;NEWEX:=ABS(EXPON);
"FOR"OLDEX:=NEWEX"WHILE"NEWEX^=0"DO"
"BEGIN"NEWEX:=OLDEX//2;
"IF"NEWEX+NEWEX^=OLDEX
"THEN"LNG MUL(C,CC,D,DD,C,CC);
"IF"NEWEX^=0
"THEN"LNG MUL(D,DD,D,DD,D,DD)
"END";
"IF"EXPON<0"THEN"LNG DIV(1,0,C,CC,C,CC)
"END" LNG POW;
"EOP"
1SECTION : 1.5.2 (JANUARY 1976) PAGE 1
AUTHORS: J. WOLLESWINKEL AND D. WINTER.
CONTRIBUTOR: J. C. P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 750501.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOURTEEN PROCEDURES FOR CALCULATING, WITH
DOUBLE LENGTH ARITHMETIC, THE (SCALAR) PRODUCTS OF SINGLE LENGTH
VECTORS AND MATRICES.
LNGVECVEC: CALCULATES THE SCALAR PRODUCT OF TWO VECTORS GIVEN IN
ONE-DIMENSIONAL ARRAYS;
LNGMATVEC: CALCULATES THE SCALAR PRODUCT OF A VECTOR GIVEN IN A
ONE-DIMENSIONAL ARRAY AND A ROW OF A MATRIX GIVEN IN A
TWO DIMENSIONAL ARRAY;
LNGTAMVEC: CALCULATES THE SCALAR PRODUCT OF A VECTOR GIVEN IN A
ONE-DIMENSIONAL ARRAY AND A COLUMN OF A MATRIX GIVEN IN
A TWO-DIMENSIONAL ARRAY;
LNGMATMAT: CALCULATES THE SCALAR PRODUCT OF A ROW OF A MATRIX AND
A COLUMN OF ANOTHER MATRIX, WHICH ARE BOTH GIVEN IN
TWO-DIMENSIONAL ARRAYS;
LNGTAMMAT: CALCULATES THE SCALAR PRODUCT OF COLUMNS OF MATRICES,
WHICH ARE BOTH GIVEN IN TWO-DIMENSIONAL ARRAYS;
LNGMATTAM: CALCULATES THE SCALAR PRODUCT OF ROWS OF MATRICES,
WHICH ARE BOTH GIVEN IN TWO-DIMENSIONAL ARRAYS;
LNGSEQVEC: CALCULATES THE SCALAR PRODUCT OF TWO VECTORS GIVEN IN
ONE-DIMENSIONAL ARRAYS, WHERE THE MUTUAL SPACINGS
BETWEEN THE INDICES OF THE FIRST VECTOR CHANGE
LINEARLY;
LNGSCAPRD1: CALCULATES THE SCALAR PRODUCT OF TWO VECTORS GIVEN IN
ONE-DIMENSIONAL ARRAYS, WHERE THE SPACINGS OF BOTH
VECTORS ARE CONSTANT;
LNGSYMMATVEC: CALCULATES THE SCALAR PRODUCT OF A VECTOR GIVEN IN A
ONE-DIMENSIONAL ARRAY AND A ROW OF A SYMMETRIC MATRIX,
WHOSE UPPER TRIANGLE IS STORED COLUMN-WISE IN A ONE-
DIMENSIONAL ARRAY;
THE ABOVE PROCEDURES HAVE THE POSSIBILITY OF ADDING A DOUBLE LENGTH
SCALAR TO THE CALCULATED SCALAR PRODUCT;
1SECTION : 1.5.2 (JANUARY 1976) PAGE 2
LNGFULMATVEC: CALCULATES THE VECTOR A * B, WHERE A IS A GIVEN
MATRIX AND B IS A VECTOR;
LNGFULTAMVEC: CALCULATES THE VECTOR A' * B, WHERE A' IS THE
TRANSPOSED OF THE MATRIX A AND B IS A VECTOR;
LNGFULSYMMATVEC: CALCULATES THE VECTOR A * B, WHERE A IS A
SYMMETRIC MATRIX, WHOSE UPPERTRIANGLE IS STORED
COLUMNWISE IN A ONE-DIMENSIONAL ARRAY, AND B IS A
VECTOR;
LNGRESVEC: CALCULATES THE RESIDUAL VECTOR A * B + X * C, WHERE A
IS A GIVEN MATRIX, B AND C ARE VECTORS AND X IS A
SCALAR;
LNGSYMRESVEC: CALCULATES THE RESIDUAL VECTOR A * B + X * C, WHERE A
IS A SYMMETRIC MATRIX, WHOSE UPPERTRIANGLE IS STORED
COLUMNWISE IN A ONE-DIMENSIONAL ARRAY, B AND C ARE
VECTORS AND, X IS A SCALAR.
IN THIS SECTION (X, XX) DENOTES A DOUBLE-LENGTH NUMBER WITH HEAD X
AND TAIL XX (SEE METHOD AND PERFORMANCE).
KEYWORDS:
ELEMENTARY OPERATIONS,
VECTOR OPERATIONS,
SCALAR PRODUCTS,
DOUBLE-LENGTH ARITHMETIC.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 3
SUBSECTION: LNGVECVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGVECVEC(L, U, SHIFT, A, B, C, CC, D, DD);
"VALUE" L, U, SHIFT, C, CC; "INTEGER" L, U, SHIFT;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34410;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
SHIFT: <ARITHMETIC EXPRESSION>;
THE INDEX-SHIFTING PARAMETER OF THE VECTOR GIVEN IN B;
A: <ARRAY IDENTIFIER>;
ONE OF THE VECTORS SHOULD BE GIVEN IN ARRAY A[L:U];
B: <ARRAY IDENTIFIER>;
THE OTHER VECTOR SHOULD BE GIVEN IN ARRAY
B[L + SHIFT : U + SHIFT];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[L:U] AND B[L + SHIFT : U + SHIFT].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 4
SUBSECTION: LNGMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGMATVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC; "INTEGER" L, U, I;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34411;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE ROW-VECTOR GIVEN IN ARRAY A;
A: <ARRAY IDENTIFIER>;
THE ROW-VECTOR SHOULD BE GIVEN IN ARRAY A[I:I, L:U];
B: <ARRAY IDENTIFIER>;
THE VECTOR SHOULD BE GIVEN IN ARRAY B[L:U];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[I:I, L:U] AND B[L:U].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 5
SUBSECTION: LNGTAMVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGTAMVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC; "INTEGER" L, U, I;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34412;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE COLUMN-VECTOR GIVEN IN ARRAY A;
A: <ARRAY IDENTIFIER>;
THE COLUMN-VECTOR SHOULD BE GIVEN IN ARRAY A[L:U, I:I];
B: <ARRAY IDENTIFIER>;
THE VECTOR SHOULD BE GIVEN IN ARRAY B[L:U];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[L:U, I:I] AND B[L:U].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 6
SUBSECTION: LNGMATMAT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGMATMAT(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34413;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE ROW-VECTOR GIVEN IN ARRAY A;
J: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE COLUMN-VECTOR GIVEN IN ARRAY B;
A: <ARRAY IDENTIFIER>;
THE ROW-VECTOR SHOULD BE GIVEN IN ARRAY A[I:I, L:U];
B: <ARRAY IDENTIFIER>;
THE COLUMN-VECTOR SHOULD BE GIVEN IN ARRAY B[L:U, J:J];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[I:I, L:U] AND B[L:U, J:J].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 7
SUBSECTION: LNGTAMMAT.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGTAMMAT(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34414;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE COLUMN-VECTOR GIVEN IN ARRAY A;
J: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE COLUMN-VECTOR GIVEN IN ARRAY B;
A: <ARRAY IDENTIFIER>;
ONE OF THE COLUMN-VECTORS SHOULD BE GIVEN IN ARRAY
A[L:U, I:I];
B: <ARRAY IDENTIFIER>;
THE OTHER COLUMN-VECTOR SHOULD BE GIVEN IN ARRAY
B[L:U, J:J];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[L:U, I:I] AND B[L:U, J:J].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 8
SUBSECTION: LNGMATTAM.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGMATTAM(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34415;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE ROW-VECTOR GIVEN IN ARRAY A;
J: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE ROW-VECTOR GIVEN IN ARRAY B;
A: <ARRAY IDENTIFIER>;
ONE OF THE ROW-VECTORS SHOULD BE GIVEN IN ARRAY A[I:I,L:U];
B: <ARRAY IDENTIFIER>;
THE OTHER ROW-VECTOR SHOULD BE GIVEN IN ARRAY B[J:J, L:U];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[I:I, L:U] AND B[J:J, L:U].
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 9
SUBSECTION: LNGSEQVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGSEQVEC(L, U, IL, SHIFT, A, B, C, CC, D, DD);
"VALUE" L, U, IL, SHIFT, C, CC; "INTEGER" L, U, IL, SHIFT;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34416;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
IL: <ARITHMETIC EXPRESSION>;
THE FIRST INDEX OF THE VECTOR GIVEN IN ARRAY A;
SHIFT: <ARITHMETIC EXPRESSION>;
THE INDEX-SHIFTING PARAMETER OF THE VECTOR GIVEN IN B;
A: <ARRAY IDENTIFIER>;
ONE OF THE VECTORS SHOULD BE GIVEN IN ARRAY A[P:Q], WHERE
P = MIN(IL + (J + L - 1) * (J - 1) // 2; J = L, ..., U) AND
Q = MAX(IL + (J + L - 1) * (J - 1) // 2; J = L, ..., U);
B: <ARRAY IDENTIFIER>;
THE OTHER VECTOR SHOULD BE GIVEN IN ARRAY
B[L + SHIFT : U + SHIFT];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[P:Q] (SEE THE MEANING OF PARAMETER A ) AND
B[L + SHIFT: U + SHIFT], WHERE THE ELEMENTS OF THE FIRST VECTOR ARE
A[IL + (J + L - 1) * (J - 1) // 2], J = L, ..., U.
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 10
SUBSECTION: LNGSCAPRD1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGSCAPRD1(LA, SA, LB, SB, N, A, B, C, CC, D, DD);
"VALUE" LA, SA, LB, SB, N, C, CC; "INTEGER" LA, SA, LB, SB, N;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34417;
THE MEANING OF THE FORMAL PARAMETERS IS:
LA: <ARITHMETIC EXPRESSION>;
THE FIRST INDEX OF THE VECTOR GIVEN IN ARRAY A;
SA: <ARITHMETIC EXPRESSION>;
THE SPACING OF THE VECTOR GIVEN IN ARRAY A;
LB: <ARITHMETIC EXPRESSION>;
THE FIRST INDEX OF THE VECTOR GIVEN IN ARRAY B;
SB: <ARITHMETIC EXPRESSION>;
THE SPACING OF THE VECTOR GIVEN IN ARRAY B;
N: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT; IF N < 1, THEN ON
EXIT THE RESULT (D, DD) WILL SATISFY: (D, DD) = (C, CC);
A: <ARRAY IDENTIFIER>;
ONE OF THE VECTORS SHOULD BE GIVEN IN ARRAY
A[MIN(LA, LA + (N - 1) * SA) : MAX(LA, LA + (N - 1) * SA)];
B: <ARRAY IDENTIFIER>;
THE OTHER VECTOR SHOULD BE GIVEN IN ARRAY
B[MIN(LB, LB + (N - 1) * SB) : MAX(LB, LB + (N - 1) * SB)];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[MIN(LA, LA + (N - 1) * SA):MAX(LA, LA + (N - 1) * SA)]
AND B[MIN(LB, LB + (N - 1) * SB) : MAX(LB, LB + (N - 1) * SB)],
WHERE THE ELEMENTS OF THE VECTORS ARE A[LA + (J - 1) * SA] AND
B[LB + (J - 1) * SB], FOR J = 1, .., N.
LANGUAGE: COMPASS.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 11
SUBSECTION: LNGSYMMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGSYMMATVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC; "INTEGER" L, U, I;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"CODE" 34418;
THE MEANING OF THE FORMAL PARAMETERS IS:
L: <ARITHMETIC EXPRESSION>;
THE LOWER BOUND OF THE RUNNING SUBSCRIPT;
U: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE RUNNING SUBSCRIPT;
I: <ARITHMETIC EXPRESSION>;
THE INDEX OF THE ROW OF THE SYMMETRIC MATRIX, WHOSE UPPER
TRIANGLE IS STORED COLUMN-WISE IN THE ONE-DIMENSIONAL ARRAY
A;
A: <ARRAY IDENTIFIER>;
THE ROW OF THE SYMMETRIC MATRIX SHOULD BE GIVEN IN ARRAY
A[P:Q], WHERE, IF I > L THEN P = (I - 1) * I // 2 + L ELSE
P = (L - 1) * L // 2 + I, AND IF I > U THEN
Q = (I - 1) * I // 2 + U ELSE Q = (U - 1) * U // 2 + I;
B: <ARRAY IDENTIFIER>;
THE VECTOR SHOULD BE GIVEN IN ARRAY B[L:U];
C, CC: <ARITHMETIC EXPRESSION>;
THE HEAD AND TAIL OF THE DOUBLE-LENGTH SCALAR THAT HAS TO
BE ADDED TO THE CALCULATED SCALAR PRODUCT; IF CC IS NOT A
TAIL TO C THEN AN ERROR MESSAGE WILL BE PRINTED (SEE METHOD
AND PERFORMANCE);
D, DD: <REAL VARIABLE>;
EXIT: THE HEAD AND TAIL OF THE CALCULATED DOUBLE-LENGTH
RESULT.
PROCEDURES USED:
DPMUL = CP31103;
LNGADD = CP31105.
DATA AND RESULTS:
(D, DD):= (C, CC) + THE SCALAR PRODUCT OF THE VECTORS GIVEN IN
THE ARRAYS A[P:Q] (SEE THE MEANING OF PARAMETER A) AND B[L:U],
WHERE THE ELEMENTS OF THE FIRST VECTOR ARE: IF L < I THEN
A[(I - 1) * I // 2 + J], J = L, ..., MIN(U, I - 1) AND
A[(J - 1) * J // 2 + I], J = I, ..., U, RESPECTIVELY, OTHERWISE
A[(J - 1) * J // 2 + I], J = L, ..., U; THE VALUES OF L, U, I
SHOULD BE POSITIVE.
LANGUAGE: ALGOL 60.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 12
SUBSECTION: LNGFULMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGFULMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31505;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B, CALCULATED WITH DOUBLE LENGTH ARITHMETIC,
IS DELIVERED IN C.
LANGUAGE: COMPASS.
(FOR THE COMPASS-VERSION THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 13
SUBSECTION: LNGFULTAMVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGFULTAMVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31506;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LR:UR]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LC:UC];
THE RESULT A' * B, CALCULATED WITH DOUBLE LENGTH ARITHMETIC,
IS DELIVERED IN C; HERE A' DENOTES THE TRANSPOSED OF THE
MATRIX A.
LANGUAGE: COMPASS.
(FOR THE COMPASS-VERSION THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 14
SUBSECTION: LNGFULSYMMATVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGFULSYMMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC, B; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"CODE" 31507;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX; LR >= 1;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX; LC >= 1;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U], WHERE:
L = MIN(LR * (LR - 1) // 2 + LC, LC * (LC - 1) // 2 + LR),
U = MAX(UR * (UR - 1) // 2 + UC, UC * (UC - 1) // 2 + UR)
AND THE (I,J)-TH ELEMENT OF THE SYMMETRIC MATRIX SHOULD BE
GIVEN IN A[J * (J - 1) // 2 + I];
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B, CALCULATED WITH DOUBLE LENGTH ARITHMETIC,
IS DELIVERED IN C.
PROCEDURES USED:
LNGSYMMATVEC = CP34418.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 15
SUBSECTION: LNGRESVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"CODE" 31508;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[LR:UR,LC:UC]; THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE VALUE OF THE MULTIPLYING SCALAR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B + X * C, CALCULATED WITH DOUBLE LENGTH
ARITHMETIC, IS OVERWRITTEN ON C.
LANGUAGE: COMPASS.
(FOR THE COMPASS-VERSION THE PARAMETER B CAN BE ASSUMED TO HAVE
BEEN SPECIFIED "VALUE".)
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 16
SUBSECTION: LNGSYMRESVEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LNGSYMRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, B, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"CODE" 31509;
THE MEANING OF THE FORMAL PARAMETERS IS:
LR, UR: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE ROW-INDEX; LR >= 1;
LC, UC: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND OF THE COLUMN-INDEX; LC >= 1;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[L:U], WHERE:
L = MIN(LR * (LR - 1) // 2 + LC, LC * (LC - 1) // 2 + LR),
U = MAX(UR * (UR - 1) // 2 + UC, UC * (UC - 1) // 2 + UR)
AND THE (I,J)-TH ELEMENT OF THE SYMMETRIC MATRIX SHOULD BE
GIVEN IN A[J * (J - 1) // 2 + I];
B: <ARRAY IDENTIFIER>;
"ARRAY" B[LC:UC]; THE VECTOR;
X: <ARITHMETIC EXPRESSION>;
THE VALUE OF THE MULTIPLYING SCALAR;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[LR:UR];
THE RESULT A * B + X * C, CALCULATED WITH DOUBLE LENGTH
ARITHMETIC, IS OVERWRITTEN ON C.
PROCEDURES USED:
DPMUL = CP31103;
LNGSYMMATVEC = CP34418.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
ELEMENTARY.
1SECTION : 1.5.2 (JANUARY 1976) PAGE 17
METHOD AND PERFORMANCE:
ALL PROCEDURES GIVEN IN THIS SECTION MAKE USE OF THE DOUBLE-LENGTH
ARITHMETIC OPERATIONS AVAILABLE IN HARDWARE ON THE CYBER 73.
LET (X, XX) DENOTE A DOUBLE-LENGTH NUMBER, THEN WE WILL SAY THAT XX
IS A TAIL TO X, WHEN THE FOLLOWING CONDITIONS HOLD:
X = FL1(X + XX),
(X, XX) = FL2(X + XX),
WHERE FL1(. + .) AND FL2(. + .) DENOTE SINGLE-LENGTH RESPECTIVELY
DOUBLE-LENGTH ADDITION WITH TRUNCATING OF THE RESULT TO 48 AND 96
BITS RESPECTIVELY.
WHEN A PROCEDURE DELIVERS A DOUBLE LENGTH RESULT IN D AND DD, THEN
THESE RESULTS ARE SUCH THAT DD IS A TAIL TO D; WHEN ONE SHOULD
PROVIDE AN INITIALIZING DOUBLE LENGTH SCALAR IN C AND CC, THEN CC
SHOULD BE A TAIL TO C, OTHERWISE THE FOLLOWING ERROR MESSAGE WILL
BE PRINTED:
DP PARAMETER TAIL ERROR
AND EXECUTION OF THE PROGRAM WILL TERMINATE IN THE USUAL WAY.
NOTE THAT CC = 0 IS A TAIL TO C FOR ALL VALUES OF C. FURTHERMORE,
IT SEEMS WORTHWHILE TO NOTE THAT THE ARRAY B MUST BE A VALUE
PARAMETER IN ALGOL 60, HOWEVER, IN THE COMPASS ROUTINE THE
DUPLICATION OF THIS ARRAY IS ONLY DONE IF NECESSARY, I.E. IF THE
ACTUAL PARAMETERS B AND C ARE THE SAME.
SOURCE TEXTS:
THE PROCEDURES IN THIS SECTION ARE WRITTEN IN COMPASS, EXCEPT FOR
LNGSYMMATVEC, LNGFULSYMMATVEC AND LNGSYMRESVEC.
WE GIVE EQUIVALENT ALGOL 60 TEXTS OF THE COMPASS ROUTINES. FOR THE
COMPASS TEXT SEE APPENDIX C, SECTION 1.5.2.
0"CODE" 34410;
"PROCEDURE" LNGVECVEC(L, U, SHIFT, A, B, C, CC, D, DD);
"VALUE" L, U, SHIFT, C, CC; "INTEGER" L,U,SHIFT;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[L], B[L + SHIFT], E, EE);
LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGVECVEC
1SECTION : 1.5.2 (JANUARY 1976) PAGE 18
;
"EOP"
0"CODE" 34411;
"PROCEDURE" LNGMATVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC; "INTEGER" L, U, I;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[I,L], B[L], E, EE); LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGMATVEC;
"EOP"
0"CODE" 34412;
"PROCEDURE" LNGTAMVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC; "INTEGER" L, U, I;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[L,I], B[L], E, EE); LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGTAMVEC;
"EOP"
0"CODE" 34413;
"PROCEDURE" LNGMATMAT(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[I,L], B[L,J], E, EE); LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGMATMAT
1SECTION : 1.5.2 (JANUARY 1976) PAGE 19
;
"EOP"
0"CODE" 34414;
"PROCEDURE" LNGTAMMAT(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[L,I], B[L,J], E, EE); LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGTAMMAT;
"EOP"
0"CODE" 34415;
"PROCEDURE" LNGMATTAM(L, U, I, J, A, B, C, CC, D, DD);
"VALUE" L, U, I, J, C, CC; "INTEGER" L, U, I, J;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[I,L], B[J,L], E, EE); LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGMATTAM;
"EOP"
0"CODE" 34416;
"PROCEDURE" LNGSEQVEC(L, U, IL, SHIFT, A, B, C, CC, D, DD);
"VALUE" L, U, IL, SHIFT, C, CC; "INTEGER" L, U, IL, SHIFT;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE;
"FOR" L:= L "STEP" 1 "UNTIL" U "DO"
"BEGIN" DPMUL(A[IL], B[L + SHIFT], E, EE); IL:= IL + L;
LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END" LNGSEQVEC
1SECTION : 1.5.2 (JANUARY 1976) PAGE 20
;
"EOP"
"CODE" 34417;
"PROCEDURE" LNGSCAPRD1(LA, SA, LB, SB, N, A, B, C, CC, D, DD);
"VALUE" LA, SA, LB, SB, N, C, CC; "INTEGER" LA, SA, LB, SB,N;
"REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "REAL" E, EE; "INTEGER" K;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" DPMUL(A[LA], B[LB], E, EE); LA:= LA + SA; LB:= LB + SB;
LNGADD(C, CC, E, EE, C, CC)
"END";
D:= C; DD:= CC
"END";
"EOP"
"CODE" 34418;
"PROCEDURE" LNGSYMMATVEC(L, U, I, A, B, C, CC, D, DD);
"VALUE" L, U, I, C, CC;
"INTEGER" L, U, I; "REAL" C, CC, D, DD; "ARRAY" A, B;
"BEGIN" "INTEGER" K, M;
M:= "IF" L > I "THEN" L "ELSE" I; K:= M * (M - 1) // 2;
LNGVECVEC(L, "IF" I <= U "THEN" I - 1 "ELSE" U,
K, B, A, C, CC, C, CC);
LNGSEQVEC(M, U, K + I, 0, A, B, C, CC, D, DD)
"END" LNGSYMMATVEC;
"EOP"
"CODE" 31505;
"PROCEDURE" LNGFULMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN" "REAL" D, DD;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"BEGIN" LNGMATVEC(LC, UC, LR, A, B, 0, 0, D, DD); C[LR]:= D + DD
"END"
"END" LNGFULMATVEC;
"EOP"
"CODE" 31506;
"PROCEDURE" LNGFULTAMVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN" "REAL" D, DD;
"FOR" LC:= LC "STEP" 1 "UNTIL" UC "DO"
"BEGIN" LNGTAMVEC(LR, UR, LC, A, B, 0, 0, D, DD); C[LC]:= D + DD
"END"
"END" LNGFULTAMVEC
1SECTION : 1.5.2 (JANUARY 1976) PAGE 21
;
"EOP"
0"CODE" 31507;
"PROCEDURE" LNGFULSYMMATVEC(LR, UR, LC, UC, A, B, C);
"VALUE" LR, UR, LC, UC, B; "INTEGER" LR, UR, LC, UC;
"ARRAY" A, B, C;
"BEGIN" "REAL" D, DD;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"BEGIN" LNGSYMMATVEC(LC, UC, LR, A, B, 0, 0, D, DD);
C[LR]:= D + DD
"END"
"END" LNGFULSYMMATVEC;
"EOP"
0"CODE" 31508;
"PROCEDURE" LNGRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"BEGIN" "REAL" D, DD, E, EE;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"BEGIN" DPMUL(C[LR], X, E, EE);
LNGMATVEC(LC, UC, LR, A, B, E, EE, D, DD); C[LR]:= D + DD
"END"
"END" LNGRESVEC;
"EOP"
0"CODE" 31509;
"PROCEDURE" LNGSYMRESVEC(LR, UR, LC, UC, A, B, C, X);
"VALUE" LR, UR, LC, UC, B, X; "INTEGER" LR, UR, LC, UC;
"REAL" X; "ARRAY" A, B, C;
"BEGIN" "REAL" D, DD, E, EE;
"FOR" LR:= LR "STEP" 1 "UNTIL" UR "DO"
"BEGIN" DPMUL(C[LR], X, E, EE);
LNGSYMMATVEC(LC, UC, LR, A, B, E, EE, D, DD); C[LR]:= D + DD
"END"
"END" LNGSYMRESVEC;
"EOP"
1SECTION : 1.5.3 (MARCH 1977) PAGE 1
AUTHORS: T.J.DEKKER & J.KOOPMAN.
CONTRIBUTOR: J.KOOPMAN.
INSTITUTE: UNIVERSITY OF AMSTERDAM.
RECIEVED: 770328
BRIEF DESCRIPTION:
LNGREATODECI CONVERTS A DOUBLE-LENGTH NUMBER TO A
NUMBER IN DECIMAL FLOATING-POINT REPRESENTATION.
THE RESULT CONSISTS OF A MANTISSA MANT OF MAGNITUDE<1
(AND >=.1) AND A DECIMAL EXPONENT EXPO.
KEYWORDS:
DOUBLE PRECISION ARITHMETIC,
CONVERSION,
DECIMAL REPRESENTATION.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LNGREATODECI(X, XX, S, MANT, EXPO);
"VALUE"X,XX,S;"INTEGER"S,EXPO;"REAL"X,XX;"INTEGER""ARRAY"MANT;
"CODE" 31100;
THE MEANING OF THE FORMAL PARAMETERS IS:
X,XX : <ARITHMETIC EXPRESSIONS>;
ENTRY: THE HEAD (X) AND TAIL (XX) OF THE NUMBER
THAT IS TO BE CONVERTED;
S : <ARITHMETIC EXPRESSION>;
ENTRY: THE DESIRED NUMBER OF SIGNIFICANT DIGITS
OF THE CONVERTED VARIABLE.
ONE SHOULD NOT CHOOSE S LARGER THAN THE NUMBER
OF DIGITS CORRESPONDING TO THE DOUBLE LENGTH
MACHINE PRECISION (FOR CDC: S<29 ). OTHERWISE,
THE LAST DIGITS ARE USELESS, AS ALL OPERATIONS
IN LNGREATODECI ARE PERFORMED IN DOUBLE-LENGTH A-
RITHMETIC; IF S IS CHOSEN NONPOSITIVE,ONLY THE SIGN
AND THE DECIMAL EXPONENT OF THE CONVERTED NUMBER
ARE DELIVERED;
1SECTION : 1.5.3 (MARCH 1977) PAGE 2
MANT : <ARRAY IDENTIFIER>;
"INTEGER""ARRAY"MANT[0:S];
EXIT: MANT[0]: THE SIGN OF THE DECIMAL NUMBER;
MANT[I] (I^=0): THE I-TH SIGNIFICANT
DIGIT OF THE MANTISSA OF THE CONVERTED NUMBER;
I.E. THE VALUE OF THE MANTISSA EQUALS
MANT[0]*(MANT[1]/10+MANT[2]/100+...MANT[S]/10**S);
EXPO : <INTEGER VARIABLE>;
EXIT: THE DECIMAL EXPONENT OF THE CONVERTED NUMBER,
I.E. THE DOUBLE-LENGTH NUMBER (X,XX) APPROXIMATELY
EQUALS MANTISSA*10**EXPO WITH THE VALUE OF
MANTISSA GIVEN IN MANT.
PROCEDURES USED:
LNG SUB = CP31106.
LNG MUL = CP31108.
DP POW = CP31109.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO LN(LN(X))+S.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
LNGREATODECI DETERMINES THE DECIMAL EXPONENT EXPO. AFTER
THAT, THE LONG REAL NUMBER (X,XX) IS DIVIDED BY 10**EXPO
IN DOUBLE PRECISION. BY TRUNCATING THE RESULT, THE
FIRST MOST SIGNIFICANT DIGIT OF THE MANTISSA IS OB-
TAINED. SUBTRACTING THIS DIGIT FROM (X,XX)/10**EXPO,
MULTIPLYING THE RESULT WITH 10, THE NEXT MOST SIGNI-
FICANT DIGIT CAN BE OBTAINED BY TRUNCATION. THIS PRO-
CESS OF SUBTRACTION, MULTIPLICATION AND TRUNCATION WILL
BE REPEATED UNTIL S DIGITS ARE OBTAINED. FINALLY,
THE MANTISSA THUS OBTAINED IS PROPERLY ROUNDED.
1SECTION : 1.5.3 (MARCH 1977) PAGE 3
EXAMPLE OF USE:
"BEGIN""COMMENT"EXAMPLE OF USE OF LNGREATODECI AND DP POW;
"INTEGER"S,EXPO;
"REAL"OP,OPL;
"PROCEDURE"PRINT(S,MANT,EXPONENT);
"VALUE"S,EXPONENT;"INTEGER"S,EXPONENT;"INTEGER""ARRAY"MANT;
"BEGIN""INTEGER"K;
OUTCHARACTER(61,"("-++")",MANT[0]+2);
"FOR"K:=1"STEP"1"UNTIL"S"DO"
"BEGIN""IF"K=1"THEN"OUTPUT(61,"(""(".")"")");
OUTPUT(61,"("D")",MANT[K])
"END";OUTPUT(61,"(""(""")",+3D")",EXPONENT)
"END"PRINT;
DP POW(2,48,OP,OPL);
"FOR"S:=0"STEP"4 "UNTIL"28"DO"
"BEGIN""INTEGER""ARRAY"MANT[0:S];
LNGREATODECI(OP,OPL,S,MANT,EXPO);
PRINT(S,MANT,EXPO);
OUTPUT(61,"("/")")
"END"
"END"
DELIVERS:
+"+015
+.2815"+015
+.28147498"+015
+.281474976711"+015
+.2814749767106560"+015
+.28147497671065600000"+015
+.281474976710656000000000"+015
+.2814749767106560000000000000"+015
1SECTION : 1.5.3 (MARCH 1977) PAGE 4
SOURCE TEXT(S):
"CODE"31100;
"PROCEDURE"LNGREATODECI(X,XX,S,MANT,EXPO);
"VALUE"X,XX,S;"INTEGER"S,EXPO;"REAL"X,XX;"INTEGER""ARRAY"MANT;
"BEGIN""INTEGER"I,K;
"REAL"P,PP;
MANT[0]:=SIGN(X);"IF"X<0"THEN""BEGIN"X:=-X;XX:=-XX"END";
"IF"X=0"THEN"EXPO:=0
"ELSE"EXPO:=ENTIER(LN(X)/LN(10))+1;
DP POW(10,-EXPO,P,PP);
LNG MUL(X,XX,P,PP,X,XX);
"FOR"I:=0"WHILE"ENTIER(X)=0 & X^=0 "DO"
"BEGIN"LNG MUL(X,XX,10,0,X,XX);EXPO:=EXPO-1"END";
"FOR"I:=1"STEP"1"UNTIL"S"DO"
"BEGIN"K:=ENTIER(X);"IF"K>9"THEN"K:=9;MANT[I]:=K;
LNG SUB(X,XX,K,0,P,PP);LNG MUL(P,PP,10,0,X,XX)
"END";
"IF"ENTIER(X)>=5
"THEN""BEGIN""FOR"I:=S"STEP"-1"UNTIL"1"DO"
"BEGIN"K:=MANT[I]+1;
"IF"K<10"THEN""BEGIN"MANT[I]:=K;"GOTO"READY
"END";
MANT[I]:=0
"END";
EXPO:=EXPO+1;
"IF"S>0"THEN"MANT[1]:=1;
READY:
"END";
EXPO:=EXPO+1
"END" LNGREATODECI;
"EOP"
1SECTION : 2.2.1.1 (OCTOBER 1975) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
INSTITUTE: REKENCENTRUM RIJKSUNIVERSITEIT GRONINGEN.
RECEIVED: 741114.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THE PROCEDURES POL, TAYPOL, NORDERPOL AND
DERPOL.
POL EVALUATES A POLYNOMIAL;
DERPOL EVALUATES THE FIRST K DERIVATIVES OF A POLYNOMIAL;
NORDERPOL EVALUATES THE FIRST K NORMALIZED DERIVATIVES OF A
POLYNOMIAL (I.E. J-TH DERIVATIVE/(J FACTORIAL),J=0,1,...,K<=DEGREE;
TAYPOL EVALUATES X**J*(J-TH DERIVATIVE)/(J FACTORIAL),
J=0,1,...,K<=DEGREE.
KEYWORDS:
POLYNOMIAL EVALUATION,
(NORMALIZED) DERIVATIVES,
DERIVATIVES OF A POLYNOMIAL.
SUBSECTION: POL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE" POL(N,X,A);
"VALUE"N,X;"INTEGER"N;"REAL"X;"ARRAY"A;
"CODE" 31040;
POL:=A[0]+A[1]*X+...+A[N]*X**N;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL
A[0]+A[1]*X+...+A[N]*X**N.
PROCEDURES USED: NONE.
1SECTION : 2.2.1.1 (OCTOBER 1975) PAGE 2
RUNNING TIME: PROPORTIONAL TO N
(N MULTIPLICATIONS AND ADDITIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE METHOD USED FOR EVALUATION IS HORNER'S RULE (SYNTHETIC
DIVISION). THE ERROR GROWTH IS GIVEN BY A LINEAR FUNCTION OF THE
DEGREE OF THE POLYNOMIAL (SEE VAN DER LAAN, STOER(1972) P. 29 (EX.
11) OR WILKINSON(1963) P. 36,37).
SUBSECTION: DERPOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" DERPOL(N,K,X,A);
"VALUE"N,K,X;"INTEGER"N,K;"REAL"X;"ARRAY"A;
"CODE" 31243;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
K: <ARITHMETIC EXPRESSION>;
ENTRY: THE FIRST K DERIVATIVES ARE TO BE CALCULATED;
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL
A[0]+A[1]*X+...+A[N]*X**N;
EXIT: THE J-TH DERIVATIVE IS DELIVERED IN A[J], J=0,1,...,K<=
DEGREE; THE OTHER ELEMENTS ARE GENERALLY ALTERED.
PROCEDURES USED :
NORDERPOL = CP31242
RUNNING TIME: THE NUMBER OF ADDITIONS IS (K+1)*(N-K/2) AND
THE NUMBER OF MULTIPLICATIONS IS N, IN FIRST ORDER.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE TAYPOL(THIS SECTION).
1SECTION : 2.2.1.1 (OCTOBER 1975) PAGE 3
SUBSECTION: NORDERPOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" NORDERPOL(N,K,X,A);
"VALUE"N,K,X;"INTEGER"N,K;"REAL"X;"ARRAY"A;
"CODE" 31242;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
K: <ARITHMETIC EXPRESSION>;
THE FIRST K NORMALIZED DERIVATIVES ARE TO BE CALCULATED
(I.E. (J-TH DERIVATIVE)/(J FACTORIAL), J=0,1,...,K<=DEGREE).
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL
A[0]+A[1]*X+...+A[N]*X**N;
EXIT: THE J-TH NORMALIZED DERIVATIVE IS DELIVERED IN A[J]
J=0,1,...,K<=DEGREE; THE OTHER ELEMENTS ARE GENERALLY
ALTERED.
PROCEDURES USED: NONE.
REQUIRED CENTRAL MEMORY: AN AUXILIARY ARRAY OF ORDER N + 1 IS DECLARED.
RUNNING TIME: THE NUMBER OF ADDITIONS IS (K+1)*(N-K/2) AND
THE NUMBER OF MULTIPLICATIONS/DIVISIONS IS 2 * N + K.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE TAYPOL(THIS SECTION).
1SECTION : 2.2.1.1 (OCTOBER 1975) PAGE 4
SUBSECTION: TAYPOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" TAYPOL(N,K,X,A);
"VALUE"N,K,X;"INTEGER"N,K;"REAL"X;"ARRAY"A;
"CODE" 31241;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
K: <ARITHMETIC EXPRESSION>;
ENTRY: THE FIRST K TERMS X**J*(J-TH DERIVATIVE)/(J FACTORIAL),
J=0,1,...,K<=DEGREE, ARE TO BE CALCULATED;
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL
A[0]+A[1]*X+...+A[N]*X**N;
EXIT: THE J-TH TERM X**J*(J-TH DERIVATIVE)/(J FACTORIAL), IS
DELIVERED IN A[J], J=0,1,...,K<=DEGREE; THE OTHER
ELEMENTS ARE GENERALLY ALTERED.
PROCEDURES USED: NONE.
RUNNING TIME: THE NUMBER OF ADDITIONS IS (K+1)*(N-K/2) AND
THE NUMBER OF MULTIPLICATIONS IS 2 * N.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE METHOD OF EVALUATION IS GIVEN BY TRAUB AND SHAW(1972,1974).
LET X**J*(J-TH DERIVATIVE OF THE POLYNOMIAL)/(J FACTORIAL)=
(J OVER J)*A[J]*X**J+(J+1 OVER J)*A[J+1]*X**(J+1)+...+(N OVER J)*
A[N]*X**N,THEN THE J-TH DERIVATIVE (UP TO A FACTOR) CAN BE OBTAINED
FROM THE BINOMIAL COEFFICIENTS FOLLOWED BY EVALUATION OF THE ABOVE
INPRODUCT. THE SHAW AND TRAUB ALGORITHM PERFORMS THE BUILDING UP OF
THE BINOMIAL COEFFICIENTS IMPLICITLY.
WE HAVE NOT IMPLEMENTED THE MORE SOPHISTICATED ALGORITHM, BASED
ON DIVISORS OF N+1, BECAUSE OF THE MORE COMPLEX APPEARANCE
OF THE IMPLEMENTATION AND BECAUSE OF THE DIFFICULTY IN CHOSING
THE MOST EFFICIENT DIVISOR. OUR (RESTRICTED) IMPLEMENTATION OF THE
ONE-PARAMETER FAMILY OF ALGORITHMS PRESERVES THE LINEAR NUMBER OF
MULTIPLICATIONS (2*N (NORDERPOL, TAYPOL) AND 3*N (DERPOL)).
THE ABSOLUTE ERROR IS OF ORDER MAX((N OVER N)*A[N]*X**(N-K),...,
(N OVER K)*A[K]), FOR THE K-TH NORMALIZED DERIVATIVE (SEE VAN DER
LAAN OR WOZNIAKOWSKI).
1SECTION : 2.2.1.1 (DECEMBER 1979) PAGE 5
REFERENCES:
[1].SHAW,M. AND J. TRAUB:
ON THE NUMBER OF MULTIPLICATIONS FOR THE EVALUATION OF A
POLYNOMIAL AND SOME OF ITS DERIVATIVES (21 P.).
JOURN. ACM, 1974, VOL. 21, NO. 1, P. 161-167.
[2].STOER,J. :
EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK 1.
SPRINGER, 1972.
[3].VAN DER LAAN C.G.:
ORTHOGONAL POLYNOMIALS IN NUMERICAL ANALYSIS.
1. ERROR ANALYSIS OF RECURRENCE RELATIONS IN FLOATING-POINT
COMPUTATION, (TO APPEAR).
[4].WILKINSON,J.H. :
ROUNDING ERRORS IN ALGEBRAIC PROCESSES.
HSO, NOTES ON APPLIED SCIENCES NO. 32, 1963.
[5].WOZNIAKOWSKI,H.:
ROUNDING ERROR ANALYSIS FOR THE EVALUATION OF A POLYNOMIAL AND
SOME OF ITS DERIVATIVES.
SIAM J. OF NUM. AN. VOL 11, NO. 4, P. 780-787.
EXAMPLE OF USE:
AS A FORMAL TEST OF THE PROCEDURE DERPOL THE DERIVATIVES OF THE
POLYNOMIAL 3*X**3-2*X**2+X-1 ARE CALCULATED AT X=1.
"BEGIN""ARRAY"A[0:3];
A[3]:=3;A[2]:=-2;A[1]:=1;A[0]:=-1;
DERPOL(3,3,1,A); OUTPUT(61,"("
"("THE 0-TH UNTIL AND INCLUDING THE 3-TH DERIVATIVES :")",
4(BZDB)")",A[0],A[1],A[2],A[3]);
"END" EXAMPLE OF USE;
THE 0-TH UNTIL AND INCLUDING THE 3-TH DERIVATIVES : 1 6 14 18
SOURCE TEXT(S):
0"CODE" 31040;
"REAL" "PROCEDURE" POL(N,X,A);
"VALUE" N,X;"INTEGER" N;"REAL" X;"ARRAY" A;
"BEGIN" "REAL" R;
R:= 0;
"FOR" N:= N "STEP" -1 "UNTIL" 0 "DO"
R:=R*X + A[N];
POL:= R
"END" POL
1SECTION : 2.2.1.1 (OCTOBER 1975) PAGE 6
;
"EOP"
0"CODE" 31241;
"PROCEDURE" TAYPOL(N,K,X,A);
"VALUE" N,K,X;
"INTEGER" N,K;"REAL" X;"ARRAY" A;
"IF" X^= 0 "THEN"
"BEGIN" "INTEGER" I,J,NM1;
"REAL" XJ,AA,H;
XJ:=1;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" XJ:=XJ*X;A[J]:=A[J]*XJ "END";
AA:=A[N];NM1:=N-1;
"FOR" J:= 0 "STEP" 1 "UNTIL" K "DO"
"BEGIN" H:=AA;
"FOR" I:= NM1 "STEP" -1 "UNTIL" J "DO"
H:= A[ I]:=A[I]+H
"END"
"END" "ELSE"
"FOR" K:= K "STEP" -1 "UNTIL" 1 "DO" A[K]:=0;
"EOP"
0"CODE" 31242;
"PROCEDURE" NORDERPOL (N,K,X,A);
"VALUE" N,K,X;
"INTEGER" N,K;"REAL" X;"ARRAY" A;
"IF" X^= 0 "THEN"
"BEGIN" "INTEGER" I,J,NM1;
"REAL" XJ,AA,H;
"ARRAY" XX[0:N];
XJ:=1;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" XJ:=XX[J]:=XJ*X;A[J]:=A[J]*XJ "END";
H:=AA:=A[N];NM1:=N-1;
"FOR" I:= NM1 "STEP" -1 "UNTIL" 0 "DO" H:= A[I]:=A[I]+H;
"FOR" J:= 1 "STEP" 1 "UNTIL" K "DO"
"BEGIN" H:=AA;
"FOR" I:= NM1 "STEP" -1 "UNTIL" J "DO"
H:= A[ I]:=A[I]+H;
A[J]:=H/XX[J]
"END"
"END" NORDERPOL ;
"EOP"
0"CODE" 31243;
"PROCEDURE" DERPOL (N,K,X,A);
"VALUE" N,K,X;
"INTEGER" N,K;"REAL" X;"ARRAY" A;
"BEGIN" "INTEGER" J; "REAL" FAC;
FAC:=1;
NORDERPOL (N,K,X,A);
"FOR" J:= 2 "STEP" 1 "UNTIL" K "DO"
"BEGIN" FAC:=FAC*J;A[J]:=A[J]*FAC "END"
"END" DERPOL ;
"EOP"
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 1
AUTHOR: C.G. VAN DER LAAN
CONTRIBUTORS: C.G. VAN DER LAAN, M. VOORINTHOLT
INSTITUTE: REKENCENTRUM RIJKSUNIVERSITEIT GRONINGEN
RECEIVED: 780601
BRIEF DESCRIPTION:
THIS SECTION CONTAINS SIX PROCEDURES FOR THE EVALUATION
OF ORTHOGONAL POLYNOMIALS:
ORTPOL,ORTPOLSYM: EVALUATE AN ORTHOGONAL POLYNOMIAL,
ALLORTPOL,ALLORTPOLSYM: EVALUATE ALL ORTHOGONAL POLYNOMIALS OF
DEGREE LESS THAN A GIVEN POSITIVE INTEGER.
SUMORTPOL,SUMORTPOLSYM: EVALUATE A SERIES OF
ORTHOGONAL POLYNOMIALS.
THE PROCEDURES ENDING WITH SYM ARE EFFICIENT VERSIONS FOR
SYMMETRICAL POLYNOMIALS (I.E. ODD INDEXED POLYNOMIALS THAT ARE ODD
FUNCTIONS AND EVEN INDEXED POLYNOMIALS THAT ARE EVEN FUNCTIONS);
KEYWORDS:
ORTHOGONAL POLYNOMIAL,
SERIES OF ORTHOGONAL POLYNOMIALS,
LINEAR THREE TERM (IN)HOMOGENEOUS RECURRENCE RELATION.
DATA AND RESULTS:
ORTHOGONAL POLYNOMIALS CAN BE CHARACTERIZED BY A RECURRENCE
RELATION OF THE FOLLOWING FORM
A1[K] * F[K+1](X) = (A2[K] + X * A3[K]) * F[K](X)
- A4[K] * F[K-1](X),
WHERE AI[K] ARE REAL NUMBERS. SEE FOR INSTANCE TABLE 22.7 IN
ABRAMOWITZ AND STEGUN (1964) FOR THE CLASSICAL ORTHOGONAL
POLYNOMIALS. BY AN ELEMENTARY TRANSFORMATION, THE COEFFICIENTS
IN THE RECURRENCE RELATION ABOVE CAN BE MADE SUCH THAT
THE RECURRENCE RELATION IS GIVEN BY
P[K+1](X) = (X - B[K]) * P[K](X) - C[K] * P[K-1](X),
P[0](X) = 1, P[1](X) = X - B[0].
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 2
IN THIS WAY WE OBTAIN A NORMALIZATION OF THE ORTHOGONAL POLYNOMIAL
SUCH THAT THE LEADING COEFFICIENT IN THE EXPLICIT REPRESENTATION OF
P[K](X) EQUALS 1.
AS A CONSEQUENCE THE FOLLOWING RELATION HOLDS BETWEEN THE VALUES
OBTAINED BY OUR PROCEDURES (E.G. ORTPOL) AND THE VALUES FROM
THE REPRESENTATION IN ABRAMOWITZ AND STEGUN (1964) (I.C. F)
N-1
ORTPOL[N](X) = PROD (A1[K]/A3[K]) * F[N](X) ,
K=0
WHERE A1[K], A3[K], F[N] ARE DETERMINED BY 22.4 AND 22.7 IN
ABRAMOWITZ AND STEGUN(1964). WE NOTICE THAT OVERFLOW/UNDERFLOW
MAY OCCUR EARLIER AS A CONSEQUENCE OF OUR NORMALIZATION.
IN ORDER TO AVOID MISTAKES WHEN OBTAINING THE RECURRENCE
COEFFICIENTS THE FOLLOWING TABLE GIVES THE RECURRENCE COEFFICIENTS
FOR THE CLASSICAL ORTHOGONAL POLYNOMIALS (NOTE THAT THE FIRST AND
SECOND POLYNOMIAL ARE DEFINED BY THE NORMALIZATION AND B[0]):
POLYNOMIAL KIND : RECURRENCE COEFFICIENTS
----------------------:--------------------------------------------
: B[K] : C[K]
:-----------------------:--------------------
CHEBYSHEV (1-ST KIND) : 0 : 1/2 , K=1
: : 1/4 , K>1
: :
CHEBYSHEV (2-ND KIND) : 0 : 1/4
: :
LEGENDRE : 0 : K**2/(4*K**2-1)
: :
JACOBI : -(ALPHA**2-BETA**2)/ : 4*(1+ALPHA)*
: ((2*K+ALPHA+BETA)* : (1+BETA)/((ALPHA+
: (2*K+ALPHA+BETA+2)) : BETA+2)**2*(ALPHA+
: : BETA+3)) , K=1
: :
: : 4*K*(K+ALPHA)*
: : (K+BETA)*(K+ALPHA+
: : BETA)/((2*K+ALPHA+
: : BETA)**2*((2*K+
: : ALPHA+BETA)**2-1))
: : , K>1
: :
: : (ALPHA,BETA > -1)
: :
LAGUERRE : 2*K+ALPHA+1 : K*(K+ALPHA)
: :
HERMITE : 0 : K/2
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 3
IN GENERAL THE RECURRENCE COEFFICIENTS MAY BE OBTAINED BY USE OF
THE PROCEDURE RECCOF. THE ABOVE TABLE IS OBTAINED BY ADAPTION
OF TABLE 22.7,ABRAMOWITZ AND STEGUN (1964) P. 782, AS FOLLOWS:
(NOTE THAT N>=1; FOR N=0 CONSULT 22.4)
B[I]:=-A2[I]/A3[I] , I=0,1,...,N-1
C[I]:=(A4[I]*A1[I-1])/(A3[I]*A3[I-1]) , I=1,2,...,N-1.
METHOD AND PERFORMANCE:
LET THE ORTHOGONAL POLYNOMIAL BE DEFINED BY
P[K+1](X)=(X-B[K])*P[K](X)-C[K]*P[K-1](X) , K=1,2,...N-1
WHERE B[0:N-1], C[1:N-1] CONTAIN THE RECURRENCE COEFFICIENTS AND
P[1](X)=X-B[0], P[0](X)=1 (SEE STOER 1972, P. 119).
THEN
N-1 / X-B[K] 1 \ / 1 \
P[N](X) = (X-B[0],1) * PROD [ ] [ ] , N=1,2,.....
K=1 \ -C[K] 0 / \ 0 /
AND
A[0]+A[1]*P[1](X)+...+A[N]*P[N](X)=
N J-1 / X-B[K] 1 \ / A[J] \
A[0] + (X-B[0],1) * SUM PROD [ ] [ ] .
J=1 K=1 \ -C[K] 0 / \ 0 /
THESE EXPRESSIONS ARE EVALUATED BY A GENERALISED HORNER RULE.
(SEE ALSO LUKE, 1969, P. 327).
REFERENCES:
ABRAMOWITZ, M. & I. STEGUN (1964):
HANDBOOK OF MATHEMATICAL FUNCTIONS,
NATIONAL BUREAU OF STANDARDS, WASHINGTON D.C.
LUKE, Y.L. (1969):
THE SPECIAL FUNCTIONS AND THEIR APPROXIMATIONS I,
ACADEMIC PRESS, LONDON, NEW YORK.
STOER, J. (1972):
EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK I,
SPRINGER VERLAG, BERLIN.
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 4
SUBSECTION: ORTPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL" "PROCEDURE" ORTPOL(N,X,B,C); "VALUE" N,X;
"INTEGER" N; "REAL" X; "ARRAY" B,C;
"CODE" 31044;
ORTPOL DELIVERS THE VALUE OF THE ORTHOGONAL POLYNOMIAL OF
DEGREE N FOR THE ARGUMENT X AS DETERMINED BY THE
RECURRENCE COEFFICIENTS.
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE ORTHOGONAL POLYNOMIAL;
B,C: <ARRAY IDENTIFIER>;
"ARRAY" B[0:N-1], C[1:N-1];
ENTRY: THE RECURRENCE COEFFICIENTS (SEE DATA AND RESULTS).
SUBSECTION: ORTPOLSYM.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL" "PROCEDURE" ORTPOLSYM(N,X,C); "VALUE" N,X;
"INTEGER" N; "REAL" X; "ARRAY" C;
"CODE" 31048;
ORTPOLSYM DELIVERS THE VALUE OF THE ORTHOGONAL POLYNOMIAL OF
DEGREE N FOR THE ARGUMENT X AS DETERMINED BY THE
RECURRENCE COEFFICIENTS.
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
X: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE ORTHOGONAL POLYNOMIAL;
C: <ARRAY IDENTIFIER>;
"ARRAY" C[1:N-1];
ENTRY: THE RECURRENCE COEFFICIENTS (SEE DATA AND RESULTS).
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 5
SUBSECTION: ALLORTPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" ALLORTPOL(N,X,B,C,P);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" B,C,P;
"CODE" 31045;
THE MEANING OF THE FORMAL PARAMETERS IS:
N,X,B,C: SEE ORTPOL (THIS SECTION);
P: <ARRAY IDENTIFIER>;
"ARRAY" P[0:N];
EXIT: P[K] CONTAINS, FOR THE ARGUMENT, THE VALUE OF THE
K-TH ORTHOGONAL POLYNOMIAL AS DEFINED BY THE
RECURRENCE COEFFICIENTS.
SUBSECTION: ALLORTPOLSYM.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" ALLORTPOLSYM(N,X,C,P);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" C,P;
"CODE" 31049;
THE MEANING OF THE FORMAL PARAMETERS IS:
N,X,C: SEE ORTPOLSYM (THIS SECTION);
P: <ARRAY IDENTIFIER>;
"ARRAY" P[0:N];
EXIT: P[K] CONTAINS, FOR THE ARGUMENT, THE VALUE OF THE
K-TH ORTHOGONAL POLYNOMIAL AS DEFINED BY THE
RECURRENCE COEFFICIENTS.
SUBSECTION: SUMORTPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL" "PROCEDURE" SUMORTPOL(N,X,B,C,A);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" B,C,A;
"CODE" 31047;
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 6
SUMORTPOL : DELIVERS THE VALUE OF THE POLYNOMIAL
A[0]+A[1]*P[1](X)+...+A[N]*P[N](X)
WHERE P[K](X) IS THE K-TH ORTHOGONAL POLYNOMIAL.
THE MEANING OF THE FORMAL PARAMETERS IS:
N,X,B,C: SEE ORTPOL (THIS SECTION);
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE SERIES EXPANSION
A[0]+A[1]*P[1](X)+...+A[N]*P[N](X)
WHERE P[K](X) IS THE K-TH ORTHOGONAL POLYNOMIAL
AS DEFINED BY THE RECURRENCE COEFFICIENTS.
SUBSECTION: SUMORTPOLSYM.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL" "PROCEDURE" SUMORTPOLSYM(N,X,C,A);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" C,A;
"CODE" 31058;
SUMORTPOLSYM : DELIVERS THE VALUE OF THE POLYNOMIAL
A[0]+A[1]*P[1](X)+...+A[N]*P[N](X)
WHERE P[K](X) IS THE K-TH ORTHOGONAL POLYNOMIAL.
THE MEANING OF THE FORMAL PARAMETERS IS:
N,X,C: SEE ORTPOLSYM (THIS SECTION);
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE SERIES EXPANSION
A[0]+A[1]*P[1](X)+...+A[N]*P[N](X)
WHERE P[K](X) IS THE K-TH ORTHOGONAL POLYNOMIAL
AS DEFINED BY THE RECURRENCE COEFFICIENTS.
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 7
EXAMPLE OF USE:
THE FOLLOWING PROGRAM DELIVERS THE VALUES OF THE
LAGUERRE POLYNOMIAL OF DEGREES 0,1,2,3,4,5 FOR X=0 BY MEANS OF THE
PROCEDURE ALLORTPOL (B[K]=2*K+1, C[K]=K**2):
"BEGIN" "ARRAY" B[0:4], C[1:4], P[0:5];
"INTEGER" I;
B[0]:=1;
"FOR" I:=1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" B[I]:=2*I+1;
C[I]:=I**2;
"END" I;
ALLORTPOL(5,0,B,C,P);
OUTPUT(61,"("2/,6(2B,+2ZD.D)")",P);
"END" PROGRAM;
RESULTS (NOTE THE DIFFERENCE WITH FIGURE 22.9 IN ABRAMOWITZ AND
STEGUN (1964) BECAUSE OF THE NORMALIZATION):
+1.0 -1.0 +2.0 -6.0 +24.0 -120.0
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 8
SOURCE TEXTS:
"CODE" 31044;
"REAL" "PROCEDURE" ORTPOL(N,X,B,C);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" B,C;
"IF" N=0 "THEN" ORTPOL:=1 "ELSE"
"BEGIN" "INTEGER" K,L; "REAL" R,S,H;
R:=X-B[0]; S:=1; L:=N-1;
"FOR" K:=1 "STEP" 1 "UNTIL" L "DO"
"BEGIN" H:=R;
R:=(X-B[K])*R-C[K]*S;
S:=H;
"END";
ORTPOL:=R;
"END" ORTPOL;
"EOP"
0
"CODE" 31048;
"REAL" "PROCEDURE" ORTPOLSYM(N,X,C);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" C;
"IF" N=0 "THEN" ORTPOLSYM:=1.0 "ELSE"
"BEGIN"
"INTEGER" K,L; "REAL" R,S,H;
R:=X; S:=1.0; L:=N-1;
"FOR" K:=1 "STEP" 1 "UNTIL" L "DO"
"BEGIN"
H:=R; R:=X*R-C[K]*S;
S:=H
"END";
ORTPOLSYM:=R
"END" ORTPOLSYM;
"EOP"
0
"CODE" 31045;
"PROCEDURE"ALLORTPOL(N,X,B,C)RESULTS:(P);
"VALUE" N,X; "INTEGER" N;"REAL" X;"ARRAY" B,C,P;
"IF" N=0 "THEN" P[0]:=1"ELSE"
"BEGIN" "INTEGER" K,K1; "REAL" R,S,H;
R:=P[1]:=X-B[0]; S:=P[0]:=1;K:=1;
"FOR" K1:=2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" H:=R; P[K1]:=R:=(X-B[K])*R-C[K]*S;
S:=H; K:=K1;
"END";
"END" ALLORTPOL
1SECTION : 2.2.2.1 (NOVEMBER 1978) PAGE 9
;
"EOP"
"CODE" 31049;
"PROCEDURE" ALLORTPOLSYM(N,X,C)RESULTS:(P);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" C,P;
"IF" N=0 "THEN" P[0]:=1.0 "ELSE"
"BEGIN"
"INTEGER" K; "REAL" R,S,H;
R:=P[1]:=X; S:=P[0]:=1.0;
"FOR" K:=2 "STEP" 1 "UNTIL" N "DO"
"BEGIN"
H:=R;
P[K]:=R:=X*R-C[K-1]*S;
S:=H
"END";
"END" ALLORTPOLSYM;
"EOP"
0
"CODE" 31047;
"REAL" "PROCEDURE" SUMORTPOL(N,X,B,C,A);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" B,C,A;
"IF" N=0 "THEN" SUMORTPOL:=A[0] "ELSE"
"BEGIN" "INTEGER" K; "REAL" H,R,S;
R:=A[N]; S:=0;
"FOR" K:=N-1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN"
H:=R;
R:=A[K]+(X-B[K])*R+S;
S:= -C[K]*H
"END";
SUMORTPOL:=A[0]+(X-B[0])*R+S
"END" SUMORTPOL;
"EOP"
0
"CODE" 31058;
"REAL" "PROCEDURE" SUMORTPOLSYM(N,X,C,A);
"VALUE" N,X; "INTEGER" N; "REAL" X; "ARRAY" C,A;
"IF" N=0 "THEN" SUMORTPOLSYM:=A[0] "ELSE"
"BEGIN"
"INTEGER" K; "REAL" H,R,S;
R:=A[N]; S:=0;
"FOR" K:=N-1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN"
H:=R;
R:=A[K]+X*R+S;
S:= -C[K]*H
"END";
SUMORTPOLSYM:=A[0]+X*R+S
"END" SUMORTPOLSYM;
"EOP"
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
CONTRIBUTORS: C.G. VAN DER LAAN, M. VOORINTHOLT.
INSTITUTE: REKENCENTRUM RIJKSUNIVERSITEIT GRONINGEN.
RECEIVED: 780601.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOUR PROCEDURES ABOUT CHEBYSHEV POLYNOMIALS
OF THE FIRST KIND:
CHEPOLSUM: EVALUATES A (FINITE) SUM OF CHEBYSHEV POLYNOMIALS,
ODDCHEPOLSUM: EVALUATES A (FINITE) SUM OF CHEBYSHEV POLYNOMIALS
OF ODD DEGREE,
CHEPOL: EVALUATES A CHEBYSHEV POLYNOMIAL,
ALLCHEPOL: EVALUATES ALL CHEBYSHEV POLYNOMIALS WITH DEGREE LESS
THAN A GIVEN POSITIVE INTEGER.
KEYWORDS:
(FINITE) SUM OF (SHIFTED) CHEBYSHEV POLYNOMIALS OF THE FIRST KIND,
GOERTZEL,WATT,CLENSHAW,GENERALIZED HORNER ALGORITHM,
LINEAR THREE TERM (INHOMOGENEOUS) RECURRENCE RELATION.
REFERENCES:
CLENSHAW, C.W. (1962):
CHEBYSHEV SERIES FOR MATHEMATICAL FUNCTIONS.
MATH. TAB. NAT. PHYS. LAB. 5, LONDON. HMO.
FOX, L. & I.B. PARKER (1972):
CHEBYSHEV POLYNOMIALS IN NUMERICAL ANALYSIS.
OXFORD UNIVERSITY PRESS.
RIVLIN, T.J. (1974):
THE CHEBYSHEV POLYNOMIALS.
WILEY.
STOER,J.(1972):
EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK 1.
HEIDELBERGER TASCHENBUECHER 105,SPRINGER-VERLAG.
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 2
SUBSECTION: CHEPOLSUM.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL""PROCEDURE"CHEPOLSUM(N,X,A);
"VALUE"N,X;"INTEGER"N;"REAL"X;"ARRAY"A;
"CODE"31046;
CHEPOLSUM:=THE VALUE OF THE CHEBYSHEV SUM
A[0] + A[1]*T[1](X) + .... + A[N]*T[N](X),
WHERE T[1](X),....,T[N](X) ARE CHEBYSHEV POLYNOMIALS
OF THE FIRST KIND, OF DEGREE 1,....,N, RESPECTIVELY.
THE MEANING OF THE FORMAL PARAMETERS IS :
N : <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL REPRESENTED BY THE
CHEBYSHEV SUM (N>=0);
X : <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE CHEBYSHEV POLYNOMIALS , ABS(X)<=1;
A : <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE CHEBYSHEV SUM MUST BE GIVEN IN
ARRAY A, WHERE A[K] IS THE COEFFICIENT OF THE CHEBYSHEV
POLYNOMIAL OF DEGREE K, 0<=K<=N.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE:
N N / 2*X 1 \ K / A[K] \
SUM A[K]*T[K](X) = (1,X) * SUM [ ] * [ ]
K=0 K=0 \ -1 0 / \ 0 /
WE USE THE CLENSHAW OR GENERALIZED HORNER ALGORITHM:
N
SUM A[K]*T[K](X) = (1,X) *
K=0
/ / A[0] \ / 2*X 1 \ / / A[1] \ / 2*X 1 \ / A[N] \ \ \
[ [ ] + [ ] * [ [ ] +..+ [ ] * [ ] ]..].
\ \ 0 / \ -1 0 / \ \ 0 / \ -1 0 / \ 0 / / /
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 3
THIS PROCEDURE MAY BE USED:
- TO EVALUATE THE SUM OF CHEBYSHEV POLYNOMIALS OF EVEN DEGREE
N
SUM A[K]*T[2*K](X) ,
K=0
BY THE CALL OF
CHEPOLSUM(N,"IF" ABS(X)<ARREB "THEN" -1 "ELSE" 2*X*X-1,A)
BECAUSE OF
T[2*K](X) = T[K](T[2](X));
(ARREB DENOTES THE MACHINE PRECISION)
- TO EVALUATE THE SUM OF SHIFTED CHEBYSHEV POLYNOMIALS FOR 0<=X<=1
N
SUM A[K]*T'[K](X) ,
K=0
BY THE CALL OF
CHEPOLSUM(N,2*X-1,A)
BECAUSE OF
T'[K](X) = T[K](2*X-1).
EXAMPLE OF USE :
THE POLYNOMIAL : 1 + 1/2*T[1](X) + 1/4*T[2](X) IS EVALUATED FOR
X = -1,0,1, WHERE T[1](X) AND T[2](X) ARE THE CHEBYSHEV POLYNOMIALS
OF FIRST AND SECOND DEGREE, RESPECTIVELY.
"BEGIN""ARRAY"A[0:2];
A[2]:=.25;A[1]:=.5;A[0]:=1;
OUTPUT(61,"("3(BZ.DD)")",CHEPOLSUM(2,-1,A),CHEPOLSUM(2,0,A),
CHEPOLSUM(2,1,A))
"END"
.75 .75 1.75
SUBSECTION: ODDCHEPOLSUM.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL""PROCEDURE" ODDCHEPOLSUM(N,X,A);
"VALUE"N,X;"INTEGER"N;"REAL"X;"ARRAY"A;
"CODE"31059;
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 4
ODDCHEPOLSUM:= THE VALUE OF THE CHEBYSHEV SUM
A[1]*T[1](X) + .... + A[N]*T[2*N+1](X),
WHERE T[1](X),....,T[2*N+1](X) ARE CHEBYSHEV POLYNOMIALS
OF THE FIRST KIND, OF (ODD) DEGREE 1,....,2*N+1.
THE MEANING OF THE FORMAL PARAMETERS IS:
N : <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL REPRESENTED BY
THE CHEBYSHEV SUM IS 2*N+1 (N>=0);
X : <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE CHEBYSHEV POLYNOMIALS, ABS(X)<=1;
A : <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE CHEBYSHEV SUM MUST BE GIVEN IN
ARRAY A, WHERE A[K] IS THE COEFFICIENT OF THE CHEBYSHEV
POLYNOMIAL OF DEGREE 2*K+1, 0<=K<=N.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE:
FROM THE REPRESENTATION, FOR ABS(X)<=1,
N N / 2*T[2](X) -1 \ K /A[K] \
SUM A[K]*T[2*K+1](X) = X * (1,-1) * SUM [ ] * [ ]
K=0 K=0 \ 1 0 / \ 0 /
WE USE THE CLENSHAW OR GENERALIZED HORNER ALGORITHM:
N / / A[0] \
SUM A[K]*T[2*K+1](X) = X * (1,-1) * [ [ ] +
K=0 \ \ 0 /
/ 2*T[2](X) -1 \ / / A[1] \ / 2*T[2](X) -1 \ / A[N] \ \ \
[ ] * [ [ ]+...+[ ] * [ ] ]..].
\ 1 0 / \ \ 0 / \ 1 0 / \ 0 / / /
THIS PROCEDURE MAY BE USED TO EVALUATE THE SUM OF SHIFTED CHEBYSHEV
POLYNOMIALS OF ODD DEGREE FOR 0<=X<=1,
N
SUM A[K]*T'[2*K+1](X) ,
K=0
BY THE CALL OF
ODDCHEPOLSUM(N,2*X-1,A)
BECAUSE OF
T'[K](X) = T[K](2*X-1).
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 5
EXAMPLE OF USE:
THE POLYNOMIAL 1/2*T[1](X) + 1/5*T[3](X) IS EVALUATED FOR X=-1,0,1,
WHERE T[1](X) AND T[3](X) ARE CHEBYSHEV POLYNOMIALS OF THE FIRST
AND THIRD DEGREE, RESPECTIVELY.
"BEGIN"
"ARRAY"A[0:1];
A[1]:=.2;A[0]:=.5;
OUTPUT(61,"("/,3(B,-Z.DD)")",ODDCHEPOLSUM(1,-1,A),
ODDCHEPOLSUM(1,0,A),
ODDCHEPOLSUM(1,1,A));
"END"
-.70 .00 .70
SUBSECTION: CHEPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"REAL""PROCEDURE"CHEPOL(N,X);
"VALUE"N,X;"INTEGER"N;"REAL"X;
"CODE"31042;
CHEPOL:=THE VALUE OF THE CHEBYSHEV POLYNOMIAL OF THE FIRST KIND OF
DEGREE N FOR THE ARGUMENT X.
THE MEANING OF THE FORMAL PARAMETERS IS:
N : <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL (N>=0);
X : <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE CHEBYSHEV POLYNOMIAL, ABS(X)<=1.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE: SEE ALLCHEPOL (NEXT SUBSECTION).
EXAMPLE OF USE: SEE NEXT SUBSECTION.
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 6
SUBSECTION: ALLCHEPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE"ALLCHEPOL(N,X,T);
"VALUE"N,X;"INTEGER"N;"REAL"X;"REAL""ARRAY"T;
"CODE"31043;
THE MEANING OF THE FORMAL PARAMETERS IS:
N : <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE LAST POLYNOMIAL (N>=0);
X : <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE CHEBYSHEV POLYNOMIALS, ABS(X)<=1;
T : <ARRAY IDENTIFIER>;
"ARRAY" T[0:N];
EXIT: THE VALUES OF THE CHEBYSHEV POLYNOMIALS OF THE FIRST
KIND OF DEGREES 0,1,...,N , FOR THE ARGUMENT X, ARE
DELIVERED IN T[0],T[1],...,T[N], RESPECTIVELY.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE:
FOR A DESCRIPTION OF THE ALGORITHM SEE STOER,1972,P.21.
THE MAXIMUM (ABSOLUTE) VALUE OF THE CHEBYSHEV POLYNOMIAL
EQUALS 1 AS A NORMALIZATION.
AN UPPER BOUND FOR THE (ABSOLUTE) ERROR IS A QUADRATIC FUNCTION
OF THE DEGREE OF THE CHEBYSHEV POLYNOMIAL.THIS UPPER BOUND IS A
ROUGH OVER-ESTIMATE FOR THE SPECIAL CASE ABS(X)<.5 (STOER,1972,
P. 21-24).
EXAMPLE OF USE :
BY THE PROCEDURE (ALL)CHEPOL THE CHEBYSHEV POLYNOMIALS OF THE FIRST
KIND OF DEGREES 0,1,2 ARE EVALUATED AT -1,0,1.
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 7
"BEGIN"
"ARRAY"T[0:2];
ALLCHEPOL(2,-1,T);OUTPUT(61,"("/,3(-DB)")",T[0],T[1],T[2]);
ALLCHEPOL(2, 0,T);OUTPUT(61,"("/,3(-DB)")",T[0],T[1],T[2]);
ALLCHEPOL(2, 1,T);OUTPUT(61,"("/,3(-DB)")",T[0],T[1],T[2]);
OUTPUT(61,"("/,3(/,-D)")",CHEPOL(2,-1),CHEPOL(2,0),CHEPOL(2,1))
"END"
1 -1 1
1 0 -1
1 1 1
1
-1
1
SOURCE TEXT(S):
"CODE"31046;
"REAL" "PROCEDURE" CHEPOLSUM(N,X,A);
"VALUE" N,X;"INTEGER" N;"REAL" X;"ARRAY" A;
"IF" N=0 "THEN" CHEPOLSUM:=A[0] "ELSE"
"IF" N=1 "THEN" CHEPOLSUM:=A[0]+A[1]*X "ELSE"
"BEGIN" "INTEGER" K;"REAL" H,R,S,TX;
TX:=X+X;R:=A[N];
H:=A[N-1]+R*TX;
"FOR" K:=N-2 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" S:=R;R:=H;
H:=A[K]+R*TX-S
"END"K;
CHEPOLSUM:=A[0]-R+H*X
"END" CHEPOLSUM
1SECTION : 2.2.2.2 (DECEMBER 1978) PAGE 8
;
"EOP"
"CODE"31059;
"REAL""PROCEDURE"ODDCHEPOLSUM(N,X,A);
"VALUE"N,X;"INTEGER"N;"REAL"X;"ARRAY"A;
"COMMENT" ODDCHEPOLSUM:=A[0]T[1](X)+A[1]T[3](X)+....+A[N]T[2N+1](X);
"IF" N=0 "THEN" ODDCHEPOLSUM:=X*A[0] "ELSE"
"IF" N=1 "THEN" ODDCHEPOLSUM:=X*(A[0]+A[1]*(4*X*X-3)) "ELSE"
"BEGIN"
"INTEGER" K;
"REAL" H,R,S,Y;
Y:=4*X*X-2;
R:=A[N];
H:=A[N-1]+R*Y;
"FOR" K:=N-2 "STEP" -1 "UNTIL" 0 "DO"
"BEGIN"
S:=R;
R:=H;
H:=A[K]+R*Y-S;
"END" K;
ODDCHEPOLSUM:=X*(H-R);
"END" ODDCHEPOLSUM;
"EOP"
"CODE"31042;
"REAL""PROCEDURE"CHEPOL(N,X);
"VALUE"N,X;"INTEGER"N;"REAL"X;
"IF" N = 0 "THEN" CHEPOL :=1 "ELSE"
"IF" N = 1 "THEN" CHEPOL :=X "ELSE"
"BEGIN""INTEGER"I;"REAL"T1,T2,H,X2;
T2:=X;T1:=1;X2:=X+X;
"FOR"I:=2"STEP"1"UNTIL"N"DO"
"BEGIN"H:=X2*T2-T1;T1:=T2;T2:=H"END";
CHEPOL:=H
"END"CHEPOL;
"EOP"
"CODE"31043;
"PROCEDURE"ALLCHEPOL(N,X,T);
"VALUE"N,X;"INTEGER"N;"REAL"X;"REAL""ARRAY"T;
"IF" N = 0 "THEN" T[0] :=1 "ELSE"
"IF" N = 1 "THEN" "BEGIN" T[0] := 1; T[1] := X "END" "ELSE"
"BEGIN""INTEGER"I;"REAL"T1,T2,H,X2;
T[0]:=T1:=1;T[1]:=T2:=X;X2:=X+X;
"FOR"I:=2"STEP"1"UNTIL"N"DO"
"BEGIN"T[I]:=H:=X2*T2-T1;T1:=T2;T2:=H"END"
"END"ALLCHEPOL;
"EOP"
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
INSTITUTE: RIJKSUNIVERSITEIT GRONINGEN.
RECEIVED: 740701.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THE PROCEDURES:
SINSER FOR EVALUATING A SINE SERIES;
COSSER FOR EVALUTING A COSINE SERIES;
FOUSER,FOUSER1,FOUSER2 FOR EVALUATING A FOURIER SERIES
(IN FOUSER THE SERIES IS RESTRICTED TO A SERIES WITH SINE
COEFFICIENTS EQUAL TO COSINE COEFFICIENTS);
COMFOUSER,COMFOUSER1,COMFOUSER2 FOR EVALUATING A COMPLEX FOURIER
SERIES
(IN COMFOUSER THE SERIES IS RESTRICTED TO A SERIES WITH REAL
COEFFICIENTS).
KEYWORDS:
FINITE FOURIER SERIES EVALUATION,
TRIGONOMETRIC POLYNOMIAL EVALUATION,
GOERTZEL,WATT,CLENSHAW,REINSCH ALGORITHM,
LINEAR THREE-TERM INHOMOGENEOUS RECURRENCE RELATION.
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 2
SUBSECTION : SINSER.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"SINSER(N,THETA,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"B;
"CODE" 31090;
SINSER:= THE VALUE OF THE SINE SERIES
B[1]*SIN(THETA)+...+B[N]*SIN(N*THETA).
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE NUMBER OF TERMS IN THE SINE SERIES;
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE SINE SERIES;
B: <ARRAY IDENTIFIER>;
"ARRAY"B[1:N];
ENTRY: THE COEFFICIENTS OF THE SINE SERIES.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: N MULTIPLICATIONS; 3N ADDITIONS;
3 SINE/COSINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 3
SUBSECTION : COSSER.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"COSSER(N,THETA,A);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A;
"CODE" 31091;
COSSER:= THE VALUE OF THE COSINE SERIES
A[0]+A[1]*COS(THETA)+...+A[N]*COS(N*THETA).
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE TRIGONOMETRIC POLYNOMIAL.
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE COSINE SERIES.
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE COSINE SERIES.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: N MULTIPLICATIONS; 3N ADDITIONS;
2 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 4
SUBSECTION : FOUSER.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"FOUSER (N,THETA,A);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A;
"CODE" 31092;
FOUSER := THE VALUE OF THE FOURIER SERIES
A[0]+A[1]*(COS(THETA)+SIN(THETA))+...+A[N]*(COS(N*THETA)
+SIN(N*THETA)).
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE TRIGONOMETRIC POLYNOMIAL;
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE COEFFICIENTS OF THE (FINITE) FOURIER SERIES.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: N MULTIPLICATIONS; 3N ADDITIONS;
3 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 5
SUBSECTION : FOUSER1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"FOUSER1(N,THETA,A,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A,B;
"CODE" 31093;
FOUSER1:= THE VALUE OF THE FOURIER SERIES
A[0]+A[1]*COS(THETA)+B[1]*SIN(THETA)+...
+A[N]*COS(N*THETA)+B[N]*SIN(N*THETA).
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE TRIGONOMETRIC POLYNOMIAL;
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
A,B: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N],B[1:N];
ENTRY: THE COEFFICIENTS OF THE (FINITE) FOURIER SERIES,
WITH A[K] COEFFICIENT OF COS(K*THETA), (K=0,...,N)
AND B[K] COEFFICIENT OF SIN(K*THETA), (K=1,...,N).
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: 4N MULTIPLICATIONS; 4N ADDITIONS;
2 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 6
SUBSECTION : FOUSER2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL""PROCEDURE"FOUSER2(N,THETA,A,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A,B;
"CODE" 31094;
FOUSER2:= THE VALUE OF THE FOURIER SERIES
A[0]+A[1]*COS(THETA)+B[1]*SIN(THETA)+...
+A[N]*COS(N*THETA)+B[N]*SIN(N*THETA).
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE TRIGONOMETRIC POLYNOMIAL;
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
A,B: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N],B[1:N];
ENTRY: THE COEFFICIENTS OF THE (FINITE) FOURIER SERIES,
WITH A[K] COEFFICIENT OF COS(K*THETA), (K=0,...,N)
AND B[K] COEFFICIENT OF SIN(K*THETA), (K=1,...,N).
PROCEDURES USED: SINSER = CP31090,
COSSER = CP31091.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: 2N MULTIPLICATIONS; 6N ADDITIONS;
6 COSINE/SINE EVALUATIONS).
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 7
SUBSECTION : COMFOUSER.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMFOUSER (N,THETA,A,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"A;
"CODE" 31095;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL IN EXP(I*THETA);
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[0:N];
ENTRY: THE REAL COEFFICIENTS A[K] (K=0,...,N) IN THE SERIES
FN(THETA)=A[0]+A[1]*EXP(I*THETA)+...+A[N]*EXP(I*
THETA)**N, MUST BE GIVEN IN ARRAY A;
RR,RI: <VARIABLE>;
EXIT: THE REAL PART AND THE IMAGINARY PART OF FN(THETA)
ARE DELIVERED IN RR AND RI, RESPECTIVELY.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: N MULTIPLICATIONS; 3N ADDITIONS;
3 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 8
SUBSECTION : COMFOUSER1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMFOUSER1(N,THETA,AR,AI,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"AR,AI;
"CODE" 31096;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL IN EXP(I*THETA);
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY"AR,AI[0:N];
ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE COMPLEX
COEFFICIENTS C[K] (K=0,...,N) IN THE SERIES
FN(THETA)=C[0]+C[1]*EXP(I*THETA)+...+C[N]*EXP(I*
THETA)**N
MUST BE GIVEN IN ARRAY AR AND AI, RESPECTIVELY;
RR,RI: <VARIABLE>;
EXIT: THE REAL PART AND THE IMAGINARY PART OF FN(THETA)
ARE DELIVERED IN RR AND RI, RESPECTIVELY.
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: 4N MULTIPLICATIONS; 4N ADDITIONS;
2 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE : SEE COMFOUSER2 (THIS SECTION).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 9
SUBSECTION : COMFOUSER2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE"COMFOUSER2(N,THETA,AR,AI,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"AR,AI;
"CODE" 31097;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL IN EXP(I*THETA);
THETA: <ARITHMETIC EXPRESSION>;
ENTRY: THE ARGUMENT OF THE FOURIER SERIES;
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY"AR,AI[0:N];
ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE COMPLEX
COEFFICIENTS C[K] (K=0,...,N) IN THE SERIES
FN(THETA)=C[0]+C[1]*EXP(I*THETA)+...+C[N]*EXP(I*
THETA)**N
MUST BE GIVEN IN ARRAY AR AND AI, RESPECTIVELY;
RR,RI: <VARIABLE>;
EXIT: THE REAL PART AND THE IMAGINARY PART OF FN(THETA)
ARE DELIVERED IN RR AND RI, RESPECTIVELY.
PROCEDURES USED: COMFOUSER= CP31095.
RUNNING TIME: PROPORTIONAL TO N
(IN FIRST ORDER: 2N MULTIPLICATIONS; 6N ADDITIONS;
6 COSINE/SINE EVALUATIONS).
LANGUAGE: ALGOL 60.
METHOD AND PERFORANCE:
FOR THE EVALUATION OF A FINITE FOURIER SERIES
(=TRIGONOMETRIC POLYNOMIAL OF DEGREE N SEE POLYA AND SZEGOE, 1971,
P. 76)
FN(THETA)=A[0]+A[1]*COS(THETA)+B[1]*SIN(THETA)+...+
A[N]*COS(N*THETA)+B[N]*SIN(N*THETA),
TWO ALGORITHMS ARE USED:
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 10
1. HORNER SCHEME
LET C[K]=A[K]+I*B[K], K=0,...,N
AND Z=EXP(-I*THETA)
THEN
FN(THETA)=RE(C[0]+C[1]*Z+...+C[N]*Z**N).
THE ALGORITHM IS GIVEN BY:
P:=C[N]
P:=P*Z+C[K], K=N-1,...,0
FN(THETA):=RE(P).
(FOUSER1)
2. A COMBINATION OF THE CLENSHAW ALGORITHM (SEE GENTLEMAN(1969,II)
, VAN DER LAAN, LUKE(1969, P.327-329) OR STOER(1972, P.62,63))
AND THE MODIFICATION OF REINSCH (SEE REINSCH(1967), VAN DER
LAAN, STOER(1972, P.64,65)).
(SINSER,COSSER,FOUSER,FOUSER2)
A MODIFICATION OF THE IDEA OF NEWBERY IS NOT IMPLEMENTED BECAUSE
OF THE INTRODUCTION OF SINE (COSINE) TERMS IN A COSINE (SINE)
SERIES AND THE INEFFICIENCY OF THE ALGORITHM (SEE VAN DER
LAAN OR NEWBERY(1973)).
FOR THE EVALUATION OF A FINITE COMPLEX FOURIER SERIES
FN(THETA)=AR[0]+I*AI[0]+(AR[1]+I*AI[1])*EXP(I*THETA)+...
+(AR[N]+I*AI[N])*EXP(I*THETA)**N,
TWO ALGORITHMS, IN REAL ARITHMETIC, ARE USED:
1. HORNER SCHEME
LET C[K]=AR[K]+I*AI[K], K=0,...,N
AND Z=EXP(I*THETA)
THEN
FN(THETA)=C[0]+C[1]*Z+...+C[N]*Z**N.
THE ALGORITHM IS GIVEN BY
P:=C[N]
P:=P*Z+C[K], K=N-1,N-2,...,0
FN(THETA):=P.
(COMFOUSER1)
2. A COMBINATION OF THE CLENSHAW ALGORITHM AND THE MODIFICATION OF
REINSCH.
LET CAR=AR[0]+AR[1]*COS(THETA)+...+AR[N]*COS(N*THETA),
SAI= AI[1]*SIN(THETA)+...+AI[N]*SIN(N*THETA),
SAR= AR[1]*SIN(THETA)+...+AR[N]*SIN(N*THETA),
CAI=AI[0]+AI[1]*COS(THETA)+...+AI[N]*COS(N*THETA)
THEN FN(THETA)=CAR-SAI+I*(SAR+CAI).
(COMFOUSER,COMFOUSER2)
THE HORNER SCHEME IS IMPLEMENTED BECAUSE OF THE SIMPLICITY OF
THE ALGORITHM (ALTHOUGH THIS ALGORITHM IS LESS EFFICIENT THAN THE
GOERTZEL/WATT/CLENSHAW/REINSCH ALGORITHM) AND THE STABLE NATURE
OF ORTHOGONAL TRANSFORMATIONS.
A COMBINATION OF THE ALGORITHM OF GOERTZEL/WATT/CLENSHAW AND THE
MODIFICATION OF REINSCH IS IMPLEMENTED BECAUSE OF THE EFFICIENCY
OF THE GWC ALGORITHM AND THE STABILITY OF THE MODIFICATION OF
REINSCH, ESPECIALLY FOR SMALL VALUES OF THE ARGUMENT (MOD. PI).
AN UPPER BOUND FOR THE ERROR GROWTH IS GIVEN BY A LINEAR FUNCTION
OF THE DEGREE FOR BOTH (IMPLEMENTED) ALGORITHMS (SEE VAN DER LAAN).
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 11
REFERENCES:
GENTLEMAN,W.M.(1969):
AN ERROR ANALYSIS OF GOERTZEL'S(WATT'S) METHOD FOR COMPUTING
FOURIER COEFFICIENTS.
COMP.J.,VOL.12,P.160-165.
LAAN,C.G.VAN DER(TO APPEAR):
ORTHOGONAL POLYNOMIALS IN NUMERICAL ANALYSIS 1.
ERROR ANALYSIS OF LINEAR TWO-TERM AND THREE-TERM RECURRENCE
RELATIONS.
LUKE,Y.L.(1969):
THE SPECIAL FUNCTIONS AND THEIR APPROXIMATIONS.VOL.1.
ACADEMIC PRESS.
NEWBERY,A.C.R.(1973):
ERROR ANALYSIS FOR FOURIER SERIES EVALUATION.
MATH.COMP.,VOL.26,P.923-924.
POLYA,G. AND G.SZEGOE(1971):
AUFGABEN UND LEHRSAETZE AUS DER ANALYSIS II.
HEIDELBERGER TASCHENBUECHER 74. SPRINGER.
REINSCH,C.(1967):
A NOTE ON TRIGONOMETRIC INTERPOLATION.
BERICHT NR. 6709.
ABTEILUNG MATHEMATIK DER TECHNISCHEN UNIVERSITAET MUENCHEN.
STOER,J.(1972):
EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK 1.
HEIDELBERGER TASCHENBUECHER 105. SPRINGER.
EXAMPLE OF USE:
THE FOURIER SERIES .5+COS(THETA)+SIN(THETA)
IS EVALUATED FOR THE ARGUMENTS 0,PI/2,PI, BY MEANS OF FOUSER
"BEGIN""REAL"THETA,PI;"ARRAY"A[0:1];
PI:=ARCTAN(1)*4;A[0]:=.5;A[1]:=1;
"FOR"THETA:=0,PI/2,PI"DO"
OUTPUT(61,"("/,B-D.DD")",FOUSER(1,THETA,A))
"END"
1.50
1.50
-0.50
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 12
SOURCE TEXTS:
"CODE" 31090;
"REAL""PROCEDURE"SINSER(N,THETA,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"B;
"BEGIN""INTEGER"K;"REAL"C,CC,LAMBDA,H,DUN,UN,UN1;
C:=COS(THETA);
"IF"C<-.5"THEN"
"BEGIN"LAMBDA:= 4*COS(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"DUN:=LAMBDA*UN-DUN+B[K] ;
UN:=DUN-UN;
"END"
"END""ELSE""IF"C> .5"THEN"
"BEGIN"LAMBDA:=-4*SIN(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"DUN:=LAMBDA*UN+DUN+B[K] ;
UN:=DUN+UN;
"END"
"END""ELSE"
"BEGIN"CC:=C+C;UN:=UN1:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"H:=CC*UN-UN1+B[K]; UN1 := UN; UN := H; "END"
"END";
SINSER:=UN*SIN(THETA)
"END"SINSER;
"EOP"
"CODE" 31091;
"REAL""PROCEDURE"COSSER(N,THETA,A);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A;
"BEGIN""INTEGER"K;"REAL"C,CC,LAMBDA,H,DUN,UN,UN1;
C:=COS(THETA);
"IF"C<-.5"THEN"
"BEGIN"LAMBDA:= 4*COS(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN-UN;
DUN:=LAMBDA*UN-DUN+A[K]
"END";COSSER:=DUN-LAMBDA/2*UN
"END""ELSE""IF"C> .5"THEN"
"BEGIN"LAMBDA:=-4*SIN(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN+UN;
DUN:=LAMBDA*UN+DUN+A[K]
"END";COSSER:=DUN-LAMBDA/2*UN
"END""ELSE"
"BEGIN"CC:=C+C;UN:=UN1:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"H:=CC*UN-UN1+A[K];
UN1:=UN;UN:=H
"END";COSSER:=A[0]+UN*C-UN1
"END"
"END"COSSER
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 13
;
"EOP"
"CODE" 31092;
"REAL""PROCEDURE"FOUSER (N,THETA,A);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A;
"BEGIN""INTEGER"K;"REAL"C,CC,LAMBDA,H,DUN,UN,UN1,C2,S2;
C:=COS(THETA);
"IF"C<-.5"THEN"
"BEGIN"C2:=COS(THETA/2);LAMBDA:=4*C2**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN-UN;
DUN:=LAMBDA*UN-DUN+A[K]
"END";FOUSER :=DUN+2*C2*(SIN(THETA/2)-C2)*UN
"END""ELSE""IF"C> .5"THEN"
"BEGIN"S2:=SIN(THETA/2);LAMBDA:=-4*S2*S2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN+UN;
DUN:=LAMBDA*UN+DUN+A[K]
"END";FOUSER :=DUN+2*S2*(S2+COS(THETA/2))*UN
"END""ELSE"
"BEGIN"CC:=C+C;UN:=UN1:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"H:=CC*UN-UN1+A[K];
UN1:=UN;UN:=H
"END";FOUSER :=A[0]-UN1+(C+SIN(THETA))*UN
"END"
"END"FOUSER;
"EOP"
"CODE" 31093;
"REAL""PROCEDURE" FOUSER1(N,THETA,A,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A,B;
"BEGIN""INTEGER"I;"REAL"R,S,H,CO,SI;
R:=S:=0;CO:=COS(THETA);SI:=SIN(THETA);
"FOR"I:=N"STEP"-1"UNTIL"1"DO"
"BEGIN" H:=CO*R+SI*S+A[I];
S:=CO*S-SI*R+B[I];
R:=H
"END";FOUSER1:=CO*R+SI*S+A[0]
"END"FOUSER1;
"EOP"
"CODE" 31094;
"REAL""PROCEDURE"FOUSER2(N,THETA,A,B);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA;"ARRAY"A,B;
"BEGIN"
FOUSER2:=COSSER(N,THETA,A)+SINSER(N,THETA,B);
"END"FOUSER2
1SECTION : 2.2.3.1 (OCTOBER 1974) PAGE 14
;
"EOP"
"CODE" 31095;
"PROCEDURE"COMFOUSER(N,THETA,A,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"A;
"BEGIN""INTEGER"K;"REAL"C,CC,LAMBDA,H,DUN,UN,UN1;
C:=COS(THETA);
"IF"C<-.5"THEN"
"BEGIN"LAMBDA:= 4*COS(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN-UN;
DUN:=LAMBDA*UN-DUN+A[K]
"END";RR :=DUN-LAMBDA/2*UN
"END""ELSE""IF"C> .5"THEN"
"BEGIN"LAMBDA:=-4*SIN(THETA/2)**2;UN:=DUN:=0;
"FOR"K:=N"STEP"-1"UNTIL"0"DO"
"BEGIN"UN:=DUN+UN;
DUN:=LAMBDA*UN+DUN+A[K]
"END";RR :=DUN-LAMBDA/2*UN
"END""ELSE"
"BEGIN"CC:=C+C;UN:=UN1:=0;
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"H:=CC*UN-UN1+A[K];
UN1:=UN;UN:=H
"END";RR :=A[0]+UN*C-UN1
"END";RI:=UN*SIN(THETA)
"END"COMFOUSER;
"EOP"
"CODE" 31096;
"PROCEDURE"COMFOUSER1(N,THETA,AR,AI,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"AR,AI;
"BEGIN""INTEGER"K;"REAL"H,HR,HI,CO,SI;
HR:=HI:=0;CO:=COS(THETA);SI:=SIN(THETA);
"FOR"K:=N"STEP"-1"UNTIL"1"DO"
"BEGIN"H:=CO*HR-SI*HI+AR[K];
HI:=CO*HI+SI*HR+AI[K];
HR:=H
"END";
RR:=CO*HR-SI*HI+AR[0];
RI:=CO*HI+SI*HR+AI[0]
"END"COMFOUSER1;
"EOP"
"CODE" 31097;
"PROCEDURE"COMFOUSER2(N,THETA,AR,AI,RR,RI);
"VALUE"N,THETA;"INTEGER"N;"REAL"THETA,RR,RI;"ARRAY"AR,AI;
"BEGIN""REAL"CAR,CAI,SAR,SAI;
COMFOUSER(N,THETA,AR,CAR,SAR);
COMFOUSER(N,THETA,AI,CAI,SAI);
RR:=CAR-SAI;
RI:=CAI+SAR
"END"COMFOUSER2;
"EOP"
1SECTION: 2.3 (MAY 1974) PAGE 1
AUTHOR : H.FIOLET
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731105.
BRIEF DESCRIPTION:
JFRAC CALCULATES A TERMINATING CONTINUED FRACTION.
KEYWORDS:
CONTINUED FRACTION,
TERMINATING CONTINUED FRACTION.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" JFRAC(N,A,B);
"VALUE" N;"INTEGER" N;"ARRAY" A,B;
"CODE" 35083;
JFRAC DELIVERS THE VALUE OF THE TERMINATING CONTINUED FRACTION:
B[0]+A[1]/(B[1]+A[2]/(B[2]+A[3]/(B[3]+ . . . + A[N]/B[N])))...))
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE UPPER INDEX OF THE ARRAYS A AND B;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N];
"ARRAY" B[0:N];
THE ELEMENTS OF THE CONTINUED FRACTION:
B[0]+A[1]/(B[1]+A[2]/(B[2]+A[3]/(B[3]+ . . . +
+ A[N]/B[N])))...)).
PROCEDURES USED: NONE.
RUNNING TIME: PROPORTIONAL TO N.
1SECTION: 2.3 (MAY 1974) PAGE 2
EXAMPLE OF USE:
"BEGIN"
"REAL" "ARRAY" P[1:10],Q[0:10];
"INTEGER" I;
"FOR" I:=1 "STEP" 1 "UNTIL" 10 "DO"
"BEGIN" P[I]:=1;Q[I]:=2 "END";
Q[0]:=1;
"FOR" I:=7 "STEP" 1 "UNTIL" 10 "DO"
OUTPUT(61,"("N/")",JFRAC(I,P,Q))
"END"
DELIVERS:
+1.4142156862745"+000
+1.4142131979695"+000
+1.4142136248949"+000
+1.4142135516461"+000 .
SOURCE TEXT:
"CODE" 35083;
"REAL" "PROCEDURE" JFRAC(N,A,B);
"VALUE" N;"INTEGER" N;"ARRAY" A,B;
"BEGIN" "REAL" D;"INTEGER" I;
D:=0;
"FOR" I:=N "STEP" -1 "UNTIL" 1 "DO" D:=A[I]/(B[I]+D);
JFRAC:=D+B[0]
"END" JFRAC;
"EOP"
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 1
AUTHOR: C.G. VAN DER LAAN.
CONTRIBUTORS: C.G. VAN DER LAAN, M. VOORINTHOLT.
INSTITUTE: REKENCENTRUM DER RIJKSUNIVERSITEIT GRONINGEN.
RECEIVED: 780601.
BRIEF DESCRIPTION:
WE CONSIDER THE REPRESENTATIONS
N
POWER SUM : SUM A[K]*X**K,
K=0
N
CHEBYSHEV SUM : SUM A[K]*T[K](X),
K=0
N
SHIFTED CHEBYSHEV SUM : SUM A[K]*T'[K](X),
K=0
N K-1
NEWTON SUM : SUM (A[K] * PROD (X-X[J])).
K=0 J=0
THE SHIFTED CHEBYSHEV POLYNOMIAL T'[N] IS
DEFINED BY T'[N](X) = T[N](2*X-1).
THIS SECTION CONTAINS THE TRANSFORMATIONS:
PROCEDURE NAME : TRANSFORMATION
---------------:------------------------------------------------
POLCHS : POWER SUM INTO CHEBYSHEV SUM
CHSPOL : CHEBYSHEV SUM INTO POWER SUM
POLSHTCHS : POWER SUM INTO SHIFTED CHEBYSHEV SUM
SHTCHSPOL : SHIFTED CHEBYSHEV SUM INTO POWER SUM
GRNNEW : POWER SUM INTO NEWTON SUM
NEWGRN : NEWTON SUM INTO POWER SUM
LINTFMPOL : POWER SUM IN X INTO POWER SUM IN Y, X=P*Y+Q
KEYWORDS:
TRANSFORMATION OF POLYNOMIAL REPRESENTATION.
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 2
RUNNING TIME: PROPORTIONAL TO THE SQUARED DEGREE OF THE POLYNOMIAL.
METHOD AND PERFORMANCE:
ALTHOUGH THE TRANSFORMATION OF REPRESENTATIONS OF POLYNOMIALS
COULD HAVE BEEN OBTAINED BY FAST EVALUATION AND FAST INTERPOLATION
WE IMPLEMENTED THE ALGORITHM OF HAMMING (1973, 474,475), BECAUSE
OF ITS SIMPLE APPEARANCE. AN EXPLANATION OF THE HAMMING ALGORITHM
IS GIVEN IN VAN DER LAAN (1977,224-229).
REFERENCES:
HAMMING, R.W. (1973):
NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS.
MCGRAW-HILL.
LAAN, C.G. VAN DER (1977):
APPROXIMATIE VAN FUNCTIES EN DATA.
IN: RIELE, H.J.J. TE (ED.);
COLLOQUIUM NUMERIEKE PROGRAMMATUUR, DEEL 2,
MC SYLLABUS 29.2, MATHEMATISCH CENTRUM AMSTERDAM.
SUBSECTION: POLCHS.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" POLCHS(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 31051;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE POWER SUM;
EXIT: THE COEFFICIENTS OF THE CHEBYSHEV SUM;
PROCEDURES USED: NONE.
EXAMPLE OF USE: SEE NEXT SUBSECTION.
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 3
SUBSECTION: CHSPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" CHSPOL(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 31052;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE CHEBYSHEV SUM;
EXIT: THE COEFFICIENTS OF THE POWER SUM;
PROCEDURES USED: NONE.
EXAMPLE OF USE:
AS AN EXAMPLE WE TRANSFORMED
THE POWER SUM: 1 + 2*X + 3*X**2
INTO
ITS CHEBYSHEV SUM;
AS A CHECK WE TRANSFORMED THE LATTER REPRESENTATION
BACK INTO THE ORIGINAL POWER SUM.
"BEGIN"
"ARRAY" A[0:2];
A[0]:=1; A[1]:=2; A[2]:=3;
OUTPUT(61,"("/,16B,"("A[0]")",4B,"("A[1]")",4B,"("A[2]")"")");
OUTPUT(61,"("/,"(" INPUT")",5B,"(":")",3(2B,+ZD.DD)")",A);
POLCHS(2,A);
OUTPUT(61,"("/,"(" POLCHS")",4B,"(":")",3(2B,+ZD.DD)")",A);
CHSPOL(2,A);
OUTPUT(61,"("/,"(" CHSPOL")",4B,"(":")",3(2B,+ZD.DD)")",A);
"END" PROGRAM
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 4
A[0] A[1] A[2]
INPUT : +1.00 +2.00 +3.00
POLCHS : +2.50 +2.00 +1.50
CHSPOL : +1.00 +2.00 +3.00
SUBSECTION: POLSHTCHS.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" POLSHTCHS(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 31053;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE POWER SUM;
EXIT: THE COEFFICIENTS OF THE SHIFTED CHEBYSHEV SUM;
PROCEDURES USED: LINTFMPOL = CP31250,
POLCHS = CP31051.
EXAMPLE OF USE: SEE NEXT SUBSECTION.
SUBSECTION: SHTCHSPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" SHTCHSPOL(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 31054;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE SHIFTED CHEBYSHEV SUM;
EXIT: THE COEFFICIENTS OF THE POWER SUM.
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 5
PROCEDURES USED: LINTFMPOL = CP31250,
CHSPOL = CP31052.
EXAMPLE OF USE:
AS AN EXAMPLE WE TRANSFORMED
THE POWER SUM: 1 + 2*X + 3*X**2
INTO
ITS SHIFTED CHEBYSHEV SUM;
AS A CHECK WE TRANSFORMED THE LATTER REPRESENTATION
BACK INTO THE ORIGINAL POWER SUM.
"BEGIN"
"ARRAY" A[0:2];
A[0]:=1; A[1]:=2; A[2]:=3;
OUTPUT(61,"("/,16B,"("A[0]")",4B,"("A[1]")",4B,"("A[2]")"")");
OUTPUT(61,"("/,"(" INPUT")",5B,"(":")",3(2B,+ZD.DD)")",A);
POLSHTCHS(2,A);
OUTPUT(61,"("/,"(" POLSHTCHS")",B,"(":")",3(2B,+ZD.DD)")",A);
SHTCHSPOL(2,A);
OUTPUT(61,"("/,"(" SHTCHSPOL")",B,"(":")",3(2B,+ZD.DD)")",A);
"END" PROGRAM
A[0] A[1] A[2]
INPUT : +1.00 +2.00 +3.00
POLSHTCHS : +3.13 +2.50 +0.38
SHTCHSPOL : +1.00 +2.00 +3.00
SUBSECTION: GRNNEW.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" GRNNEW(N,X,A);
"VALUE" N; "INTEGER" N; "ARRAY" X,A;
"CODE" 31055;
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 6
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
X: <ARRAY IDENTIFIER>;
"ARRAY" X[0:N-1];
ENTRY: THE INTERPOLATION POINTS;
A: <ARRAY IDENTIFIER>;
ENTRY: THE COEFFICIENTS OF THE POWER SUM;
EXIT: THE COEFFICIENTS OF THE NEWTON SUM;
PROCEDURES USED: NONE.
EXAMPLE OF USE: SEE NEXT SUBSECTION.
SUBSECTION: NEWGRN.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" NEWGRN(N,X,A);
"VALUE" N; "INTEGER" N; "ARRAY" X,A;
"CODE" 31050;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
X: <ARRAY IDENTIFIER>;
"ARRAY" X[0:N-1];
ENTRY: THE INTERPOLATION POINTS;
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE NEWTON SUM;
EXIT: THE COEFFICIENTS OF THE POWER SUM;
PROCEDURES USED : ELMVEC = CP34020.
EXAMPLE OF USE:
AS AN EXAMPLE WE TRANSFORMED
THE POWER SUM: 1 + 2*X + 3*X**2
INTO
ITS NEWTON SUM WITH
INTERPOLATION POINTS: X[0]:=1.0, X[1]:=2.0;
AS A CHECK WE TRANSFORMED THE LATTER REPRESENTATION
BACK INTO THE ORIGINAL POWER SUM.
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 7
"BEGIN"
"ARRAY" X[0:1], A[0:2];
A[0]:=1; A[1]:=2; A[2]:=3;
X[0]:=1; X[1]:=2;
OUTPUT(61,"("/,16B,"("A[0]")",4B,"("A[1]")",4B,"("A[2]")"")");
OUTPUT(61,"("/,"(" INPUT")",5B,"(":")",3(2B,+ZD.DD)")",A);
GRNNEW(2,X,A);
OUTPUT(61,"("/,"(" GRNNEW")",4B,"(":")",3(2B,+ZD.DD)")",A);
NEWGRN(2,X,A);
OUTPUT(61,"("/,"(" NEWGRN")",4B,"(":")",3(2B,+ZD.DD)")",A);
"END" PROGRAM
A[0] A[1] A[2]
INPUT : +1.00 +2.00 +3.00
GRNNEW : +6.00 +11.00 +3.00
NEWGRN : +1.00 +2.00 +3.00
SUBSECTION: LINTFMPOL.
CALLING SEQUENCE:
THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS:
"PROCEDURE" LINTFMPOL(P,Q,N,A); "VALUE" N,P,Q;
"INTEGER" N; "REAL" P,Q; "ARRAY" A;
"CODE" 31250;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
ENTRY: THE DEGREE OF THE POLYNOMIAL;
P,Q: <ARITHMETIC EXPRESSION>;
ENTRY: DEFINING THE LINEAR TRANSFORMATION OF
THE INDEPENDENT VARIABLE X=P*Y+Q;
(P=0 GIVES THE VALUE OF THE POLYNOMIAL
WITH ARGUMENT Q.)
A: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N];
ENTRY: THE COEFFICIENTS OF THE POWER SUM IN X;
EXIT: THE COEFFICIENTS OF THE POWER SUM IN Y;
PROCEDURES USED: NORDERPOL = CP31242.
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 8
EXAMPLE OF USE:
AS AN EXAMPLE WE TRANSFORMED
THE POWER SUM: 1 + 2*X + 3*X**2
INTO
ITS POWER SUM IN Y WITH
X = 2*Y + 3;
AS A CHECK WE TRANSFORMED THE LATTER REPRESENTATION
BACK INTO THE ORIGINAL POWER SUM.
"BEGIN"
"ARRAY" A[0:2];
A[0]:=1; A[1]:=2; A[2]:=3;
OUTPUT(61,"("/,16B,"("A[0]")",4B,"("A[1]")",4B,"("A[2]")"")");
OUTPUT(61,"("/,"(" INPUT")",5B,"(":")",3(2B,+ZD.DD)")",A);
LINTFMPOL(2,3,2,A);
OUTPUT(61,"("/,"(" LINTFMPOL")",B,"(":")",3(2B,+ZD.DD)")",A);
LINTFMPOL(1/2,-3/2,2,A);
OUTPUT(61,"("/,"(" LINTFMPOL")",B,"(":")",3(2B,+ZD.DD)")",A);
"END" PROGRAM
A[0] A[1] A[2]
INPUT : +1.00 +2.00 +3.00
LINTFMPOL : +34.00 +40.00 +12.00 (POWER SUM IN Y)
LINTFMPOL : +1.00 +2.00 +3.00 (POWER SUM IN X)
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 9
SOURCE TEXTS:
"CODE" 31051;
"PROCEDURE" POLCHS(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"IF" N>1 "THEN"
"BEGIN"
"COMMENT" SCALING;
"INTEGER" K,L,TWOPOW;
TWOPOW:=2;
"FOR" K:=1 "STEP" 1 "UNTIL" N-2 "DO"
"BEGIN"
A[K]:=A[K]/TWOPOW; TWOPOW:=TWOPOW*2;
"END";
A[N-1]:=2*A[N-1]/TWOPOW;
A[N]:=A[N]/TWOPOW;
A[N-2]:=A[N-2]+A[N];
"COMMENT" N<=2 READY;
"FOR" K:=N-2 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN"
A[K-1]:=A[K-1]+A[K+1]; A[K]:=A[K]*2 + A[K+2];
"FOR" L:=K+1 "STEP" 1 "UNTIL" N-2 "DO"
A[L]:=A[L]+A[L+2];
"END";
"END" POLCHS;
"EOP"
"CODE" 31052;
"PROCEDURE" CHSPOL(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"IF" N>1 "THEN"
"BEGIN"
"INTEGER" K,L,TWOPOW;
"FOR" K:=0 "STEP" 1 "UNTIL" N-2 "DO"
"BEGIN"
"FOR" L:=N-2 "STEP" -1 "UNTIL" K "DO"
A[L]:=A[L]-A[L+2]; A[K+1]:=A[K+1]/2;
"END";
TWOPOW:=2;
"FOR" K:=1 "STEP" 1 "UNTIL" N-2 "DO"
"BEGIN"
A[K]:=A[K]*TWOPOW; TWOPOW:=TWOPOW*2;
"END";
A[N-1]:=TWOPOW*A[N-1];
A[N]:=TWOPOW*A[N];
"END" CHSPOL
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 10
;
"EOP"
"CODE" 31053;
"PROCEDURE" POLSHTCHS(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"BEGIN"
LINTFMPOL(.5,.5,N,A);
POLCHS(N,A);
"END" POLSHTCHS;
"EOP"
"CODE" 31054;
"PROCEDURE" SHTCHSPOL(N,A);
"VALUE" N; "INTEGER" N; "ARRAY" A;
"BEGIN"
CHSPOL(N,A);
LINTFMPOL(2,-1,N,A);
"END" SHTCHSPOL;
"EOP"
"CODE" 31055;
"PROCEDURE" GRNNEW(N,X,A);
"VALUE" N; "INTEGER" N; "ARRAY" X,A;
"BEGIN"
"PROCEDURE" ELMCEV(L,U,SHIFT,A,B,X);
"VALUE" L,U,SHIFT,X; "INTEGER" L,U,SHIFT;
"REAL" X; "ARRAY" A,B;
"FOR" L:=L "STEP" -1 "UNTIL" U "DO" A[L]:=A[L]+B[L+SHIFT]*X;
"INTEGER" K;
"FOR" K:=N-1 "STEP" -1 "UNTIL" 0 "DO"
ELMCEV(N-1,N-1-K,1,A,A,X[N-1-K]);
"END" GRNNEW
1SECTION : 2.4.1 (DECEMBER 1978) PAGE 11
;
"EOP"
"CODE" 31050;
"PROCEDURE" NEWGRN(N,X,A);
"VALUE" N; "INTEGER" N; "ARRAY" X,A;
"BEGIN"
"INTEGER" K;
"FOR" K:=N-1 "STEP" -1 "UNTIL" 0 "DO"
ELMVEC(K,N-1,1,A,A,-X[K]);
"END" NEWGRN;
"EOP"
"CODE" 31250;
"PROCEDURE" LINTFMPOL(P,Q,N,A);
"VALUE" N,P,Q; "INTEGER" N; "REAL" P,Q; "ARRAY" A;
"BEGIN"
"INTEGER" K;
"REAL" PPOWER;
NORDERPOL(N,N,Q,A);
PPOWER:=P;
"FOR" K:=1 "STEP" 1 "UNTIL" N "DO"
"BEGIN"
A[K]:=PPOWER*A[K];
PPOWER:=P*PPOWER;
"END";
"END" LINTFMPOL;
"EOP"
1SECTION : 2.4.3 (OCTOBER 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
INSTITUTE : RIJKSUNIVERSITEIT GRONINGEN.
RECEIVED : 740131.
BRIEF DESCRIPTION :
INTCHS COMPUTES THE INDEFINITE INTEGRAL OF A GIVEN CHEBYSHEV
SERIES.
KEYWORDS :
INDEFINITE INTEGRATION,
CHEBYSHEV SERIES.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE"INTCHS(N,A,B);
"VALUE"N;"INTEGER"N;"ARRAY"A,B;
"CODE" 31248;
THE MEANING OF THE FORMAL PARAMETERS IS :
N : <ARITHMETIC EXPRESSION>;
ENTRY:
THE DEGREE OF THE POLYNOMIAL REPRESENTED BY THE CHEBYSHEV
SERIES;
A,B: <ARRAY IDENTIFIER>;
"ARRAY" A[0:N],B[1:N+1];
ENTRY:
THE COEFFICIENTS OF THE CHEBYSHEV SERIES,A[0]+A[1]*T1(X)+...+
+A[N]*TN(X),SHOULD BE GIVEN IN ARRAY A.
EXIT:
THE COEFFICIENTS OF THE INTEGRAL CHEBYSHEV SERIES,
B[1]*T1(X)+...+B[N+1]*TN+1(X), ARE DELIVERED IN ARRAY B.
(T1(X),...TN+1(X) DENOTE CHEBYSHEV POLYNOMIALS OF THE FIRST
KIND,OF DEGREE 1,...N+1,RESPECTIVELY).
1SECTION : 2.4.3 (OCTOBER 1974) PAGE 2
METHOD AND PERFORMANCE :
FOR A DESCRIPTION OF THE ALGORITHM SEE AMONG OTHERS :
CLENSHAW,1962,P.11,OR FOX AND PARKER,1968,P.59.
REFERENCES :
BROUCKE,R.(1973):
TEN SUBROUTINES FOR THE MANIPULATION OF CHEBYSHEV SERIES.
ALGORITHM 446.(FORTRAN).
COMM.ACM,VOL.16,1,P.254-256.
CLENSHAW,C.W.(1962):
CHEBYSHEV SERIES FOR MATHEMATICAL FUNCTIONS.
MATH.TAB.NAT.PHYS.LAB. 5,LONDON.
H.M. STATIONARY OFFICE.
FOX,L.&I.B.PARKER(1968):
CHEBYSHEV POLYNOMIALS IN NUMERICAL ANALYSIS.
OXFORD UNIVERSITY PRESS.
EXAMPLE OF USE :
AS A FORMAL TEST OF THE PROCEDURE INTCHS THE CHEBYSHEV SERIES :
1+1/2*T1(X)+1/5*T2(X)+1/10*T3(X) IS TRANSFORMED INTO ITS INTEGRAL.
"BEGIN""ARRAY"A[0:3],B[1:4];
A[0]:=1;A[1]:=.5;A[2]:=.2;A[3]:="-1;
INTCHS(3,A,B);
OUTPUT(61,"("/,4(BZ.4D)")",B[1],B[2],B[3],B[4]);
"END"
.9000 .1000 .0333 .0125
1SECTION : 2.4.3 (OCTOBER 1974) PAGE 3
SOURCE TEXT(S):
"CODE"31248;
"PROCEDURE"INTCHS(N,A,B);
"VALUE"N;"INTEGER"N;"ARRAY"A,B;
"COMMENT"
INTCHS DELIVERS THE COEFFICIENTS B[I],I=1,...N+1, OF THE INTEGRAL
CHEBYSHEV SERIES B[1]*T1(X)+...+B[N]*TN(X)+B[N+1]*TN+1(X).
THESE COEFFICIENTS ARE OBTAINED BY MEANS OF INDEFINITE INTEGRATION
OF THE CHEBYSHEV SERIES A[0]+A[1]*T1(X)+...+A[N]*TN(X).
T1(X),...TN+1(X) DENOTE CHEBYSHEV POLYNOMIALS OF THE FIRST KIND,
OF DEGREE 1,...N+1,RESPECTIVELY;
"IF"N=0"THEN"B[1]:=A[0]
"ELSE""IF"N=1"THEN""BEGIN"B[2]:=A[1]/4;B[1]:=A[0]"END"
"ELSE""BEGIN""INTEGER"I;"REAL"H,L,DUM;
H:=A[N];DUM:=A[N-1];B[N+1]:=H/((N+1)*2);B[N]:=DUM/(N*2);
"FOR"I:=N-1"STEP"-1"UNTIL"2"DO"
"BEGIN"L:=A[I-1];B[I]:=(L-H)/(2*I);H:=DUM;DUM:=L
"END";B[1]:=A[0]-H/2
"END"INTCHS;
"EOP"
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 1
AUTHORS: J. C. P. BUS AND T. J. DEKKER.
CONTRIBUTOR: J.C.P. BUS AND P. A. BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730830.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS SIX PROCEDURES:
A: DEC PERFORMS A TRIANGULAR DECOMPOSITION WITH PARTIAL PIVOTING;
B: GSSELM PERFORMS A TRIANGULAR DECOMPOSITION WITH A COMBINATION OF
PARTIAL AND COMPLETE PIVOTING;
C: ONENRMINV DELIVERS THE 1-NORM OF THE INVERSE OF A MATRIX WHOSE
TRIANGULARLY DECOMPOSED FORM HAS BEEN DELIVERED BY DEC OR GSSELM;
D:ERBELM CALCULATES A ROUGH UPPERBOUND FOR THE SOLUTION OF A LINEAR
SYSTEM WHOSE MATRIX IS TRIANGULARLY DECOMPOSED BY GSSELM;
E: GSSERB PERFORMS A TRIANGULAR DECOMPOSTION OF THE MATRIX OF A
LINEAR SYSTEM AND CALCULATES AN UPPERBOUND FOR THE RELATIVE ERROR
OF THE SOLUTION OF THAT SYSTEM;
F: GSSNRI PERFORMS A TRIANGULAR DECOMPOSITION AND CALCULATES THE
1-NORM OF THE INVERSE MATRIX;
THE METHOD USED IN DEC AND GSSELM YIELDS A LOWER-TRIANGULAR MATRIX
L AND A UNIT UPPER-TRIANGULAR MATRIX U SUCH THAT THE PRODUCT
LU EQUALS THE GIVEN MATRIX WITH PERMUTED ROWS (DEC) OR ROWS AND
COLUMNS (GSSELM); IN DEC, ONLY PARTIAL PIVOTING IS USED ([3],
[4, P.115], [5, P.201]); THE PIVOTING STRATEGY IN GSSELM IS
A COMBINATION OF PARTIAL AND COMPLETE PIVOTING ([2], [1]); IN THIS
STRATEGY THE PROCESS WILL SWITCH TO COMPLETE PIVOTING IF PARTIAL
PIVOTING MIGHT NOT YIELD STABLE RESULTS; SO IN GSSELM THE
EFFICIENCY OF PARTIAL PIVOTING IS COMBINED WITH THE STABILITY OF
COMPLETE PIVOTING;
SINCE, IN EXCEPTIONAL CASES, PARTIAL PIVOTING MAY YIELD USELESS
RESULTS, EVEN FOR WELL-CONDITIONED MATRICES, THE USER IS ADVISED TO
USE GSSELM; HOWEVER, IF THE NUMBER OF VARIABLES IS SMALL RELATIVE
TO THE NUMBER OF BINARY DIGITS IN THE MANTISSA (48 FOR THE CYBER ),
THEN DEC MAY ALSO BE USED;
KEYWORDS:
LU DECOMPOSITION,
TRIANGULAR DECOMPOSITION,
GAUSSIAN ELIMINATION.
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 2
SUBSECTION: DEC.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DEC(A, N, AUX, P); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX; "INTEGER" "ARRAY" P;"CODE" 34300;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE MATRIX;
EXIT:THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:3];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION; SEE
METHOD AND PERFORMANCE;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS BROKEN OFF BECAUSE
THE SELECTED PIVOT IS TOO SMALL RELATIVE TO
THE MAXIMUM OF THE EUCLIDEAN NORMS OF THE ROWS OF
THE GIVEN MATRIX;
P: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" P[1:N];
EXIT:THE PIVOTAL INDICES.
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 3
PROCEDURES USED:
MATMAT = CP34013,
MATTAM = CP34015,
ICHROW = CP34032.
REQUIRED CENTRAL MEMORY:
A REAL ARRAY OF ORDER N IS DECLARED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
THE METHOD USED IN DEC IS TRIANGULAR DECOMPOSITION WITH STABILIZING
ROW INTERCHANGES, ALSO CALLED "PARTIAL PIVOTING"; SEE ALSO [3,P.19]
AND [5,P.201].
EXAMPLE OF USE: SEE DECSOL (SECTION 3.1.1.1.1.1.3).
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 4
SUBSECTION: GSSELM .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSELM(A, N, AUX, RI, CI); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX; "INTEGER" "ARRAY" RI, CI;"CODE" 34231;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT:THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:7];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION; SEE
METHOD AND PERFORMANCE;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING;
USUALLY, AUX[4] = 8 WILL GIVE GOOD RESULTS; SEE
METHOD AND PERFORMANCE;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS HAS BEEN BROKEN OFF,
BECAUSE THE SELECTED PIVOT IS TOO SMALL RELATIVE TO
THE MAXIMUM OF THE MODULI OF ELEMENTS OF THE GIVEN
MATRIX;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX WHICH HAD BEEN GIVEN
IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (I. E. THE MODULUS OF
AN ELEMENT WHICH IS OF MAXIMUM ABSOLUTE VALUE FOR
THE MATRICES OCCURRING DURING ELIMINATION);
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
EXIT: THE PIVOTAL ROW-INDICES;
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
EXIT: THE PIVOTAL COLUMN-INDICES;
1SECTION: 3.1.1.1.1.1.1 (DECEMBER 1979) PAGE 5
PROCEDURES USED:
ROWCST = CP31132,
ELMROW = CP34024,
MAXELMROW = CP34025,
ICHCOL = CP34031,
ICHROW = CP34032,
ABSMAXMAT = CP31069.
REQUIRED CENTRAL MEMORY: NO EXTRA ARRAYS ARE DECLARED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
THE PROCESS OF GAUSSIAN ELIMINATION IS PERFORMED IN AT MOST N STEPS
, WHERE N DENOTES THE ORDER OF THE MATRIX; PARTIAL PIVOTING WILL BE
USED AS LONG AS THE CALCULATED UPPER BOUND FOR THE GROWTH ([2],
[1]), IS LESS THAN A CRITICAL VALUE THAT EQUALS AUX[4] * N TIMES
THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM ABSOLUTE VALUE FOR
THE GIVEN MATRIX;
IN THE PARTIAL PIVOTING STRATEGY, THAT ELEMENT IS CHOSEN AS
PIVOT IN THE K-TH STEP, WHOSE ABSOLUTE VALUE IS MAXIMAL
FOR THE K-TH COLUMN OF THE LOWER-TRIANGULAR MATRIX L; HOWEVER, IF
THE UPPER BOUND FOR THE GROWTH EXCEEDS THIS CRITICAL VALUE IN THE
K-TH STEP, THEN A PIVOT IS SELECTED IN THE J-TH STEP (J = K,..., N)
, IN SUCH A WAY, THAT ITS ABSOLUTE VALUE IS MAXIMAL FOR THE
REMAINING SUBMATRIX OF ORDER N - K + 1 (COMPLETE PIVOTING);
SINCE IN PRACTICE, IF WE CHOOSE AUX[4] PROPERLY, THE UPPER BOUND
FOR THE GROWTH RARELY EXCEEDS THIS CRITICAL VALUE ([2], [4]), WE
WILL USUALLY TAKE ADVANTAGE OF THE GREATER SPEED OF PARTIAL
PIVOTING (ORDER N - K + 1 IN THE K-TH STEP), WHILE IN A FEW
DOUBTFUL CASES NUMERICAL DIFFICULTIES WILL BE RECOGNIZED AND THE
PROCESS WILL SWITCH TO COMPLETE PIVOTING (ORDER
(N - K + 1) ** 2 IN THE K-TH STEP);
USING GSSELM,THE UPPER BOUND FOR THE RELATIVE ERROR IN THE SOLUTION
OF A LINEAR SYSTEM ([4], [5]), WILL BE AT MOST AUX[4] * N TIMES THE
UPPER BOUND USING GAUSSIAN ELIMINATION WITH COMPLETE PIVOTING ONLY;
USUALLY, HOWEVER, THIS WILL BE A CRUDE OVERESTIMATE;
THE CHOICE AUX[4] < 1 / N WILL RESULT IN COMPLETE PIVOTING ONLY
, WHILE PARTIAL PIVOTING WILL BE USED IN EVERY STEP IF WE CHOOSE
AUX[4] > (2 ** (N - 1)) / N; USUALLY, AUX[4] = 8 WILL GIVE
GOOD RESULTS ([2], [1]);
THE PROCESS WILL ALSO SWITCH TO COMPLETE PIVOTING IF THE MODULUS OF
THE PIVOT OBTAINED WITH PARTIAL PIVOTING IS LESS THAN A CERTAIN
TOLERANCE, WHICH EQUALS THE GIVEN RELATIVE TOLERANCE AUX[2] TIMES
THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM ABSOLUTE VALUE FOR
THE GIVEN MATRIX; IF ALL ELEMENTS IN THE REMAINING SUBMATRIX ARE
SMALLER IN ABSOLUTE VALUE THAN THIS TOLERANCE THEN THE PROCESS IS
BROKEN OFF AND THE PREVIOUS STEPNUMBER IS DELIVERED IN AUX[3];
IN CONTRAST WITH THE METHOD USED IN DEC (THIS SECTION), NO
EQUILIBRATING IS DONE IN THIS PIVOTING STRATEGY; THE USER HIMSELF
HAS TO TAKE CARE FOR A REASONABLE SCALING OF THE MATRIX ELEMENTS.
EXAMPLE OF USE: SEE GSSSOL (SECTION 3.1.1.1.1.1.3).
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 6
SUBSECTION: ONENRMINV.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS
"REAL" "PROCEDURE" ONENRMINV(A, N); "VALUE" N;
"INTEGER" N; "ARRAY" A; "CODE" 34240;
ONENRMINV:= THE 1-NORM OF THE CALCULATED INVERSE OF THE MATRIX,
WHOSE TRIANGULARLY DECOMPOSED FORM IS GIVEN IN ARRAY A;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:
THE TRIANGULARLY DECOMPOSED FORM OF A MATRIX, AS DELIVERED
BY GSSELM OR DEC (THIS SECTION);
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX, WHOSE TRIANGULARLY DECOMPOSED
FORM HAS BEEN GIVEN IN ARRAY A.
PROCEDURES USED:
MATVEC = CP34011.
REQUIRED CENTRAL MEMORY:
ONE REAL ARRAY OF ORDER N IS DECLARED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
THE INVERSE OF THE MATRIX WHOSE TRIANGULARLY DECOMPOSED FORM, AS
DELIVERED BY GSSELM OR DEC,HAS BEEN GIVEN IN ARRAY A, IS CALCULATED
WITH FORWARD AND BACK SUBSTITUTION ([3],[4],[5]);ONLY THE 1-NORM OF
THIS INVERSE IS DELIVERED BY ONENRMINV; THE ELEMENTS OF ARRAY A
REMAIN UNALTERED.
EXAMPLE OF USE: SEE GSSSOLERB (SECTION 3.1.1.1.1.1.3).
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 7
SUBSECTION: ERBELM.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" ERBELM(N, AUX, NRMINV); "VALUE" N, NRMINV;
"INTEGER" N; "REAL" NRMINV; "ARRAY" AUX; "CODE" 34241;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE LINEAR SYSTEM IN CONSIDERATION;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:11];
ENTRY:
AUX[0]: THE MACHINE PRECISION;
AUX[5]: THE MODULUS OF AN ELEMENT, WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX OF THE LINEAR SYSTEM;
THIS VALUE IS DELIVERED BY GSSELM IN AUX[5] (THIS
SECTION);
AUX[6]: AN UPPER BOUND FOR THE RELATIVE ERROR IN THE
ELEMENTS OF THE MATRIX OF THE LINEAR SYSTEM;
AUX[7]: AN UPPER BOUND FOR THE GROWTH DURING GAUSSIAN
ELIMINATION; THIS VALUE IS DELIVERED IN AUX[7] BY
GSSELM (THIS SECTION);
EXIT:
AUX[9]: THE VALUE OF NRMINV;
AUX[11]: A ROUGH UPPER BOUND FOR THE RELATIVE ERROR IN THE
SOLUTION OF A LINEAR SYSTEM WHEN GAUSSIAN
ELIMINATION IS USED FOR THE CALCULATION OF THIS
SOLUTION; IF NO USE CAN BE MADE OF THE FORMULA FOR
THE ERROR BOUND (SEE: METHOD AND PERFORMANCE),
BECAUSE OF A VERY BAD CONDITION OF THE MATRIX, THEN
AUX[11]:= -1;
NRMINV: <ARITHMETIC EXPRESSION>;
THE 1-NORM OF THE INVERSE OF THE MATRIX OF THE LINEAR
SYSTEM MUST BE GIVEN IN NRMINV; THIS VALUE MAY BE OBTAINED
BY ONENRMINV (THIS SECTION).
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 8
PROCEDURES USED: NONE.
REQUIRED CENTRAL MEMORY: NO EXTRA ARAYS ARE DECLARED.
METHOD AND PERFORMANCE:
WHEN CALLED AFTER GSSELM, ERBELM WILL CALCULATE A ROUGH UPPER BOUND
FOR THE RELATIVE ERROR IN THE SOLUTION OF THE LINEAR SYSTEM, WHOSE
MATRIX HAS BEEN DECOMPOSED INTO TRIANGULAR FORM BY GSSELM (THIS
SECTION), BY ([3], [4], [5]):
NORM(DX) / NORM(X) <= P / (1 - P),
WHERE : P = Q * NORM(C) / (1 - Q * NORM(C)),
Q = G * (.75 * N ** 3 + 4.5 * N ** 2) * EPS + EPSA,
C IS THE CALCULATED INVERSE OF THE MATRIX,
G THE UPPER BOUND FOR THE GROWTH DURING GAUSSIAN
ELIMINATION, AS DELIVERED BY GSSELM (THIS SECTION),
N THE ORDER OF THE MATRIX,
EPSA AN UPPER BOUND FOR THE RELATIVE ERROR IN THE MATRIX
ELEMENTS,
EPS THE MACHINE PRECISION AND
NORM(.) DENOTES THE 1-NORM.
THIS PROCEDURE IS USED IN E.G. GSSERB (THIS SECTION) AND GSSINVERB
(SECTION 3.1.1.1.1.1.4)
EXAMPLE OF USE: SEE GSSSOLERB (SECTION 3.1.1.1.1.1.3).
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 9
SUBSECTION: GSSERB .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSERB(A, N, AUX, RI, CI); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX; "INTEGER""ARRAY" RI, CI; "CODE" 34242;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE MATRIX TO BE DECOMPOSED;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT UPPER
TRIANGULAR MATRIX,WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:11];
ENTRY:(SEE ALSO GSSELM IN THIS SECTION);
AUX[0]: THE MACHINE PRECISION;
AUX[2]: A RELATIVE TOLERANCE;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING;
AUX[6]: AN UPPER BOUND FOR THE RELATIVE PRECISION OF THE
MATRIX ELEMENTS;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED,
AUX[1] EQUALS 1 IF THE DETERMINANT OF THE PRINCIPAL
SUBMATRIX OF ORDER R IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: THE MODULUS OF AN ELEMENT, WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX WHICH HAS BEEN GIVEN
IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH;
AUX[9]: IF AUX[3] = N, THEN AUX[9] WILL EQUAL THE 1-NORM OF
THE INVERSE MATRIX, ELSE AUX[9] WILL BE UNDEFINED;
AUX[11]: IF AUX[3] = N, THEN THE VALUE OF AUX[11] WILL BE A
ROUGH UPPER BOUND FOR THE RELATIVE ERROR IN THE
SOLUTION OF LINEAR SYSTEMS WITH A MATRIX AS GIVEN
IN ARRAY A, ELSE AUX[11] WILL BE UNDEFINED; IF NO
USE CAN BE MADE OF THE FORMULA FOR THE ERROR BOUND
AS GIVEN ABOVE (SUBSECTION ERBELM), BECAUSE OF A
VERY BAD CONDITION OF THE MATRIX, THEN AUX[11]:=-1;
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
EXIT: THE PIVOTAL ROW-INDICES.
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
EXIT: THE PIVOTAL COLUMN-INDICES.
PROCEDURES USED:
GSSELM = CP34231,
ONENRMINV = CP34240,
ERBELM = CP34241.
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 10
REQUIRED CENTRAL MEMORY: NO EXTRA ARRAYS ARE DECLARED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
GSSERB USES GSSELM (THIS SECTION) TO PERFORM THE TRIANGULAR
DECOMPOSITION OF THE MATRIX GIVEN IN ARRAY A AND ERBELM AND
ONENRMINV (THIS SECTION) TO CALCULATE AN UPPER BOUND FOR THE
RELATIVE ERROR IN THE SOLUTION OF LINEAR SYSTEMS WITH A MATRIX AS
GIVEN IN ARRAY A; IF AUX[3] < N, THEN THE EFFECT OF GSSERB IS
MERELY THAT OF GSSELM.
EXAMPLE OF USE: SEE GSSSOLERB (SECTION 3.1.1.1.1.1.3).
SUBSECTION: GSSNRI .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSNRI(A, N, AUX, RI, CI); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX; "INTEGER""ARRAY" RI, CI; "CODE" 34252;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE MATRIX TO BE DECOMPOSED;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT UPPER
TRIANGULAR MATRIX,WITH ITS DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:9];
ENTRY:(SEE ALSO GSSELM IN THIS SECTION);
AUX[2]: A RELATIVE TOLERANCE;
AUX[4]: A VALUE USED FOR CONTROLLING PIVOTING;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED,
THEN AUX[1] EQUALS 1 IF THE DETERMINANT OF THE
PRINCIPAL SUBMATRIX OF ORDER R IS POSITIVE, ELSE
AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: THE MODULUS OF AN ELEMENT, WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX WHICH HAD BEEN GIVEN
IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH;
AUX[9]: IF AUX[3] = N, THEN AUX[9] WIL EQUAL THE 1-NORM OF
THE INVERSE MATRIX, ELSE AUX[9] WILL BE UNDEFINED;
1SECTION: 3.1.1.1.1.1.1 (FEBRUARY 1979) PAGE 11
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
EXIT: THE PIVOTAL ROW INDICES.
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
EXIT:THE PIVOTAL COLUMN INDICES.
PROCEDURES USED:
GSSELM = CP34231,
ONENRMINV = CP34240.
REQUIRED CENTRAL MEMORY: NO EXTRA ARRAYS ARE DECLARED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
GSSNRI USES GSSELM (THIS SECTION) TO PERFORM THE TRIANGULAR
DECOMPOSITION OF THE MATRIX GIVEN IN ARRAY A AND ONENRMINV (THIS
SECTION) TO CALCULATE THE 1-NORM OF THE INVERSE MATRIX; IF
AUX[3] < N, THEN THE EFFECT OF GSSNRI IS MERELY THAT OF GSSELM
(THIS SECTION).
EXAMPLE OF USE: SEE GSSITISOLERB (SECTION 3.1.1.1.1.1.5).
REFERENCES:
[1] BUS, J. C. P.
LINEAR SYSTEMS WITH CALCULATION OF ERROR BOUNDS AND ITERATIVE
REFINEMENT (DUTCH).
MATHEMATICAL CENTRE, AMSTERDAM, LR 3. 4. 19 (1972).
[2] BUSINGER, P. A.
MONITORING THE NUMERICAL STABILITY OF GAUSSIAN ELIMINATION.
NUMER. MATH. 16, 360-361 (1971).
[3] DEKKER, T. J.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MATHEMATICAL CENTRE, AMSTERDAM, TRACT 22 (1968).
[4] WILKINSON, J. H.
ROUNDING ERRORS IN ALGEBRAIC PROCESSES.
LONDON (1963).
[5] WILKINSON, J. H.
THE ALGEBRAIC EIGENVALUE PROBLEM.
OXFORD (1965).
1SECTION: 3.1.1.1.1.1.1 (MAY 1974) PAGE 12
SOURCE TEXT(S):
0"CODE" 34300;
"PROCEDURE" DEC(A, N, AUX, P); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX; "INTEGER" "ARRAY" P;
"BEGIN" "INTEGER" I, K, K1, PK, D;
"REAL" R, S, EPS;
"ARRAY" V[1:N];
R:= -1;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:= SQRT(MATTAM(1,N,I,I,A,A));
"IF" S > R "THEN" R:= S; V[I]:= 1/S
"END";
EPS:= AUX[2] * R; D:= 1;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= - 1; K1:= K - 1;
"FOR" I:= K "STEP" 1 "UNTIL" N "DO"
"BEGIN" A[I,K]:= A[I,K] - MATMAT(1, K1, I, K, A, A);
S:= ABS(A[I,K]) * V[I]; "IF" S > R "THEN"
"BEGIN" R:= S; PK:= I "END"
"END" LOWER;
P[K]:= PK; V[PK]:= V[K]; S:= A[PK,K];
"IF" ABS(S) < EPS "THEN" "GOTO" END;
"IF" S < 0 "THEN" D:= -D; "IF" PK ^= K "THEN"
"BEGIN" D:= - D; ICHROW(1, N, K, PK, A) "END";
"FOR" I:= K + 1 "STEP" 1 "UNTIL" N "DO"
A[K,I]:= (A[K,I] - MATMAT(1, K1, K, I, A, A)) / S
"END" LU;
K:= N + 1;
END: AUX[1]:= D; AUX[3]:= K - 1
"END" DEC;
"EOP"
"CODE" 34231;
"PROCEDURE" GSSELM(A, N, AUX, RI, CI); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"INTEGER" "ARRAY" RI, CI;
"BEGIN" "INTEGER" I, J, P, Q, R, R1, JPIV, RANK, SIGNDET;
"REAL" CRIT, PIVOT, RGROW, MAX, AID, MAX1, EPS;
"BOOLEAN" PARTIAL;
"COMMENT"
1SECTION: 3.1.1.1.1.1.1 (DECEMBER 1979) PAGE 13
;
AUX[5]:= RGROW:= ABSMAXMAT(1, N, 1, N, I, J, A);
CRIT:= N * RGROW * AUX[4]; EPS:= RGROW * AUX[2]; MAX:= 0;
RANK:= N; SIGNDET:= 1; PARTIAL:= RGROW ^= 0;
"FOR" Q:= 1 "STEP" 1 "UNTIL" N "DO" "IF" Q ^= J "THEN"
"BEGIN" AID:= ABS(A[I,Q]);
"IF" AID > MAX "THEN" MAX:= AID
"END";
RGROW:= RGROW + MAX;
"FOR" R:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R1:= R + 1; "IF" I ^= R "THEN"
"BEGIN" SIGNDET:= - SIGNDET; ICHROW(1, N, R, I, A) "END";
"IF" J ^= R "THEN"
"BEGIN" SIGNDET:= - SIGNDET; ICHCOL(1, N, R, J, A) "END";
RI[R]:= I; CI[R]:= J; PIVOT:= A[R,R];
"IF" PIVOT < 0 "THEN" SIGNDET:= - SIGNDET;
"IF" PARTIAL "THEN"
"BEGIN" MAX:= MAX1:= 0; J:= R1;
ROWCST(R1, N, R, A, 1 / PIVOT);
"FOR" P:= R1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" ELMROW(R1, N, P, R, A, A, - A[P,R]);
AID:= ABS(A[P,R1]); "IF" MAX < AID "THEN"
"BEGIN" MAX:= AID; I:= P "END";
"END";
"FOR" Q:= R1 + 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" AID:= ABS(A[I,Q]);
"IF" MAX1 < AID "THEN" MAX1:= AID
"END";
AID:= RGROW; RGROW:= RGROW + MAX1;
"IF" RGROW > CRIT "OR" MAX < EPS "THEN"
"BEGIN" PARTIAL:= "FALSE"; RGROW:= AID;
MAX:= ABSMAXMAT(R1, N, R1, N, I, J, A)
"END"
"END" PARTIAL PIVOTINGSTEP
"ELSE"
"BEGIN" "IF" MAX <= EPS "THEN"
"BEGIN" RANK:= R - 1;
"IF" PIVOT < 0 "THEN" SIGNDET:= - SIGNDET;"GOTO"OUT
"END";
MAX:= - 1;
ROWCST(R1, N, R, A, 1 / PIVOT);
"FOR" P:= R1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" JPIV:= MAXELMROW(R1, N, P, R, A, A, - A[P,R]);
AID:= ABS(A[P,JPIV]); "IF" MAX < AID "THEN"
"BEGIN" MAX:= AID; I:= P; J:= JPIV "END"
"END";
"IF" RGROW < MAX "THEN" RGROW:= MAX
"END" COMPLETE PIVOTINGSTEP
"END" ELIMINATIONSTEP;
OUT: AUX[1]:= SIGNDET; AUX[3]:= RANK; AUX[7]:= RGROW
"END" GSSELM
1SECTION: 3.1.1.1.1.1.1 (MAY 1974) PAGE 14
;
"EOP"
"CODE" 34240;
"REAL" "PROCEDURE" ONENRMINV(A, N); "VALUE" N; "INTEGER" N;
"ARRAY" A;
"BEGIN" "INTEGER" I, J;
"REAL" NORM, MAX, AID;
"ARRAY" Y[1:N];
NORM:= 0;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" Y[I]:= "IF" I < J
"THEN" 0 "ELSE" "IF" I = J "THEN" 1 / A[I,I] "ELSE"
- MATVEC(J, I - 1, I, A, Y) / A[I,I];
MAX:= 0;
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" AID:= Y[I]:= Y[I] - MATVEC(I + 1, N, I, A, Y);
MAX:= MAX + ABS(AID)
"END";
"IF" NORM < MAX "THEN" NORM:= MAX
"END";
ONENRMINV:= NORM
"END" ONENRMINV;
"EOP"
"CODE" 34241;
"PROCEDURE" ERBELM(N, AUX, NRMINV); "VALUE" N, NRMINV;
"INTEGER" N; "REAL" NRMINV;
"ARRAY" AUX;
"BEGIN" "REAL" AID, EPS;
EPS:= AUX[0]; AID:= (1.06 * EPS * (.75 * N + 4.5) * N ** 2
* AUX[7] + AUX[5] * AUX[6]) * NRMINV;
AUX[11]:= "IF" 2 * AID >= (1 - EPS) "THEN" - 1 "ELSE"
AID / (1 - 2 * AID); AUX[9]:= NRMINV
"END" ERBELM;
"EOP"
"CODE" 34242;
"PROCEDURE" GSSERB(A, N, AUX, RI, CI); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"INTEGER" "ARRAY" RI, CI;
"BEGIN"
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" ERBELM(N, AUX, ONENRMINV(A, N))
"END" GSSERB;
"EOP"
"CODE" 34252;
"PROCEDURE" GSSNRI(A, N, AUX, RI, CI); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"INTEGER" "ARRAY" RI, CI;
"BEGIN"
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" AUX[9]:= ONENRMINV(A, N)
"END" GSSNRI;
"EOP"
1SECTION: 3.1.1.1.1.1.2 (MAY 1974) PAGE 1
AUTHOR: J. C. P. BUS.
CONTRIBUTOR: J.C.P. BUS AND P. A. BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730831.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS A PROCEDURE FOR CALCULATING THE
DETERMINANT OF A TRIANGULARLY DECOMPOSED MATRIX;
KEYWORDS:
DETERMINANT.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"REAL" "PROCEDURE" DETERM(A, N, SIGN); "VALUE" N, SIGN;
"INTEGER" N, SIGN; "ARRAY" A;
"CODE" 34303;
DETERM: DELIVERS THE CALCULATED VALUE OF THE DETERMINANT OF THE
MATRIX;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE DIAGONAL ELEMENTS OF THE
LOWER-TRIANGULAR MATRIX L, OBTAINED BY TRIANGULAR
DECOMPOSITION OF THE MATRIX, HAVE TO BE GIVEN IN
A[I,I], I= 1, ..., N;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX, WHOSE DETERMINANT HAS TO BE
CALCULATED;
SIGN: <ARITHMETIC EXPRESSION>;
ENTRY: IF THE DETERMINANT OF THE MATRIX IS POSITIVE THEN
THE VALUE OF SIGN SHOULD BE +1, ELSE -1; THIS VALUE
IS DELIVERED BY GSSELM OR DEC IN AUX[1], (SECTION
3.1.1.1.1.1.1).
PROCEDURES USED: NONE.
1SECTION: 3.1.1.1.1.1.2 (MAY 1974) PAGE 2
RUNNING TIME: PROPORTIONAL TO N.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
A LOWER-TRIANGULAR MATRIX L HAS TO BE
GIVEN, SUCH THAT FOR SOME UNIT UPPER-TRIANGULAR MATRIX U THE
PRODUCT LU EQUALS THE MATRIX (WITH PERMUTED ROWS AND COLUMNS);
THE SIGN OF THE DETERMINANT ALSO HAS TO BE GIVEN; THESE DATA ARE
DELIVERED IN THE MATRIX AND AUX[1] BY THE PROCEDURES GSSELM OR DEC
(SECTION 3.1.1.1.1.1.1) AND THE PROCEDURES GSSERB, GSSNRI (SECTION
3.1.1.1.1.1.1), DECSOL, GSSSOL, GSSSOLERB (SECTION 3.1.1.1.1.1.3),
GSSITISOL AND GSSITISOLERB (SECTION 3.1.1.1.1.1.5), WHICH MAKE USE
OF GSSELM OR DEC.
THE CALCULATION OF THE DETERMINANT IS DONE STRAIGHT ON BY
CALCULATING THE PRODUCT OF THE DIAGONAL ELEMENTS OF THE
LOWER-TRIANGULAR MATRIX GIVEN IN ARRAY A; THE USER IS WARNED, THAT
OVERFLOW MAY OCCUR IF THE ORDER OF THE MATRIX IS LARGE.
EXAMPLE OF USE:
THE DETERMINANT OF THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX
MAY BE OBTAINED BY THE FOLLOWING PROGRAM:
"BEGIN" "INTEGER" I, J; "REAL" D; "INTEGER" "ARRAY" RI, CI[1:4];
"ARRAY" A[1:4, 1:4], AUX[1:7];
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" A[I, J]:= 1 / (I + J - 1);
AUX[2]:= "-14; AUX[4]:= 8;
GSSELM(A, 4, AUX, RI, CI);
D:= "IF" AUX[3] = 4 "THEN" DETERM(A, 4, AUX[1]) "ELSE" 0;
OUTPUT(71, "(""("DETERMINANT =")"B+.15D"+3D")", D)
"END"
RESULT:
DETERMINANT = +.165343915345370"-006
SOURCE TEXT(S):
"CODE" 34303;
"REAL" "PROCEDURE" DETERM(A, N, SIGN); "VALUE" N, SIGN;
"INTEGER" N, SIGN; "ARRAY" A;
"BEGIN" "INTEGER" I; "REAL" DET;
DET:= 1;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DET:= A[I, I] * DET;
DETERM:= SIGN * ABS(DET)
"END" DETERM;
"EOP"
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 1
AUTHORS: J. C. P. BUS AND T. J. DEKKER.
CONTRIBUTOR: J.C.P. BUS AND P. A. BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730915.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FIVE PROCEDURES:
SOL SOLVES THE LINEAR SYSTEM WHOSE MATRIX HAS BEEN TRIANGULARLY
DECOMPOSED BY DEC;
DECSOL SOLVES A LINEAR SYSTEM WHOSE ORDER IS SMALL RELATIVE TO THE
NUMBER OF BINARY DIGITS IN THE NUMBER REPRSENTATION;
SOLELM SOLVES A LINEAR SYSTEM WHOSE MATRIX HAS BEEN TRIANGULARLY
DECOMPOSED BY GSSELM OR GSSERB(SECTION 3.1.1.1.1.1.1.).
GSSSOL SOLVES A LINEAR SYSTEM;
GSSSOLERB SOLVES A LINEAR SYSTEM AND CALCULATES A ROUGH
UPPERBOUND FOR THE RELATIVE ERROR IN THE CALCULATED SOLUTION;
THE
DIFFERENCE BETWEEN DECSOL ON THE ONE SIDE AND GSSSOL AND GSSSOLERB
ON THE OTHER SIDE LIES IN THE METHOD USED FOR TRIANGULAR
DECOMPOSITION, PARTICULARLY IN THE PIVOTING STRATEGY; DECSOL USES
DEC, GSSSOL AND GSSSOLERB USE GSSELM TO PERFORM THE TRIANGULAR
DECOMPOSITION (SECTION 3.1.1.1.1.1.1); SINCE, IN EXCEPTIONAL CASES,
DEC MAY YIELD USELESS RESULTS, ONE IS ADVISED TO USE GSSSOL OR
GSSSOLERB; HOWEVER, IF THE ORDER OF THE LINEAR SYSTEM IS VERY SMALL
RELATIVE TO THE NUMBER OF BINARY DIGITS IN THE NUMBER
REPRESENTATION, THEN DECSOL ALSO MAY BE USED.
KEYWORDS:
ALGEBRAIC EQUATIONS,
LINEAR SYSTEMS.
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 2
SUBSECTION: SOL .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" SOL(A, N, P, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, B; "INTEGER" "ARRAY" P;
"CODE" 34051;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX OF
THE LINEAR SYSTEM AS PRODUCED BY DEC (SECTION
3.1.1.1.1.1.1);
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
P: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" P[1:N];
ENTRY:THE PIVOTAL INDICES, AS PRODUCED BY DEC.
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED:
MATVEC = CP34011.
RUNNING TIME: PROPORTIONAL TO N ** 2.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SOL SHOULD BE CALLED AFTER DEC (SECTION 3.1.1.1.1.1.1) AND SOLVES
THE LINEAR SYSTEM WITH A MATRIX, WHOSE TRIANGULARLY DECOMPOSED FORM
AS PRODUCED BY DEC IS GIVEN IN ARRAY A, AND A RIGHT-HAND SIDE AS
GIVEN IN ARRAY B; SOL LEAVES A AND P UNALTERED, SO, AFTER ONE CALL
OF DEC, SEVERAL CALLS OF SOL MAY FOLLOW FOR SOLVING SEVERAL SYSTEMS
HAVING THE SAME MATRIX BUT DIFFERENT RIGHT-HAND SIDES.
EXAMPLE OF USE: SEE DECSOL (THIS SECTION).
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 3
SUBSECTION: DECSOL .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECSOL(A, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX, B;
"CODE" 34301;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:3];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS TERMINATED AND NO
SOLUTION WILL BE CALCULATED;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE CALCULATED SOLUTION OF THE
LINEAR SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
PROCEDURES USED:
DEC = CP34300,
SOL = CP34051.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: DECSOL DECLARES AN AUXILIARY ARRAY OF TYPE
INTEGER AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 4
METHOD AND PERFORMANCE:
DECSOL USES DEC TO PERFORM THE TRIANGULAR DECOMPOSITION OF THE
MATRIX AND SOL TO CALCULATE THE SOLUTION WITH FORWARD AND BACK
SUBSTITUTION; SINCE DECSOL MAY YIELD USELESS RESULTS, EVEN FOR
WELL-CONDITIONED MATRICES (SEE DEC, SECTION 3.1.1.1.1.1.1), DECSOL
SHOULD ONLY BE USED IF THE ORDER OF THE MATRIX IS SMALL RELATIVE TO
THE NUMBER OF BINARY DIGITS IN THE NUMBER REPRESENTATION; IF
AUX[3] < N, THEN THE EFFECT OF DECSOL IS MERELY THAT OF DEC.
EXAMPLE OF USE:
LET A BE THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX AND B THE
THIRD COLUMN OF A, THEN THE SOLUTION OF THE LINEAR SYSTEM
AX = B IS GIVEN BY THE THIRD UNIT VECTOR AND MAY BE CALCULATED BY
THE FOLLOWING PROGRAM:
"BEGIN" "INTEGER" I, J;
"ARRAY" A[1:4, 1:4], B[1:4], AUX[1:3];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(B[I]);
"FOR" I:= 1 "STEP" 2 "UNTIL" 3 "DO" ITEM(AUX[I])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("/, "("SOLUTION:")"B+.15D"+3D,/,3(10B+.15D"+3D,/),
"("SIGN(DET) = ")"+D,/,"("NUMBER OF ELIMINATIONSTEPS = ")"
+D")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
A[I,J]:= 1 / (I + J - 1); B[I]:= A[I,3]
"END";
AUX[2]:= "-14;
DECSOL(A, 4, AUX, B);
OUTLIST(71, LAYOUT, LIST)
"END"
RESULTS:
SOLUTION: +.000000000000000"+000
+.000000000000000"+000
+.100000000000000"+001
+.000000000000000"+000
SIGN(DET) = +1
NUMBER OF ELIMINATIONSTEPS = +4
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 5
SUBSECTION: SOLELM .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" SOLELM(A, N, RI, CI, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, B; "INTEGER" "ARRAY" RI, CI;
"CODE" 34061;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX OF
THE LINEAR SYSTEM AS PRODUCED BY GSSELM (SECTION
3.1.1.1.1.1.1);
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
ENTRY:THE PIVOTAL ROW INDICES, AS PRODUCED BY GSSELM;
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
ENTRY:THE PIVOTAL COLUMN INDICES, AS PRODUCED BY GSSELM;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED:
SOL = CP34051.
RUNNING TIME: PROPORTIONAL TO N ** 2.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SOLELM SHOULD BE CALLED AFTER GSSELM OR GSSERB (SECTION
3.1.1.1.1.1.1) AND SOLVES THE LINEAR SYSTEM WITH THE MATRIX, WHOSE
TRIANGULARLY DECOMPOSED FORM AS PRODUCED BY GSSELM IS GIVEN IN
ARRAY A, AND A RIGHT-HAND SIDE AS GIVEN IN ARRAY B; SOLELM LEAVES
A, RI AND CI UNALTERED, SO, AFTER ONE CALL OF GSSELM OR GSSERB,
SEVERAL CALLS OF SOLELM MAY FOLLOW FOR SOLVING SEVERAL SYSTEMS
HAVING THE SAME MATRIX BUT DIFFERENT RIGHT-HAND SIDES.
EXAMPLE OF USE: SEE GSSSOL OR GSSSOLERB (THIS SECTION).
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 6
SUBSECTION: GSSSOL .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSSOL(A, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX, B;
"CODE" 34232;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:7];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS TERMINATED AND NO
SOLUTION WILL HAVE BEEN CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX GIVEN IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE CALCULATED SOLUTION OF THE
LINEAR SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
PROCEDURES USED:
SOLELM = CP34061,
GSSELM = CP34231.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: GSSSOL DECLARES TWO AUXILIARY ARRAYS OF
TYPE INTEGER AND ORDER N.
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 7
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
GSSSOL USES GSSELM (SECTION 3.1.1.1.1.1.1) TO PERFORM A TRIANGULAR
DECOMPOSITION OF THE MATRIX AND SOLELM (THIS SECTION) TO CALCULATE
THE SOLUTION OF THE GIVEN LINEAR SYSTEM; IF AUX[3] < N, THEN THE
EFFECT OF GSSSOL IS MERELY THAT OF GSSELM.
EXAMPLE OF USE:
LET A BE THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX AND B THE
THIRD COLUMN OF A, THEN THE SOLUTION OF THE LINEAR SYSTEM
AX = B IS GIVEN BY THE THIRD UNIT VECTOR AND MAY BE CALCULATED BY
THE FOLLOWING PROGRAM:
"BEGIN" "INTEGER" I, J;
"ARRAY" A[1:4, 1:4], B[1:4], AUX[1:7];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(B[I]);
"FOR" I:= 1 "STEP" 2 "UNTIL" 7 "DO" ITEM(AUX[I])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("/, "("SOLUTION:")"B+.15D"+3D,/,3(10B+.15D"+3D,/),
"("SIGN(DET) = ")"+D,/,"("NUMBER OF ELIMINATIONSTEPS = ")"
+D,/,"("MAX(ABS(A[I,J]))= ")"+.15D"+3D,/,
"("UPPER BOUND GROWTH: ")"+.15D"+3D")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
A[I,J]:= 1 / (I + J - 1); B[I]:= A[I,3]
"END";
AUX[2]:= "-14; AUX[4]:= 8;
GSSSOL(A, 4, AUX, B);
OUTLIST(71, LAYOUT, LIST)
"END"
RESULTS:
SOLUTION: +.888178419700120"-014
-.497379915032070"-013
+.100000000000010"+001
+.000000000000000"+000
SIGN(DET) = +1
NUMBER OF ELIMINATIONSTEPS = +4
MAX(ABS(A[I,J]))= +.100000000000000"+001
UPPER BOUND GROWTH: +.159619047619050"+001
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 8
SUBSECTION: GSSSOLERB.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSSOLERB(A, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX, B;
"CODE" 34243;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:11];
ENTRY:
AUX[0]: THE MACHINE PRECISION;
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
AUX[6]: AN UPPER BOUND FOR THE RELATIVE PRECISION OF THE
GIVEN MATRIX ELEMENTS;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS TERMINATED AND NO
SOLUTION OR ERROR BOUND WILL HAVE BEEN CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX GIVEN IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
AUX[9]: IF AUX[3] = N, THEN AUX[9] WILL EQUAL THE 1-NORM OF
THE INVERSE MATRIX, ELSE AUX[9] WILL BE UNDEFINED;
AUX[11]: IF AUX[3] = N THEN THE VALUE OF AUX[11] WILL BE A
ROUGH UPPER BOUND FOR THE RELATIVE ERROR IN THE
CALCULATED SOLUTION OF THE GIVEN LINEAR SYSTEM,
ELSE AUX[11] WILL BE UNDEFINED; IF NO USE CAN BE
MADE OF THE FORMULA FOR THE ERROR BOUND AS GIVEN IN
SECTION 3.1.1.1.1.1.1 (SUBSECTION ERBELM), BECAUSE
OF A VERY BAD CONDITION OF THE MATRIX, THEN
AUX[11]:= -1;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE CALCULATED SOLUTION OF THE
LINEAR SYSTEM, ELSE B REMAINS UNALTERED.
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 9
PROCEDURES USED:
SOLELM = CP34061,
GSSERB = CP34242.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: GSSSOLERB DECLARES TWO AUXILIARY ARRAYS OF
TYPE INTEGER AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
GSSSOLERB USES GSSERB (SECTION 3.1.1.1.1.1.1) TO PERFORM THE
TRIANGULAR DECOMPOSITION OF THE MATRIX AND TO CALCULATE AN UPPER
BOUND FOR THE RELATIVE ERROR IN THE CALCULATED SOLUTION OF THE
GIVEN LINEAR SYSTEM, AND SOLELM (THIS SECTION) TO CALCULATE THIS
SOLUTION; IF AUX[3] < N, THEN THE EFFECT OF GSSSOLERB IS MERELY
THAT OF GSSELM (SECTION 3.1.1.1.1.1.1).
EXAMPLE OF USE:
LET A BE THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX AND B THE
THIRD COLUMN OF A, THEN THE SOLUTION OF THE LINEAR SYSTEM
AX = B IS GIVEN BY THE THIRD UNIT VECTOR AND THIS SOLUTION, AS
WELL AS AN UPPER BOUND FOR THE RELATIVE ERROR IN THE CALCULATED ONE
, MAY BE OBTAINED BY THE FOLLOWING PROGRAM:
"BEGIN" "INTEGER" I, J;
"ARRAY" A[1:4, 1:4], B[1:4], AUX[0:11];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(B[I]);
"FOR" I:= 1 "STEP" 2 "UNTIL" 11 "DO" ITEM(AUX[I])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("/, "("SOLUTION:")"B+.15D"+3D,/,3(10B+.15D"+3D,/),
"("SIGN(DET) = ")"+D,/,"("NUMBER OF ELIMINATIONSTEPS = ")"
+D,/,"("MAX(ABS(A[I,J]))= ")"+.15D"+3D,/,
"("UPPER BOUND GROWTH: ")"+.15D"+3D,/,
"("1-NORM OF THE INVERSE MATRIX:")"B+.15D"+3D,/,
"("UPPER BOUND REL. ERR. IN THE CALC. SOL.")"
B+.15D"+3D")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
A[I,J]:= 1 / (I + J - 1); B[I]:= A[I,3]
"END";
AUX[0]:= AUX[2]:= "-14; AUX[4]:= 8; AUX[6]:= "-14;
GSSSOLERB(A, 4, AUX, B);
OUTLIST(71, LAYOUT, LIST)
"END"
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 10
RESULTS:
SOLUTION: +.888178419700120"-014
-.497379915032070"-013
+.100000000000010"+001
+.000000000000000"+000
SIGN(DET) = +1
NUMBER OF ELIMINATIONSTEPS = +4
MAX(ABS(A[I,J]))= +.100000000000000"+001
UPPER BOUND GROWTH: +.159619047619050"+001
1-NORM OF THE INVERSE MATRIX: +.136199999998790"+005
UPPER BOUND REL. ERR. IN THE CALC. SOL. +.277896269157090"-007
REFERENCES:
[1] BUS, J. C. P.
LINEAR SYSTEMS WITH CALCULATION OF ERROR BOUNDS AND ITERATIVE
REFINEMENT (DUTCH).
MATHEMATICAL CENTRE, AMSTERDAM, LR 3. 4. 19 (1972).
[2] DEKKER, T. J.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MATHEMATICAL CENTRE, AMSTERDAM, TRACT 22 (1968).
SOURCE TEXT(S):
"CODE" 34051;
"PROCEDURE" SOL(A, N, P, B); "VALUE" N; "INTEGER" N; "ARRAY" A, B;
"INTEGER" "ARRAY" P;
"BEGIN" "INTEGER" K, PK;
"REAL" R;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= B[K]; PK:= P[K];
B[K]:= (B[PK] - MATVEC(1, K - 1, K, A, B)) / A[K,K];
"IF" PK ^= K "THEN" B[PK]:= R
"END";
"FOR" K:= N "STEP" - 1 "UNTIL" 1 "DO"
B[K]:= B[K] - MATVEC(K + 1, N, K, A, B)
"END" SOL
1SECTION: 3.1.1.1.1.1.3 (MAY 1974) PAGE 11
;
"EOP"
0"CODE" 34301;
"PROCEDURE" DECSOL(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN" "INTEGER" "ARRAY" P[1:N];
DEC(A, N, AUX, P);
"IF" AUX[3] = N "THEN" SOL(A, N, P, B)
"END" DECSOL;
"EOP"
0"CODE" 34061;
"PROCEDURE" SOLELM(A, N, RI, CI, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, B;
"INTEGER" "ARRAY" RI, CI;
"BEGIN" "INTEGER" R, CIR;
"REAL" W;
SOL(A, N, RI, B);
"FOR" R:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" CIR:= CI[R]; "IF" CIR ^= R "THEN"
"BEGIN" W:= B[R]; B[R]:= B[CIR]; B[CIR]:= W "END"
"END"
"END" SOLELM;
"EOP"
"CODE" 34232;
"PROCEDURE" GSSSOL(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN" "INTEGER" "ARRAY" RI, CI[1:N];
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" SOLELM(A, N, RI, CI, B)
"END" GSSSOL;
"EOP"
0"CODE" 34243;
"PROCEDURE" GSSSOLERB(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN" "INTEGER" "ARRAY" RI, CI[1:N];
GSSERB(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" SOLELM(A, N, RI, CI, B)
"END" GSSSOLERB;
"EOP"
1SECTION: 3.1.1.1.1.1.4 (DECEMBER 1975) PAGE 1
AUTHORS: J. C. P. BUS AND T. J. DEKKER.
CONTRIBUTOR: J.C.P. BUS AND P.A. BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730920.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FIVE PROCEDURES FOR INVERSION OF MATRICES:
INV CALCULATES THE INVERSE OF A MATRIX THAT HAS BEEN TRIANGULARLY
DECOMPOSED BY DEC;
DECINV CALCULATES THE INVERSE OF A MATRIX WHOSE ORDER IS SMALL
RELATIVE TO THE NUMBER OF BINARY DIGITS IN THE NUMBER
REPRESENTATION;
INV1 CALCULATES THE INVERSE OF A MATRIX THAT HAS BEEN TRIANGULARLY
DECOMPOSED BY GSSELM OR GSSERB.THE 1-NORM OF THE INVERSE MATRIX
MIGHT ALSO BE CALCULATED
GSSINV CALCULATES THE INVERSE OF A MATRIX;
GSSINVERB CALCULATES THE INVERSE OF A MATRIX AND ITS 1-NORM.
A ROUGH UPPERBOUND FOR THE RELATIVE ERROR IN THE CALCULATED INVERSE
MATRIX IS ALSO GIVEN;
THE DIFFERENCE
BETWEEN DECINV ON THE ONE SIDE AND GSSINV AND GSSINVERB ON THE
OTHER SIDE LIES IN THE METHOD USED FOR TRIANGULAR DECOMPOSITION,
PARTICULARLY IN THE PIVOTING STRATEGY; DECINV USES DEC, GSSINV AND
GSSINVERB USE GSSELM TO PERFORM THE TRIANGULAR DECOMPOSITION; THE
USER IS ADVISED TO USE GSSINV OR GSSINVERB (SEE SECTION
3.1.1.1.1.1.1).
KEYWORDS:
MATRIX INVERSION.
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 2
SUBSECTION: INV .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" INV(A, N, P); "VALUE" N;
"INTEGER" N; "ARRAY" A; "INTEGER" "ARRAY" P;
"CODE" 34053;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX AS
PRODUCED BY DEC (SECTION 3.1.1.1.1.1.1);
EXIT: THE CALCULATED INVERSE MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
P: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY P[1:N];
ENTRY:THE PIVOTAL INDICES, AS PRODUCED BY DEC;
PROCEDURES USED:
MATMAT = CP34013,
ICHCOL = CP34031,
DUPCOLVEC = CP31034.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: INV DECLARES AN AUXILIARY ARRAY OF TYPE
REAL AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
INV SHOULD BE CALLED AFTER DEC (SECTION 3.1.1.1.1.1.1) AND
CALCULATES THE INVERSE OF THE MATRIX, WHOSE TRIANGULARLY DECOMPOSED
FORM AS PRODUCED BY DEC IS GIVEN IN ARRAY A; THE INVERSE MATRIX IS
OVERWRITTEN ON A.
EXAMPLE OF USE: SEE DECINV (THIS SECTION).
1SECTION: 3.1.1.1.1.1.4 (DECEMBER 1975) PAGE 3
SUBSECTION: DECINV .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECINV(A, N, AUX); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX;
"CODE" 34302;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE MATRIX, WHOSE INVERSE HAS TO BE CALCULATED;
EXIT: IF AUX[3] = N, THEN THE CALCULATED INVERSE MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:3];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N, THEN THE PROCESS IS TERMINATED AND NO
INVERSE WILL HAVE BEEN CALCULATED.
PROCEDURES USED:
DEC = CP34300,
INV = CP34053.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: DECINV DECLARES AN AUXILIARY ARRAY OF TYPE
INTEGER AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 4
METHOD AND PERFORMANCE:
DECINV USES DEC (SECTION 3.1.1.1.1.1.1) TO PERFORM THE TRIANGULAR
DECOMPOSITION OF A MATRIX AND INV TO CALCULATE ITS INVERSE; DECINV
SHOULD ONLY BE USED IF THE ORDER OF THE MATRIX IS SMALL RELATIVE TO
THE NUMBER OF BINARY DIGITS IN THE NUMBER REPRESENTATION (SEE DEC);
IF AUX[3] < N, THEN THE EFFECT OF DECINV IS MERELY THAT OF DEC.
EXAMPLE OF USE:
THE FOLLOWING PROGRAM CALCULATES THE INVERSE OF THE INPUT MATRIX
AND PRINTS THE RESULTS:
"BEGIN"
"ARRAY" A[1:4, 1:4], AUX[1:3];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(A[I,J])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("4(4B,4(B+ZDB),/),/")");
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" INREAL(70, A[I,J])
"END" IN LIST;
AUX[2]:= "-14;
OUTPUT(71,"("/,"("INPUT:")",///")");
OUTLIST(71, LAYOUT, LIST);
OUTPUT(71,"("/,"("CALCULATED INVERSE:")",/")");
DECINV(A, 4, AUX);
OUTLIST(71, LAYOUT, LIST);
OUTPUT(71, "(""("AUX[1]=")"B+D,/,"("AUX[3]=")"B+D")",
AUX[1], AUX[3])
"END"
INPUT:
+ 4 + 2 + 4 + 1
+30 +20 +45 +12
+20 +15 +36 +10
+35 +28 +70 +20
RESULTS:
CALCULATED INVERSE:
+4 -2 +4 -1
-30 +20 -45 +12
+20 -15 +36 -10
-35 +28 -70 +20
AUX[1]= +1
AUX[3]= +4
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 5
SUBSECTION: INV1 .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"REAL" "PROCEDURE" INV1(A, N, RI, CI, WITHNORM);
"VALUE" N, WITHNORM; "INTEGER" N; "BOOLEAN" WITHNORM; "ARRAY" A;
"INTEGER" "ARRAY" RI, CI;
"CODE" 34235;
INV1: IF THE VALUE OF WITHNORM IS TRUE, THEN THE VALUE OF INV1
WILL EQUAL THE 1-NORM OF THE CALCULATED INVERSE MATRIX,
ELSE INV1:= 0;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX AS
PRODUCED BY GSSELM (SECTION 3.1.1.1.1.1.1);
EXIT: THE CALCULATED INVERSE MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
ENTRY:THE PIVOTAL ROW INDICES, AS PRODUCED BY GSSELM;
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
ENTRY:THE PIVOTAL COLUMN INDICES, AS PRODUCED BY GSSELM;
WITHNORM: <BOOLEAN EXPRESSION>;
IF THE VALUE OF WITHNORM IS TRUE, THEN THE 1-NORM OF THE
INVERSE MATRIX WILL BE CALCULATED AND ASSIGNED TO INV1,
ELSE INV1:= 0;
PROCEDURES USED:
ICHROW = CP34032,
INV = CP34053.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
INV1 SHOULD BE CALLED AFTER GSSELM OR GSSERB (SECTION
3.1.1.1.1.1.1),WHICH DELIVERS THE TRIANGULARLY DECOMPOSED FORM OF A
MATRIX; INV1 CALCULATES THE INVERSE MATRIX AND ALSO ITS 1-NORM
MIGHT BE CALCULATED; THE INVERSE MATRIX IS OVERWRITTEN ON A.
EXAMPLE OF USE: SEE GSSINV AND GSSINVERB (THIS SECTION).
1SECTION: 3.1.1.1.1.1.4 (DECEMBER 1975) PAGE 6
SUBSECTION: GSSINV .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSINV(A, N, AUX); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX;
"CODE" 34236;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE MATRIX, WHOSE INVERSE HAS TO BE CALCULATED;
EXIT: IF AUX[3] = N, THEN THE CALCULATED INVERSE MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:9];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS TERMINATED AND NO
SOLUTION WILL BE CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX GIVEN IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
AUX[9]: IF AUX[3] = N, THEN AUX[9] WILL EQUAL THE 1-NORM OF
THE CALCULATED INVERSE MATRIX, ELSE AUX[9] WILL BE
UNDEFINED.
PROCEDURES USED:
INV1 = CP34235,
GSSELM = CP34231.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: GSSINV DECLARES TWO AUXILIARY ARRAYS OF
TYPE INTEGER AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 7
METHOD AND PERFORMANCE:
GSSINV USES GSSELM (SECTION 3.1.1.1.1.1.1) TO PERFORM A TRIANGULAR
DECOMPOSITION OF THE MATRIX AND INV1 (THIS SECTION) TO CALCULATE
THE INVERSE MATRIX; IF AUX[3] < N, THEN THE EFFECT OF GSSINV IS
MERELY THAT OF GSSELM.
EXAMPLE OF USE:
THE FOLLOWING PROGRAM CALCULATES THE INVERSE OF THE INPUT MATRIX
AND PRINTS THE RESULTS:
"BEGIN" "ARRAY" A[1:4, 1:4], AUX[1:9];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(A[I,J])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("4(4B,4(B+ZDB),/),/")");
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" INREAL(70, A[I,J])
"END" IN LIST;
AUX[2]:= "-14; AUX[4]:= 8;
OUTPUT(71,"("/,"("INPUT:")",///")");
OUTLIST(71, LAYOUT, LIST);
GSSINV(A, 4, AUX);
OUTPUT(71,"("/,"("CALCULATED INVERSE:")",/")");
OUTLIST(71, LAYOUT, LIST);
OUTPUT(71, "("4B"("AUX ELEMENTS:")",/,2(4B+D,/),
3(4B+.15D"+3D,/)")", AUX[1], AUX[3], AUX[5], AUX[7], AUX[9])
"END"
INPUT:
+ 4 + 2 + 4 + 1
+30 +20 +45 +12
+20 +15 +36 +10
+35 +28 +70 +20
RESULTS:
CALCULATED INVERSE:
+4 -2 +4 -1
-30 +20 -45 +12
+20 -15 +36 -10
-35 +28 -70 +20
AUX ELEMENTS:
+1
+4
+.700000000000000"+002
+.112528571428570"+003
+.154999999999730"+003
1SECTION: 3.1.1.1.1.1.4 (DECEMBER 1975) PAGE 8
SUBSECTION: GSSINVERB.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSINVERB(A, N, AUX); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX;
"CODE" 34244;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N, 1:N];
ENTRY: THE MATRIX, WHOSE INVERSE HAS TO BE CALCULATED;
EXIT: IF AUX[3] = N, THEN THE CALCULATED INVERSE MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:11];
ENTRY:
AUX[0]: THE MACHINE PRECISION;
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
AUX[6]: AN UPPER BOUND FOR THE RELATIVE PRECISION OF THE
GIVEN MATRIX ELEMENTS;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS IS TERMINATED AND NO
SOLUTION WILL HAVE BEEN CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX GIVEN IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
AUX[9]: IF AUX[3] = N, THEN AUX[9] WILL EQUAL THE 1-NORM OF
THE INVERSE MATRIX, ELSE AUX[9] WILL BE UNDEFINED;
AUX[11]: IF AUX[3] = N THEN THE VALUE OF AUX[11] WILL BE A
ROUGH UPPER BOUND FOR THE RELATIVE ERROR IN THE
CALCULATED INVERSE MATRIX, ELSE AUX[11] WILL BE
BE UNDEFINED; IF NO USE CAN BE MADE OF THE FORMULA
FOR THE ERROR BOUND AS GIVEN IN SECTION
3.1.1.1.1.1.1 (SUBSECTION ERBELM), BECAUSE OF A
VERY BAD CONDITION OF THE MATRIX, THEN AUX[11]:=-1.
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 9
PROCEDURES USED:
INV1 = CP34235,
GSSELM = CP34231,
ERBELM = CP34241.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: GSSINVERB DECLARES TWO AUXILIARY ARRAYS OF
TYPE INTEGER AND ORDER N.
RUNNING TIME: PROPORTIONAL TO N ** 3.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
GSSINVERB USES GSSELM (SECTION 3.1.1.1.1.1.1) TO PERFORM THE
TRIANGULAR DECOMPOSITION OF THE MATRIX, INV1 (THIS SECTION) TO
CALCULATE THE INVERSE MATRIX AND ITS 1-NORM AND ERBELM (SECTION
3.1.1.1.1.1.1) TO CALCULATE AN UPPER BOUND FOR THE RELATIVE ERROR
IN THE CALCULATED INVERSE; IF AUX[3] < N, THEN THE EFFECT OF
GSSINVERB IS MERELY THAT OF GSSELM.
EXAMPLE OF USE:
THE FOLLOWING PROGRAM CALCULATES THE INVERSE OF THE INPUT MATRIX
WITH AN UPPER BOUND FOR THE RELATIVE ERROR IN IT AND PRINTS THE
RESULTS:
"BEGIN" "ARRAY" A[1:4, 1:4], AUX[0:11];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(A[I,J])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("4(4B,4(B+ZDB),/),/")");
"BEGIN" "INTEGER" I, J;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO" INREAL(70, A[I,J])
"END" IN LIST;
AUX[0]:= AUX[2]:= AUX[6]:= "-14;
OUTPUT(71,"("/,"("INPUT:")",///")");
OUTLIST(71, LAYOUT, LIST);
AUX[4]:= 8; GSSINVERB(A, 4, AUX);
OUTPUT(71,"("/,"("CALCULATED INVERSE:")",/")");
OUTLIST(71, LAYOUT, LIST);
OUTPUT(71, "("4B"("AUX ELEMENTS:")",/,2(4B+D,/),
4(4B+.15D"+3D,/)")", AUX[1], AUX[3], AUX[5], AUX[7], AUX[9],
AUX[11])
"END"
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 10
INPUT:
+ 4 + 2 + 4 + 1
+30 +20 +45 +12
+20 +15 +36 +10
+35 +28 +70 +20
RESULTS:
CALCULATED INVERSE:
+4 -2 +4 -1
-30 +20 -45 +12
+20 -15 +36 -10
-35 +28 -70 +20
AUX ELEMENTS:
+1
+4
+.700000000000000"+002
+.112528571428570"+003
+.154999999999730"+003
+.222946341369190"-007
REFERENCES:
[1] BUS, J. C. P.
LINEAR SYSTEMS WITH CALCULATION OF ERROR BOUNDS AND ITERATIVE
REFINEMENT (DUTCH).
MATHEMATICAL CENTRE, AMSTERDAM, LR 3. 4. 19 (1972).
[2] DEKKER, T. J.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MATHEMATICAL CENTRE, AMSTERDAM, TRACT 22 (1968).
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 11
SOURCE TEXT(S):
"CODE" 34053;
"PROCEDURE" INV(A, N, P); "VALUE" N; "INTEGER" N; "ARRAY" A;
"INTEGER" "ARRAY" P;
"BEGIN" "INTEGER" J, K, K1;
"REAL" R;
"ARRAY" V[1:N];
"FOR" K:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" K1:= K + 1;
"FOR" J:= N "STEP" - 1 "UNTIL" K1 "DO"
"BEGIN" A[J,K1]:= V[J];
V[J]:= - MATMAT(K1, N, K, J, A, A)
"END";
R:= A[K,K];
"FOR" J:= N "STEP" - 1 "UNTIL" K1 "DO"
"BEGIN" A[K,J]:= V[J];
V[J]:= - MATMAT(K1, N, J, K, A, A) / R
"END";
V[K]:= (1 - MATMAT(K1, N, K, K, A, A)) / R
"END";
DUPCOLVEC(1, N, 1, A, V);
"FOR" K:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" K1:= P[K]; "IF" K1 ^= K "THEN"
ICHCOL(1, N, K, K1, A)
"END"
"END" INV;
"EOP"
0"CODE" 34302;
"PROCEDURE" DECINV(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" "ARRAY" P[1:N];
DEC(A, N, AUX, P); "IF" AUX[3] = N "THEN" INV(A, N, P)
"END" DECINV
1SECTION: 3.1.1.1.1.1.4 (MAY 1974) PAGE 12
;
"EOP"
"CODE" 34235;
"REAL" "PROCEDURE" INV1(A, N, RI, CI, WITHNORM);
"VALUE" N, WITHNORM; "INTEGER" N; "BOOLEAN" WITHNORM;
"ARRAY" A; "INTEGER" "ARRAY" RI, CI;
"BEGIN" "INTEGER" L, K, K1;
"REAL" AID, NRMINV;
INV(A, N, RI); NRMINV:= 0; "IF" WITHNORM "THEN"
"FOR" L:= 1 "STEP" 1 "UNTIL" N "DO"
NRMINV:= NRMINV + ABS(A[L,N]);
"FOR" K:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" "IF" WITHNORM "THEN"
"BEGIN" AID:= 0;
"FOR" L:= 1 "STEP" 1 "UNTIL" N "DO"
AID:= AID + ABS(A[L,K]);
"IF" NRMINV < AID "THEN" NRMINV:= AID
"END";
K1:= CI[K]; "IF" K1 ^= K "THEN" ICHROW(1, N, K, K1, A)
"END";
INV1:= NRMINV
"END" INV1;
"EOP"
0"CODE" 34236;
"PROCEDURE" GSSINV(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" "ARRAY" RI, CI[1:N];
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" AUX[9]:= INV1(A, N, RI, CI, "TRUE")
"END" GSSINV;
"EOP"
0"CODE" 34244;
"PROCEDURE" GSSINVERB(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" "ARRAY" RI, CI[1:N];
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN"
ERBELM(N, AUX, INV1(A, N, RI, CI, "TRUE"))
"END" GSSINVERB;
"EOP"
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 1
AUTHOR: J. C. P. BUS.
CONTRIBUTOR: J.C.P. BUS AND P. A. BEENTJES.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731008.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOUR PROCEDURES FOR CALCULATING AN
ITERATIVELY IMPROVED SOLUTION OF A SYSTEM OF LINEAR EQUATIONS:
ITISOL SOLVES A LINEAR SYSTEM WHOSE MATRIX HAS BEEN TRIANGULARLY
DECOMPOSED BY GSSELM OR GSSERB. THIS SOLUTION THUS OBTAINED
NUMERICALLY, IS IMPROVED ITERATIVELY;
GSSITISOL SOLVES A LINEAR SYSTEM AND THIS SOLUTION THUS OBTAINED
NUMERICALLY, IS IMPROVED ITERATIVELY;
ITISOLERB SOLVES A LINEAR SYSTEM WHOSE MATRIX HAS BEEN TRIANGULARLY
DECOMPOSED BY GSSNRI.THIS SOLUTION IS IMPROVED ITERATIVELY.MOREOVER
A REALISTIC UPPERBOUND FOR THE RELATIVE ERROR IN THE SOLUTION IS
CALCULATED.
GSSITISOLERB SOLVES A LINEAR SYSTEM.THIS SOLUTION IS IMPROVED
ITERATIVELY AND A REALISTIC UPPERBOUND FOR THE RELATIVE ERROR IN
THE SOLUTION IS CALCULATED;
KEYWORDS:
ALGEBRAIC EQUATIONS,
LINEAR SYSTEMS,
ITERATIVE REFINEMENT.
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 2
SUBSECTION: ITISOL .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" ITISOL(A, LU, N, AUX, RI, CI, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, LU, AUX, B; "INTEGER" "ARRAY" RI, CI;
"CODE" 34250;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE MATRIX OF THE LINEAR SYSTEM;
LU: <ARRAY IDENTIFIER>;
"ARRAY" LU[1:N, 1:N];
ENTRY:THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX GIVEN
IN A, AS DELIVERED BY GSSELM (SECTION 3.1.1.1.1.1.1);
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[10:13];
ENTRY:
AUX[10]: A RELATIVE TOLERANCE FOR THE SOLUTION VECTOR; IF
THE 1-NORM OF THE VECTOR OF CORRECTIONS TO THE
SOLUTION, DIVIDED BY THE 1-NORM OF THE CALCULATED
SOLUTION, IS SMALLER THAN AUX[10], THEN THE PROCESS
WILL STOP; THE USER SHOULD NOT CHOOSE THE VALUE OF
AUX[10] SMALLER THAN THE RELATIVE PRECISION OF THE
ELEMENTS OF THE MATRIX AND THE RIGHT-HAND SIDE OF
THE LINEAR SYSTEM;
AUX[12]: THE MAXIMUM NUMBER OF ITERATIONS ALLOWED FOR THE
REFINEMENT OF THE SOLUTION; IF THE NUMBER OF
ITERATIONS EXCEEDS THE VALUE OF AUX[12], THEN THE
PROCESS WILL BE BROKEN OFF; USUALLY AUX[12] = 5
WILL GIVE GOOD RESULTS;
EXIT:
AUX[11]: THE 1-NORM OF THE VECTOR OF CORRECTIONS TO THE
SOLUTION IN THE LAST ITERATION STEP, DIVIDED BY THE
1-NORM OF THE CALCULATED SOLUTION;
IF AUX[11] > AUX[10], THEN THE PROCESS HAS BEEN
BROKEN OFF, BECAUSE THE NUMBER OF ITERATIONS
EXCEEDED THE VALUE GIVEN IN AUX[12];
AUX[13]: THE 1-NORM OF THE RESIDUAL VECTOR (SEE METHOD AND
PERFORMANCE;
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
ENTRY:THE PIVOTAL ROW-INDICES, AS PRODUCED BY GSSELM;
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
ENTRY:THE PIVOTAL COLUMN-INDICES, AS PRODUCED BY GSSELM;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE CALCULATED SOLUTION OF THE LINEAR SYSTEM.
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 3
PROCEDURES USED:
SOLELM = CP34061,
INIVEC = CP31010,
DUPVEC = CP31030,
LNGMATVEC = CP34411.
REQUIRED CENTRAL MEMORY:
TWO REAL ARRAYS, BOTH OF ORDER N, ARE DECLARED.
RUNNING TIME: THE NUMBER OF ARITHMETICAL OPERATIONS IN EACH ITERATION
STEP IS PROPORTIONAL TO N ** 2.
METHOD AND PERFORMANCE:
ITISOL SHOULD BE CALLED AFTER GSSELM OR GSSERB (SECTION
3.1.1.1.1.1.1) AND SOLVES THE LINEAR SYSTEM WITH A MATRIX GIVEN
IN ARRAY A, AND A RIGHT-HAND SIDE GIVEN IN ARRAY B; ONCE A
SOLUTION IS CALCULATED WITH SOLELM (SECTION 3.1.1.1.1.1.3), THIS
SOLUTION WILL BE REFINED ITERATIVELY UNTIL THE CALCULATED RELATIVE
CORRECTION TO THIS SOLUTION WILL BE LESS THAN A PRESCRIBED VALUE
(SEE AUX[10]);
EACH ITERATION OF THE REFINEMENT PROCESS CONSISTS OF THE FOLLOWING
THREE STEPS (SEE [1], [2], [3]):
1 CALCULATE, IN DOUBLE PRECISION, THE RESIDUAL VECTOR R, DEFINED
BY: R = AX - B,
WHERE X DENOTES THE SOLUTION, OBTAINED IN THE PREVIOUS
ITERATION, B THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM, GIVEN
IN B[1:N], AND A THE MATRIX GIVEN IN A[1:N, 1:N];
2 CALCULATE THE SOLUTION C, SAY, OF THE LINEAR SYSTEM: AC = R,
WITH THE AID OF THE TRIANGULARLY DECOMPOSED MATRIX AS GIVEN IN
LU[1:N, 1:N];
3 CALCULATE THE NEW SOLUTION: XNEW = X - C;
CONDITION OF THE MATRIX IS NOT TOO BAD, THEN THE PRECISION OF THE
CALCULATED SOLUTION WILL BE OF THE ORDER OF THE PRECISION ASKED FOR
IN AUX[10]; HOWEVER, IF THE CONDITION OF THE MATRIX IS VERY BAD,
THEN THIS PROCESS WILL POSSIBLY NOT CONVERGE OR, IN EXCEPTIONAL
CASES, CONVERGE TO A USELESS RESULT; IF THE USER WANTS TO MAKE
CERTAIN ABOUT THE PRECISION OF THE CALCULATED SOLUTION, THEN HE
HAS TO USE ITISOLERB (THIS SECTION), WHICH NEEDS THE CALCULATION
(OF ORDER N ** 3) OF THE INVERSE MATRIX TO GET AN UPPER BOUND FOR
THE CONDITION NUMBER AND WHICH GIVES A REALISTIC UPPER BOUND FOR
THE RELATIVE ERROR IN THE CALCULATED SOLUTION;
ITISOL LEAVES A, LU, RI AND CI UNALTERED, SO AFTER ONE CALL OF
GSSELM SEVERAL CALLS OF ITISOL MAY FOLLOW TO CALCULATE THE SOLUTION
OF SEVERAL LINEAR SYSTEMS WITH THE SAME MATRIX BUT DIFFERENT RIGHT-
HAND SIDES.
EXAMPLE OF USE: SEE GSSITISOL (THIS SECTION).
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 4
SUBSECTION: GSSITISOL .
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSITISOL(A, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX, B; "CODE" 34251;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPERTRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[1:13];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE FOR THE PROCESS OF TRIANGULAR
DECOMPOSITION;A REASONABLE CHOICE FOR THIS VALUE IS
AN ESTIMATE OF THE RELATIVE PRECISION OF THE MATRIX
ELEMENTS; HOWEVER,IT SHOULD NOT BE CHOSEN SMALLER
THANN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
AUX[10]: A RELATIVE TOLERANCE FOR THE SOLUTION VECTOR; IF
THE 1-NORM OF THE VECTOR OF CORRECTIONS TO THE
SOLUTION, DIVIDED BY THE 1-NORM OF THE CALCULATED
SOLUTION, IS SMALLER THAN AUX[10], THEN THE PROCESS
WILL STOP; THE USER SHOULD NOT CHOOSE THE VALUE OF
AUX[10] SMALLER THAN THE RELATIVE PRECISION OF THE
ELEMENTS OF THE MATRIX AND THE RIGHT-HAND SIDE OF
THE LINEAR SYSTEM;
AUX[12]: THE MAXIMUM NUMBER OF ITERATIONS ALLOWED FOR THE
REFINEMENT OF THE SOLUTION; IF THE NUMBER OF
ITERATIONS EXCEEDS THE VALUE OF AUX[12], THEN THE
PROCESS WILL BE BROKEN OFF; USUALLY AUX[12] = 5
WILL GIVE GOOD RESULTS;
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS HAS BEEN BROKEN OFF AND
NO SOLUTION WILL HAVE BEEN CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX WHICH HAD BEEN GIVEN
IN ARRAY A;
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 5
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
AUX[11]: IF AUX[3] < N, THEN AUX[11] WILL BE UNDEFINED,
ELSE AUX[11] EQUALS THE 1-NORM OF THE VECTOR OF
CORRECTIONS TO THE SOLUTION IN THE LAST STEP,
DIVIDED BY THE 1-NORM OF THE CALCULATED SOLUTION;
IF AUX[11] > AUX[10], THEN THE PROCESS HAS BEEN
BROKEN OFF, BECAUSE THE NUMBER OF ITERATIONS
EXCEEDED THE VALUE GIVEN IN AUX[12];
AUX[13]: IF AUX[3] = N, THEN THE VALUE OF AUX[13] WILL
EQUAL THE 1-NORM OF THE RESIDUAL VECTOR (SEE ITISOL
IN THIS SECTION), ELSE AUX[13] WILL BE UNDEFINED;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE CALCULATED SOLUTION OF THE
LINEAR SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
PROCEDURES USED:
DUPMAT = CP31035,
GSSELM = CP34231,
ITISOL = CP34250.
REQUIRED CENTRAL MEMORY:
THREE REAL ARRAYS, TWO OF ORDER N AND ONE OF ORDER N ** 2, ARE
DECLARED. FURTHERMORE, TWO INTEGER ARRAYS OF ORDER N ARE USED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
GSSITISOL USES GSSELM (SECTION 3.1.1.1.1.1.1) TO PERFORM A
TRIANGULAR DECOMPOSITION OF THE MATRIX AND ITISOL (THIS SECTION) TO
CALCULATE AN ITERATIVELY REFINED SOLUTION OF THE GIVEN LINEAR
SYSTEM; IF AUX[3] < N, THEN THE EFFECT OF GSSITISOL IS MERELY THAT
OF GSSELM; IF THE CONDITION OF THE MATRIX IS VERY BAD, THEN, IN
EXCEPTIONAL CASES, THE CALCULATED SOLUTION MAY BE USELESS (SEE
ITISOL, IN THIS SECTION).
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 6
EXAMPLE OF USE:
LET A BE THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX,
MULTIPLIED WITH 840 TO GET INTEGER ELEMENTS, AND B THE THIRD COLUMN
OF A, THEN THE SOLUTION OF THE LINEAR SYSTEM AX = B IS GIVEN BY
THE THIRD UNIT VECTOR AND MAY BE OBTAINED BY THE FOLLOWING
PROGRAM:
"BEGIN" "INTEGER" I, J;
"ARRAY" A[1:4, 1:4], B[1:4], AUX[1:13];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(B[I]);
"FOR" I:= 1 "STEP" 2 "UNTIL" 7 "DO" ITEM(AUX[I]);
ITEM(AUX[11]); ITEM(AUX[13])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("*, "("SOLUTION:")"B+.15D"+3D,/,3(10B+.15D"+3D,/),
"("SIGN(DET) = ")"+D,/,"("NUMBER OF ELIMINATIONSTEPS = ")"
+D,/,"("MAX(ABS(A[I,J]))= ")"+.15D"+3D,/,
"("UPPER BOUND GROWTH: ")"+.15D"+3D,/,
"("NORM LAST CORRECTION VECTOR: ")"+.15D"+3D,/,
"("NORM RESIDUAL VECTOR: ")"+.15D"+3D")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
A[I,J]:= 840 // (I + J - 1); B[I]:= A[I,3]
"END";
AUX[2]:= "-14; AUX[4]:= 8; AUX[10]:= "-14; AUX[12]:= 5;
GSSITISOL(A, 4, AUX, B);
OUTLIST(71, LAYOUT, LIST)
"END"
RESULTS:
SOLUTION: +.000000000000000"+000
+.000000000000000"+000
+.100000000000000"+001
+.000000000000000"+000
SIGN(DET) = +1
NUMBER OF ELIMINATIONSTEPS = +4
MAX(ABS(A[I,J]))= +.840000000000000"+003
UPPER BOUND GROWTH: +.134080000000000"+004
NORM LAST CORRECTION VECTOR: +.000000000000000"+000
NORM RESIDUAL VECTOR: +.000000000000000"+000
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 7
SUBSECTION: ITISOLERB.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" ITISOLERB(A, LU, N, AUX, RI, CI, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, LU, AUX, B; "INTEGER" "ARRAY" RI, CI;
"CODE" 34253;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"[1:N,1:N];
ENTRY:THE MATRIX OF THE LINEAR SYSTEM;
LU: <ARRAY IDENTIFIER>;
"ARRAY" LU[1:N,1:N];
ENTRY:THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX GIVEN
IN A AS DELIVERED BY GSSNRI (SECTION 3.1.1.1.1.1.1);
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:13];
ENTRY:
AUX[0]: THE MACHINE PRECISION;
AUX[5]: THE MODULUS OF AN ELEMENT, WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX OF THE LINEAR SYSTEM;
THIS VALUE IS DELIVERED BY GSSNRI (SECTION
3.1.1.1.1.1.1) IN AUX[5];
AUX[6]: AN UPPER BOUND FOR THE RELATIVE ERROR IN THE
ELEMENTS OF THE MATRIX OF THE LINEAR SYSTEM;
AUX[7]: AN UPPER BOUND FOR THE GROWTH DURING GAUSSIAN
ELIMINATION; THIS VALUE IS DELIVERED BY GSSNRI IN
AUX[7];
AUX[8]: AN UPPER BOUND FOR THE RELATIVE ERROR IN THE
ELEMENTS OF THE RIGHT-HAND SIDE OF THE LINEAR
SYSTEM;
AUX[9]: THE 1-NORM OF THE INVERSE MATRIX; THIS VALUE IS
DELIVERED BY GSSNRI IN AUX[9];
AUX[10]: A RELATIVE TOLERANCE FOR THE SOLUTION VECTOR; IF
THE 1-NORM OF THE VECTOR OF CORRECTIONS TO THE
SOLUTION, DIVIDED BY THE 1-NORM OF THE CALCULATED
SOLUTION, IS SMALLER THAN AUX[10], THEN THE PROCESS
WILL STOP; THE USER SHOULD NOT CHOOSE THE VALUE OF
AUX[10] SMALLER THAN THE RELATIVE PRECISION OF THE
ELEMENTS OF THE MATRIX AND THE RIGHT-HAND SIDE OF
THE LINEAR SYSTEM, GIVEN IN AUX[6] AND AUX[8];
AUX[12]: THE MAXIMUM NUMBER OF ITERATIONS ALLOWED FOR THE
REFINEMENT OF THE SOLUTION; IF THE NUMBER OF
ITERATIONS EXCEEDS THE VALUE OF AUX[12], THEN THE
PROCESS WILL BE BROKEN OFF; USUALLY AUX[12] = 5
WILL GIVE GOOD RESULTS;
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 8
EXIT:
AUX[11]:A REALISTIC UPPERBOUND FOR THE RELATIVE ERROR IN
THE CALCULATED SOLUTION; HOWEVER, IF NO USE CAN BE
MADE OF THE ERROR-FORMULA, THEN AUX[11] := -1;
AUX[13]:THE 1-NORM OF THE RESIDUAL VECTOR (SEE METHOD AND
PERFORMANCE);
RI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" RI[1:N];
ENTRY:THE PIVOTAL ROW-INDICES, AS PRODUCED BY GSSNRI;
CI: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:N];
EXIT:THE PIVOTAL COLUMN-INDICES, AS PRODUCED BY GSSNRI;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED:
ITISOL = CP34250.
REQUIRED CENTRAL MEMORY:TWO REAL AARRYS, BOTH OF ORDER N, ARE DECLARED.
RUNNING TIME: THE NUMBER OF ARITHMETICAL OPERATIONS IN EACH ITERATION
STEP OF THE REFINEMENT PROCESS IS PROPORTIONAL TO N ** 2.
METHOD AND PERFORMANCE:
ITISOLERB SHOULD BE CALLED AFTER GSSNRI (SECTION 3.1.1.1.1.1.1),
WHICH DELIVERS THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX AND
THE PROPER VALUES FOR THE ODD ELEMENTS OF ARRAY AUX; ITISOLERB
CALCULATES, WITH THE USE OF ITISOL (THIS SECTION), AN ITERATIVELY
IMPROVED SOLUTION OF THE LINEAR SYSTEM WITH A MATRIX AS GIVEN IN
ARRAY A AND A RIGHT-HAND SIDE AS GIVEN IN B; MOREOVER, ITISOLERB
CALCULATES A REALISTIC UPPER BOUND FOR THE RELATIVE ERROR IN THE
CALCULATED SOLUTION, BY (SEE [1], [2]):
NORM(DX) / NORM(X) <= P / (1 - P),
WHERE : P = ( NORM(R) / NORM(X) + DB / NORM(X) + DA )
* NORM(C) / (1 - Q * NORM(C) )
FOR Q SEE SECTION 3.1.1.1.1.1.1 (SUBSECTION ERBELM),
R IS THE RESIDUAL VECTOR (SEE ITISOL IN THIS SECTION),
X IS THE CALCULATED SOLUTION,
DB IS THE UPPER BOUND FOR THE RELATIVE ERROR IN THE RIGHT-HAND
SIDE ELEMENTS,
DA IS THE UPPER BOUND FOR THE RELATIVE ERROR IN THE MATRIX
ELEMENTS, C IS THE CALCULATED INVERSE MATRIX,
AND THE 1-NORM OF A VECTOR OR A MATRIX IS DENOTED BY: NORM(.)
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 9
IF 1 - P < AUX[0], THEN THE VALUE -1 IS DELIVERED IN AUX[11];
ITISOLERB LEAVES A, LU, RI AND CI UNALTERED, SO AFTER ONE CALL OF
GSSNRI SEVERAL CALLS OF ITISOLERB MAY FOLLOW, TO CALCULATE THE
SOLUTION OF SEVERAL LINEAR SYSTEMS WITH THE SAME MATRIX BUT
DIFFERENT RIGHT-HAND SIDES.
EXAMPLE OF USE: SEE GSSITISOLERB (THIS SECTION).
SUBSECTION: GSSITISOLERB.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" GSSITISOLERB(A, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" A, AUX, B; "CODE" 34254;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY:THE N-TH ORDER MATRIX;
EXIT: THE CALCULATED LOWER-TRIANGULAR MATRIX AND UNIT
UPPER-TRIANGULAR MATRIX WITH ITS UNIT DIAGONAL OMITTED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[0:13];
ENTRY:
AUX[0]: THE MACHINE PRECISION;
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS; HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
AUX[4]: A VALUE WHICH IS USED FOR CONTROLLING PIVOTING (SEE
GSSELM, SECTION 3.1.1.1.1.1.1);
AUX[6]: AN UPPER BOUND FOR THE RELATIVE ERROR IN THE MATRIX
ELEMENTS OF THE LINEAR SYSTEM;
AUX[8]: AN UPPER BOUND FOR THE RELATIVE ERROR IN THE
ELEMENTS OF THE RIGHT-HAND SIDE;
AUX[10]: A RELATIVE TOLERANCE FOR THE SOLUTION VECTOR; IF
THE 1-NORM OF THE VECTOR OF CORRECTIONS TO THE
SOLUTION, DIVIDED BY THE 1-NORM OF THE CALCULATED
SOLUTION, IS SMALLER THAN AUX[10], THEN THE PROCESS
WILL STOP; THE USER SHOULD NOT CHOOSE THE VALUE OF
AUX[10] SMALLER THAN THE RELATIVE PRECISION OF THE
ELEMENTS OF THE MATRIX AND THE RIGHT-HAND SIDE OF
THE LINEAR SYSTEM (AUX[10] >= AUX[2]);
AUX[12]: THE MAXIMUM NUMBER OF ITERATIONS ALLOWED FOR THE
REFINEMENT OF THE SOLUTION; IF THE NUMBER OF
ITERATIONS EXCEEDS THE VALUE OF AUX[12], THEN THE
PROCESS WILL BE BROKEN OFF; USUALLY AUX[12] = 5
WILL GIVE GOOD RESULTS;
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 10
EXIT:
AUX[1]: IF R IS THE NUMBER OF ELIMINATION STEPS PERFORMED
(SEE AUX[3]), THEN AUX[1] EQUALS 1 IF THE
DETERMINANT OF THE PRINCIPAL SUBMATRIX OF ORDER R
IS POSITIVE, ELSE AUX[1] = -1;
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED; IF
AUX[3] < N THEN THE PROCESS HAS BEEN BROKEN OFF
AND NO SOLUTION WILL HAVE BEEN CALCULATED;
AUX[5]: THE MODULUS OF AN ELEMENT WHICH IS OF MAXIMUM
ABSOLUTE VALUE FOR THE MATRIX WHICH HAD BEEN GIVEN
IN ARRAY A;
AUX[7]: AN UPPER BOUND FOR THE GROWTH (SEE GSSELM, SECTION
3.1.1.1.1.1.1);
AUX[9]: IF AUX[3] = N THEN AUX[9] EQUALS THE 1-NORM OF THE
CALCULATED INVERSE MATRIX, ELSE AUX[9] WILL BE
UNDEFINED;
AUX[11]: IF AUX[3] < N, THEN AUX[11] WILL BE UNDEFINED,
ELSE THE VALUE OF AUX[11] EQUALS A REALISTIC UPPER
BOUND FOR THE RELATIVE ERROR IN THE CALCULATED
SOLUTION; HOWEVER, IF NO USE CAN BE MADE OF THE
ERROR FORMULA (SEE ITISOLERB IN THIS SECTION), THEN
AUX[11]:= -1;
AUX[13]: IF AUX[3] = N, THEN AUX[13] EQUALS THE 1-NORM OF
THE RESIDUAL VECTOR (SEE ITISOL IN THIS SECTION),
ELSE AUX[13] WILL BE UNDEFINED;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE CALCULATED SOLUTION OF THE
LINEAR SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
PROCEDURES USED:
DUPMAT = CP31035,
GSSNRI = CP34252,
ITISOLERB = CP34253.
REQUIRED CENTRAL MEMORY:
THREE REAL ARRAYS, TWO OF ORDER N AND ONE OF ORDER N ** 2 , ARE
USED. FURTHERMORE, TWO INTEGER ARRAYS OF ORDER N ARE USED.
RUNNING TIME: PROPORTIONAL TO N ** 3.
METHOD AND PERFORMANCE:
GSSITISOLERB USES GSSNRI (SECTION 3.1.1.1.1.1.1) TO PERFORM A
TRIANGULAR DECOMPOSITION OF THE MATRIX AND ITISOLERB (THIS SECTION)
TO CALCULATE AN ITERATIVELY REFINED SOLUTION OF THE GIVEN LINEAR
SYSTEM AND A REALISTIC UPPER BOUND FOR THE RELATIVE ERROR IN THIS
SOLUTION; IF AUX[3] < N, THEN THE EFFECT OF GSSITISOLERB IS MERELY
THAT OF GSSELM (SECTION 3.1.1.1.1.1.1).
1SECTION: 3.1.1.1.1.1.5 (FEBRUARY 1979) PAGE 11
EXAMPLE OF USE:
LET A BE THE FOURTH ORDER SEGMENT OF THE HILBERT MATRIX,
MULTIPLIED WITH 840 TO GET INTEGER ELEMENTS, AND B THE THIRD COLUMN
OF A, THEN THE SOLUTION OF THE LINEAR SYSTEM AX = B IS GIVEN BY
THE THIRD UNIT VECTOR AND THIS SOLUTION, AS WELL AS A REALISTIC
UPPER BOUND FOR THE RELATIVE ERROR IN IT, MAY BE OBTAINED BY THE
FOLLOWING PROGRAM:
"BEGIN" "INTEGER" I, J;
"ARRAY" A[1:4, 1:4], B[1:4], AUX[0:13];
"PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" ITEM(B[I]);
"FOR" I:= 1 "STEP" 2 "UNTIL" 13 "DO" ITEM(AUX[I])
"END" LIST;
"PROCEDURE" LAYOUT;
FORMAT("("*, "("SOLUTION:")"B+.15D"+3D,/,3(10B+.15D"+3D,/),
"("SIGN(DET) = ")"+D,/,"("NUMBER OF ELIMINATIONSTEPS = ")"
+D,/,"("MAX(ABS(A[I,J]))= ")"+.15D"+3D,/,
"("UPPER BOUND GROWTH: ")"+.15D"+3D,/,
"("NORM CALCULATED INVERSE MATRIX: ")"+.15D"+3D,/,
"("UPPER BOUND FOR THE RELATIVE ERROR: ")"+.15D"+3D,/,
"("NORM RESIDUAL VECTOR: ")"+.15D"+3D")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
A[I,J]:= 840 // (I + J - 1); B[I]:= A[I,3]
"END";
AUX[0]:= AUX[2]:= "-14; AUX[4]:= 8; AUX[6]:= AUX[8]:= 0;
AUX[10]:= "-14; AUX[12]:= 5;
GSSITISOLERB(A, 4, AUX, B);
OUTLIST(71, LAYOUT, LIST)
"END"
RESULTS:
SOLUTION: +.000000000000000"+000
+.000000000000000"+000
+.100000000000000"+001
+.000000000000000"+000
SIGN(DET) = +1
NUMBER OF ELIMINATIONSTEPS = +4
MAX(ABS(A[I,J]))= +.840000000000000"+003
UPPER BOUND GROWTH: +.134080000000000"+004
NORM CALCULATED INVERSE MATRIX: +.162142857143540"+002
UPPER BOUND FOR THE RELATIVE ERROR: +.000000000000000"+000
NORM RESIDUAL VECTOR: +.000000000000000"+000
1SECTION: 3.1.1.1.1.1.5 (MAY 1974) PAGE 12
REFERENCES:
[1] BUS, J. C. P.
LINEAR SYSTEMS WITH CALCULATION OF ERROR BOUNDS AND ITERATIVE
REFINEMENT (DUTCH).
MATHEMATICAL CENTRE, AMSTERDAM, LR 3.4.19, (1972).
[2] DEKKER, T. J.
NUMERICAL ALGEBRA (DUTCH).
MATHEMATICAL CENTRE, AMSTERDAM, SYLLABUS 12, (1971).
[3] WILKINSON, J. H.
THE ALGEBRAIC EIGENVALUE PROBLEM.
OXFORD (1965).
SOURCE TEXT(S):
"CODE" 34250;
"PROCEDURE" ITISOL(A, LU, N, AUX, RI, CI, B); "VALUE" N;
"INTEGER" N;
"ARRAY" A, LU, AUX, B; "INTEGER" "ARRAY" RI, CI;
"BEGIN" "INTEGER" I, ITER, MAXITER;
"REAL" MAXERX, ERX, NRMRES, NRMSOL, R, RR;
"ARRAY" RES, SOL[1:N];
MAXERX:= ERX:= AUX[10]; MAXITER:= AUX[12];
INIVEC(1, N, SOL, 0); DUPVEC(1, N, 0, RES, B);
"FOR" ITER:= 1, ITER + 1 "WHILE" ITER <= MAXITER &
MAXERX < ERX "DO"
"BEGIN" SOLELM(LU, N, RI, CI, RES); ERX:= NRMSOL:= NRMRES:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= RES[I]; ERX:= ERX + ABS(R); RR:= SOL[I] + R;
SOL[I]:= RR; NRMSOL:= NRMSOL + ABS(RR)
"END";
ERX:= ERX / NRMSOL;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" LNGMATVEC(1, N, I, A, SOL, - B[I], 0, R, RR);
R:= - (R + RR); RES[I]:= R; NRMRES:= NRMRES + ABS(R)
"END"
"END" ITERATION;
DUPVEC(1, N, 0, B, SOL);
AUX[11]:= ERX; AUX[13]:= NRMRES
"END" ITISOL
1SECTION: 3.1.1.1.1.1.5 (MAY 1974) PAGE 13
;
"EOP"
0"CODE" 34251;
"PROCEDURE" GSSITISOL(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN" "INTEGER" I, J;
"ARRAY" AA[1:N,1:N];
"INTEGER" "ARRAY" RI, CI[1:N];
DUPMAT(1, N, 1, N, AA, A);
GSSELM(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" ITISOL(AA, A, N, AUX, RI, CI, B)
"END" GSSITISOL;
"EOP"
"CODE" 34253;
"PROCEDURE" ITISOLERB(A, LU, N, AUX, RI, CI, B); "VALUE" N;
"INTEGER" N;
"ARRAY" A, LU, AUX, B; "INTEGER" "ARRAY" RI, CI;
"BEGIN" "INTEGER" I;
"REAL" NRMSOL, NRMINV, NRMB, ALFA, TOLA, EPS;
EPS:= AUX[0];
NRMINV:= AUX[9]; TOLA:= AUX[5] * AUX[6]; NRMB:= NRMSOL:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" NRMB:= NRMB + ABS(B[I]);
ITISOL(A, LU, N, AUX, RI, CI, B);
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
NRMSOL:= NRMSOL + ABS(B[I]);
ALFA:= 1 - (1.06 * EPS * AUX[7] * (.75 * N + 4.5) * N ** 2
+ TOLA) * NRMINV;
"IF" ALFA < EPS "THEN" AUX[11]:= - 1 "ELSE"
"BEGIN" ALFA:= ((AUX[13] + AUX[8] * NRMB) / NRMSOL + TOLA) *
NRMINV / ALFA;
AUX[11]:= "IF" 1 - ALFA < EPS "THEN" - 1 "ELSE"
ALFA / (1 - ALFA)
"END"
"END" ITISOLERB;
"EOP"
0"CODE" 34254;
"PROCEDURE" GSSITISOLERB(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN" "INTEGER" I, J;
"ARRAY" AA[1:N,1:N];
"INTEGER" "ARRAY" RI, CI[1:N];
DUPMAT(1, N, 1, N, AA, A);
GSSNRI(A, N, AUX, RI, CI);
"IF" AUX[3] = N "THEN" ITISOLERB(AA, A, N, AUX, RI, CI, B)
"END" GSSITISOLERB;
"EOP"
1SECTION: 3.1.1.1.1.2.1 (MAY 1974) PAGE 1
AUTHOR: T.J. DEKKER.
CONTRIBUTORS: S.P.N. VAN KAMPEN, J. KOK.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731015.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES:
A) CHLDEC2 CALCULATES THE CHOLESKY DECOMPOSITION OF A POSITIVE
DEFINITE SYMMETRIC MATRIX WH0SE UPPER TRIANGLE IS GIVEN IN A TWO-
DIMENSIONAL ARRAY;
B) CHLDEC1 CALCULATES THE CHOLESKY DECOMPOSITION OF A POSITIVE
DEFINITE SYMMETRIC MATRIX WHOSE UPPER TRIANGLE IS GIVEN COLUMNWISE
IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS:
LINEAR EQUATIONS,
POSITIVE DEFINITE SYMMETRIC MATRIX,
CHOLESKY DECOMPOSITION.
SUBSECTION: CHLDEC2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDEC2(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"CODE" 34310;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER TRIANGLE OF THE POSITIVE DEFINITE MATRIX
MUST BE GIVEN IN THE UPPER-TRIANGULAR PART OF A (THE
ELEMENTS A[I,J], I <= J);
EXIT: THE CHOLESKY DECOMPOSITION OF THE MATRIX IS
DELIVERED IN THE UPPER TRIANGLE OF A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS;
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
1SECTION: 3.1.1.1.1.2.1 (MAY 1974) PAGE 2
PROCEDURES USED: TAMMAT = CP34014.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
CHLDEC2 PERFORMS THE CHOLESKY DECOMPOSITION OF A SYMMETRIC
POSITIVE DEFINITE MATRIX.
THE METHOD USED IS CHOLESKY'S SQUARE ROOT METHOD WITHOUT
PIVOTING (SEE REF[1] AND [2]). IF THE GIVEN SYMMETRIC MATRIX IS
POSITIVE DEFINITE, THE METHOD YIELDS AN UPPER-TRIANGULAR MATRIX U
SUCH THAT U'U EQUALS THE GIVEN MATRIX.
THE PROCESS IS TERMINATED AT STAGE K, IF THE K-TH DIAGONAL ELEMENT
OF THE GIVEN MATRIX MINUS THE SUM OF THE SQUARED ELEMENTS OF THE
K-TH COLUMN OF U IS LESS THAN A TOLERANCE (AUX[2]) TIMES THE
MAXIMAL DIAGONAL ELEMENT OF THE GIVEN MATRIX. IN THIS CASE THE
MATRIX, POSSIBLY MODIFIED BY ROUNDING ERRORS, IS NOT POSITIVE
DEFINITE.
REFERENCES:
[1]. T.J. DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MC TRACT 22, 1968, MATH. CENTR., AMSTERDAM.
[2]. J.H. WILKINSON.
THE ALGEBRAIC EIGENVALUE PROBLEM.
CLARENDON PRESS, OXFORD, 1965.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLINV2, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.1 (MAY 1974) PAGE 3
SUBSECTION: CHLDEC1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDEC1(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"CODE" 34311;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1 : (N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE POSITIVE DEFINITE
SYMMETRIC MATRIX MUST BE GIVEN COLUMNWISE IN ARRAY A
(THE (I,J)-TH ELEMENT OF THE MATRIX MUST BE GIVEN IN
A[(J - 1) * J // 2 + I] FOR 1 <= I <= J <= N);
EXIT: THE CHOLESKY DECOMPOSITION OF THE MATRIX IS
DELIVERED COLUMNWISE IN A.
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS;
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
PROCEDURES USED: VECVEC = CP34010.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
CHLDEC1 PERFORMS THE CHOLESKY DECOMPOSITION OF A SYMMETRIC
POSITIVE DEFINITE MATRIX, WHOSE UPPER TRIANGLE IS STORED IN A ONE-
DIMENSIONAL ARRAY, BY CHOLESKY'S SQUARE ROOT METHOD WITHOUT
PIVOTING.
SEE ALSO METHOD AND PERFORMANCE OF CHLDEC2, (THIS SECTION).
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLINV1, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.1 (DECEMBER 1975) PAGE 4
SOURCE TEXT(S) :
"CODE" 34310;
"PROCEDURE" CHLDEC2(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" K, J; "REAL" R, EPSNORM;
R:= 0;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"IF" A[K,K] > R "THEN" R:= A[K,K];
EPSNORM:= AUX[2] * R;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= A[K,K] - TAMMAT(1, K - 1, K, K, A, A);
"IF" R <= EPSNORM "THEN"
"BEGIN" AUX[3]:= K - 1; "GOTO" END "END";
A[K,K]:= R:= SQRT(R);
"FOR" J:= K + 1 "STEP" 1 "UNTIL" N "DO"
A[K,J]:= (A[K,J] - TAMMAT(1, K - 1, J, K, A, A)) / R
"END";
AUX[3]:= N;
END:
"END" CHLDEC2;
"EOP"
"CODE" 34311;
"PROCEDURE" CHLDEC1(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" J, K, KK, KJ, LOW, UP; "REAL" R, EPSNORM;
R:= 0; KK:= 0;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + K; "IF" A[KK] > R "THEN" R:= A[KK] "END";
EPSNORM:= AUX[2] * R; KK:= 0;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + K; LOW:= KK - K + 1; UP:= KK - 1;
R:= A[KK] - VECVEC(LOW, UP, 0, A, A);
"IF" R <= EPSNORM "THEN"
"BEGIN" AUX[3]:= K - 1; "GOTO" END "END";
A[KK]:= R:= SQRT(R); KJ:= KK + K;
"FOR" J:= K + 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" A[KJ]:= (A[KJ] -
VECVEC(LOW, UP, KJ - KK, A, A)) / R;
KJ:= KJ + J
"END"
"END";
AUX[3]:= N;
END:
"END" CHLDEC1;
"EOP"
1SECTION: 3.1.1.1.1.2.2 (FEBRUARY 1979) PAGE 1
CONTRIBUTORS: S.P.N. VAN KAMPEN, J. KOK.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731015.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES:
CHLDETERM2, FOR THE CALCULATION OF THE DETERMINANT OF A
SYMMETRIC POSITIVE DEFINITE MATRIX WHOSE CHOLESKY MATRIX IS GIVEN
IN THE UPPER TRIANGLE OF A TWO-DIMENSIONAL ARRAY;
CHLDETERM1, FOR THE CALCULATION OF THE DETERMINANT OF A
SYMMETRIC POSITIVE DEFINITE MATRIX WHOSE CHOLESKY MATRIX IS GIVEN
COLUMNWISE IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS:
DETERMINANT,
POSITIVE DEFINITE SYMMETRIC MATRIX,
CHOLESKY DECOMPOSITION.
SUBSECTION: CHLDETERM2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"REAL" "PROCEDURE" CHLDETERM2(A, N); "VALUE" N; "INTEGER" N;
"ARRAY" A; "CODE" 34312;
CHLDETERM2 := THE DETERMINANT OF THE SYMMETRIC POSITIVE
DEFINITE MATRIX OF WHICH THE CHOLESKY MATRIX IS STORED IN A;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC2 (SECTION 3.1.1.1.1.2.1) OR
CHLDECSOL2 (SECTION 3.1.1.1.1.2.3) MUST BE GIVEN
IN THE UPPER TRIANGLE OF A;
EXIT: THE CONTENTS OF A ARE NOT CHANGED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX.
1SECTION: 3.1.1.1.1.2.2 (FEBRUARY 1979) PAGE 2
PROCEDURES USED: NONE.
REQUIRED CENTRAL MEMORY: NO EXTRA ARRAYS ARE DECLARED.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDETERM2 SHOULD BE CALLED AFTER A SUCCESSFUL CALL
OF CHLDEC2 OR CHLDECSOL2, I.E. IF AUX[3] = N;
CHLDETERM2 SHOULD NOT BE CALLED IF OVERFLOW IS TO BE EXPECTED.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLDECINV2 (SECTION 3.1.1.1.1.2.4)
SUBSECTION: CHLDETERM1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"REAL" "PROCEDURE" CHLDETERM1(A, N); "VALUE" N; "INTEGER" N;
"ARRAY" A; "CODE" 34313;
CHLDETERM1 := THE DETERMINANT OF THE SYMMETRIC POSITIVE
DEFINITE MATRIX OF WHICH THE CHOLESKY MATRIX IS STORED IN A;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1 : (N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC1 (SECTION 3.1.1.1.1.2.1) OR
CHLDECSOL1 (SECTION 3.1.1.1.1.2.3) MUST BE GIVEN
COLUMNWISE IN ARRAY A;
EXIT: THE CONTENTS OF A ARE NOT CHANGED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX.
1SECTION: 3.1.1.1.1.2.2 (FEBRUARY 1979) PAGE 3
PROCEDURES USED: NONE.
REQUIRED CENTRAL MEMORY: NO EXTRA ARRAYS ARE DECLARED.
RUNNING TIME: PROPORTIONAL TO N.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDETERM1 SHOULD BE CALLED AFTER A SUCCESSFUL CALL
OF CHLDEC1 OR CHLDECSOL1, I.E. IF AUX[3] = N;
CHLDETERM1 SHOULD NOT BE CALLED IF OVERFLOW IS TO BE EXPECTED.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLDECINV1, SECTION 3.1.1.1.1.2.4.
SOURCE TEXT(S) :
"CODE" 34312;
"REAL" "PROCEDURE" CHLDETERM2(A, N); "VALUE" N; "INTEGER" N;
"ARRAY" A;
"BEGIN" "INTEGER" K; "REAL" D;
D:= 1;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO" D:= A[K,K] * D;
CHLDETERM2:= D * D
"END" CHLDETERM2;
"EOP"
"CODE" 34313;
"REAL" "PROCEDURE" CHLDETERM1(A, N); "VALUE" N; "INTEGER" N;
"ARRAY" A;
"BEGIN" "INTEGER" K, KK; "REAL" D;
D:= 1; KK:= 0;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + K; D:= A[KK] * D "END";
CHLDETERM1:= D * D
"END" CHLDETERM1;
"EOP"
1SECTION: 3.1.1.1.1.2.3 (MAY 1974) PAGE 1
AUTHOR: T.J. DEKKER.
CONTRIBUTORS: S.P.N. VAN KAMPEN, J. KOK.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731015.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOUR PROCEDURES:
A) CHLSOL2, FOR THE SOLUTION OF A SYSTEM OF LINEAR EQUATIONS IF THE
COEFFICIENT MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE PROCEDURE
CHLDEC2, SECTION 3.1.1.1.1.2.1., OR CHLDECSOL2;
B) CHLSOL1, FOR THE SOLUTION OF A SYSTEM OF LINEAR EQUATIONS IF THE
COEFFICIENT MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE PROCEDURE
CHLDEC1, SECTION 3.1.1.1.1.2.1., OR CHLDECSOL1;
C) CHLDECSOL2, FOR THE SOLUTION OF A SYSTEM OF LINEAR EQUATIONS BY
CHOLESKY'S SQUARE ROOT METHOD;
THE COEFFICIENT MATRIX HAS TO BE SYMMETRIC POSITIVE DEFINITE AND
MUST BE GIVEN IN THE UPPER TRIANGLE OF A TWO-DIMENSIONAL ARRAY;
D) CHLDECSOL1, FOR THE SOLUTION OF A SYSTEM OF LINEAR EQUATIONS BY
CHOLESKY'S SQUARE ROOT METHOD;
THE COEFFICIENT MATRIX HAS TO BE SYMMETRIC POSITIVE DEFINITE AND
MUST BE GIVEN COLUMNWISE IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS:
LINEAR EQUATIONS,
POSITIVE DEFINITE SYMMETRIC MATRIX,
CHOLESKY DECOMPOSITION.
1SECTION: 3.1.1.1.1.2.3 (MAY 1974) PAGE 2
SUBSECTION: CHLSOL2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLSOL2(A, N, B); "VALUE" N; "INTEGER" N; "ARRAY" A, B;
"CODE" 34390;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC2, SECTION 3.1.1.1.1.2.1., OR
CHLDECSOL2 (THIS SECTION), MUST BE GIVEN IN THE
UPPER TRIANGLE OF A;
EXIT: THE CONTENTS OF A ARE NOT CHANGED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT: THE SOLUTION OF THE SYSTEM.
PROCEDURES USED:
MATVEC = CP34011,
TAMVEC = CP34012.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLSOL2 CALCULATES THE SOLUTION OF A SYSTEM OF LINEAR
EQUATIONS, PROVIDED THAT THE COEFFICIENT MATRIX HAS BEEN DECOMPOSED
BY A SUCCESSFUL CALL OF CHLDEC2 OR CHLDECSOL2;
THE SOLUTION IS OBTAINED BY CARRYING OUT THE FORWARD AND BACK
SUBSTITUTION WITH THE CHOLESKY MATRIX AND THE RIGHT HAND SIDE.
THE RIGHT HAND SIDE IS OVERWRITTEN BY THE SOLUTION BUT THE
ELEMENTS OF THE CHOLESKY MATRIX ARE NOT CHANGED, THUS SEVERAL
SYSTEMS OF LINEAR EQUATIONS WITH THE SAME COEFFICIENT MATRIX BUT
DIFFERENT RIGHT HAND SIDES CAN BE SOLVED BY SUCCESSIVE CALLS OF
CHLSOL2. SEE ALSO REF [1].
REFERENCES:
[1]. T.J. DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MC TRACT 22, 1968, MATH. CENTR., AMSTERDAM.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLINV2, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.3 (DECEMBER 1975) PAGE 3
SUBSECTION: CHLSOL1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLSOL1(A, N, B); "VALUE" N; "INTEGER" N; "ARRAY" A, B;
"CODE" 34391;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1 : (N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC1, SECTION 3.1.1.1.1.2.1., OR
CHLDECSOL1 (THIS SECTION), MUST BE GIVEN COLUMNWISE
IN ARRAY A;
EXIT: THE CONTENTS OF A ARE NOT CHANGED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT: THE SOLUTION OF THE SYSTEM.
PROCEDURES USED:
VECVEC = CP34010,
SEQVEC = CP34016.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLSOL1 CALCULATES THE SOLUTION OF A SYSTEM OF LINEAR
EQUATIONS, PROVIDED THAT THE COEFFICIENT MATRIX HAS BEEN DECOMPOSED
BY A SUCCESSFUL CALL OF CHLDEC1 OR CHLDECSOL1;
SEVERAL SYSTEMS WITH THE SAME COEFFICIENT MATRIX BUT DIFFERENT
RIGHT HAND SIDES CAN BE SOLVED BY SUCCESSIVE CALLS OF CHLSOL1.
SEE ALSO METHOD AND PERFORMANCE OF CHLSOL2 (THIS SECTION).
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLINV1, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.3 (DECEMBER 1975) PAGE 4
SUBSECTION: CHLDECSOL2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDECSOL2(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"CODE" 34392;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER TRIANGLE OF THE POSITIVE DEFINITE MATRIX
MUST BE GIVEN IN THE UPPER-TRIANGULAR PART OF A (THE
ELEMENTS A[I,J], I <= J);
EXIT: THE CHOLESKY DECOMPOSITION OF THE MATRIX IS
DELIVERED IN THE UPPER TRIANGLE OF A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS; (SEE METHOD
AND PERFORMANCE OF CHLDEC2, SECTION 3.1.1.1.1.2.1);
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT: THE SOLUTION OF THE SYSTEM.
PROCEDURES USED:
CHLDEC2 = CP34310,
CHLSOL2 = CP34390.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDECSOL2 SOLVES A SYSTEM OF LINEAR EQUATIONS WITH
A SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX BY CALLING
CHLDEC2, SECTION 3.1.1.1.1.2.1., AND, IF THIS CALL WAS
SUCCESSFUL, CHLSOL2 (THIS SECTION).
SEE ALSO CHLDEC2, SECTION 3.1.1.1.1.2.1.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLDECINV2, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.3 (DECEMBER 1975) PAGE 5
SUBSECTION: CHLDECSOL1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDECSOL1(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"CODE" 34393;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1 : (N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE POSITIVE DEFINITE
SYMMETRIC MATRIX MUST BE GIVEN COLUMNWISE IN ARRAY A
(THE (I,J)-TH ELEMENT OF THE MATRIX MUST BE GIVEN IN
A[(J - 1) * J // 2 + I] FOR 1 <= I <= J <= N);
EXIT: THE CHOLESKY DECOMPOSITION OF THE MATRIX IS
DELIVERED COLUMNWISE IN A.
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS;
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT: THE SOLUTION OF THE SYSTEM.
PROCEDURES USED:
CHLDEC1 = CP34311,
CHLSOL1 = CP34391.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDECSOL1 SOLVES A SYSTEM OF LINEAR EQUATIONS WITH
A SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX BY CALLING
CHLDEC1, SECTION 3.1.1.1.1.2.1., AND, IF THIS CALL WAS
SUCCESSFUL, CHLSOL1 (THIS SECTION).
THE UPPER TRIANGLE OF THE COEFFICIENT MATRIX MUST BE STORED COLUMN-
WISE IN A ONE-DIMENSIONAL ARRAY.
SEE ALSO CHLDEC1, SECTION 3.1.1.1.1.2.1.
EXAMPLE OF USE:
SEE EXAMPLE OF USE OF CHLDECINV1, SECTION 3.1.1.1.1.2.4.
1SECTION: 3.1.1.1.1.2.3 (MAY 1974) PAGE 6
SOURCE TEXT(S) :
"CODE" 34390;
"PROCEDURE" CHLSOL2(A, N, B); "VALUE" N; "INTEGER" N; "ARRAY" A, B;
"BEGIN" "INTEGER" I;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
B[I]:= (B[I] - TAMVEC(1, I - 1, I, A, B)) / A[I,I];
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
B[I]:= (B[I] - MATVEC(I + 1, N, I, A, B)) / A[I,I]
"END" CHLSOL2;
"EOP"
"CODE" 34391;
"PROCEDURE" CHLSOL1(A, N, B); "VALUE" N; "INTEGER" N; "ARRAY" A, B;
"BEGIN" "INTEGER" I, II;
II:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" II:= II + I;
B[I]:= (B[I] - VECVEC(1, I - 1, II - I, B, A)) / A[II]
"END";
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" B[I]:= (B[I] -
SEQVEC(I + 1, N, II + I, 0, A, B)) / A[II];
II:= II - I
"END"
"END" CHLSOL1;
"EOP"
"CODE" 34392;
"PROCEDURE" CHLDECSOL2(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN"
CHLDEC2(A, N, AUX);
"IF" AUX[3] = N "THEN" CHLSOL2(A, N, B)
"END" CHLDECSOL2;
"EOP"
"CODE" 34393;
"PROCEDURE" CHLDECSOL1(A, N, AUX, B); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX, B;
"BEGIN"
CHLDEC1(A, N, AUX);
"IF" AUX[3] = N "THEN" CHLSOL1(A, N, B)
"END" CHLDECSOL1;
"EOP"
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 1
AUTHOR : T.J. DEKKER.
CONTRIBUTORS: S.P.N. VAN KAMPEN, J. KOK.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731015.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOUR PROCEDURES:
A) CHLINV2, FOR THE INVERSION OF A SYMMETRIC POSITIVE DEFINITE
MATRIX, IF THE MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE
PROCEDURE CHLDEC2, SECTION 3.1.1.1.1.2.1., OR CHLDECSOL2,
SECTION 3.1.1.1.1.2.3.;
B) CHLINV1, FOR THE INVERSION OF A SYMMETRIC POSITIVE DEFINITE
MATRIX, IF THE MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE
PROCEDURE CHLDEC1, SECTION 3.1.1.1.1.2.1., OR CHLDECSOL1,
SECTION 3.1.1.1.1.2.3.;
C) CHLDECINV2, FOR THE INVERSION OF A MATRIX BY CHOLESKY'S SQUARE
ROOT METHOD;
THE COEFFICIENT MATRIX HAS TO BE SYMMETRIC POSITIVE DEFINITE AND
MUST BE GIVEN IN THE UPPER TRIANGLE OF A TWO-DIMENSIONAL ARRAY;
D) CHLDECINV1, FOR THE INVERSION OF A MATRIX BY CHOLESKY'S SQUARE
ROOT METHOD;
THE COEFFICIENT MATRIX HAS TO BE SYMMETRIC POSITIVE DEFINITE AND
MUST BE GIVEN COLUMNWISE IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS:
MATRIX INVERSION,
POSITIVE DEFINITE SYMMETRIC MATRIX,
CHOLESKY DECOMPOSITION.
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 2
SUBSECTION: CHLINV2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLINV2(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 34400;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC2, SECTION 3.1.1.1.1.2.1., OR
CHLDECSOL2, SECTION 3.1.1.1.1.2.3., MUST BE GIVEN
IN THE UPPER TRIANGLE OF A;
EXIT: THE UPPER-TRIANGULAR PART OF THE INVERSE MATRIX IS
DELIVERED IN THE UPPER TRIANGLE OF A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX.
PROCEDURES USED:
MATVEC = CP34011,
TAMVEC = CP34012,
DUPVECROW = CP31031.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: N.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLINV2 CALCULATES THE INVERSE OF A MATRIX, PROVIDED
THAT THE MATRIX HAS BEEN DECOMPOSED BY A SUCCESSFUL CALL OF CHLDEC2
OR CHLDECSOL2;
THE INVERSE, X, OF U'U, WHERE U IS THE CHOLESKY MATRIX,
IS OBTAINED FROM THE CONDITIONS THAT X BE SYMMETRIC AND UX BE
A LOWER-TRIANGULAR MATRIX WHOSE MAIN DIAGONAL ELEMENTS ARE THE
RECIPROCALS OF THE DIAGONAL ELEMENTS OF U. HEREWITH THE UPPER-
TRIANGULAR ELEMENTS OF X ARE CALCULATED BY BACK SUBSTITUTION.
THE UPPER TRIANGLE OF THE INVERSE MATRIX IS DELIVERED IN THE UPPER
TRIANGLE OF THE GIVEN ARRAY. SEE ALSO REF[1].
REFERENCES:
[1]. T.J. DEKKER.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1.
MC TRACT 22, 1968, MATH. CENTR., AMSTERDAM.
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 3
EXAMPLE OF USE:
THE SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX (THE PASCAL
MATRIX OF ORDER 4) OF THE SYSTEM OF EQUATIONS
X1 + X2 + X3 + X4 = 2
X1 + 2 * X2 + 3 * X3 + 4 * X4 = 4
X1 + 3 * X2 + 6 * X3 + 10 * X4 = 8
X1 + 4 * X2 + 10 * X3 + 20 * X4 = 16
IS STORED IN THE TWO-DIMENSIONAL ARRAY PASCAL2.
THE INVERSE OF THE COEFFICIENT MATRIX AND THE SOLUTION OF THE
LINEAR SYSTEM ARE CALCULATED BY THE FOLLOWING PROGRAM:
"BEGIN" "COMMENT" TEST CHLDEC2, CHLSOL2 AND CHLINV2;
"INTEGER" I, J;
"ARRAY" PASCAL2[1:4,1:4], B[1:4], AUX[2:3];
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL2[1,J]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL2[I,J]:= "IF" I = J "THEN" PASCAL2[I-1,J] * 2 "ELSE"
PASCAL2[I,J-1] + PASCAL2[I-1,J];
B[J]:= 2 ** J
"END";
AUX[2]:= "-11;
CHLDEC2(PASCAL2, 4, AUX);
"IF" AUX[3] = 4 "THEN"
"BEGIN" CHLSOL2(PASCAL2, 4, B); CHLINV2(PASCAL2, 4) "END"
"ELSE" OUTPUT(61, "(""("MATRIX NOT POSITIVE DEFINITE")", /")");
OUTPUT(61, "("4B")");
OUTPUT(61, "(""("SOLUTION WITH CHLDEC2 AND CHLSOL2:")", /")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
OUTPUT(61, "("4B+D.5D")", B[I]);
OUTPUT(61, "("//, 4B")");
OUTPUT(61, "(""("INVERSE MATRIX WITH CHLINV2:")", /, 4B")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"IF" J < I "THEN" OUTPUT(61, "("12B")") "ELSE"
OUTPUT(61, "("+ZD.5D3B")", PASCAL2[I,J]);
OUTPUT(61, "("/, 4B")")
"END"
"END"
THIS PROGRAM DELIVERS:
SOLUTION WITH CHLDEC2 AND CHLSOL2:
+0.00000 +4.00000 -4.00000 +2.00000
INVERSE MATRIX WITH CHLINV2:
+4.00000 -6.00000 +4.00000 -1.00000
+14.00000 -11.00000 +3.00000
+10.00000 -3.00000
+1.00000
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 4
SUBSECTION: CHLINV1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLINV1(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 34401;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:(N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE CHOLESKY MATRIX
AS PRODUCED BY CHLDEC1, SECTION 3.1.1.1.1.2.1., OR
CHLDECSOL1, SECTION 3.1.1.1.1.2.3., MUST BE GIVEN
COLUMNWISE IN ARRAY A;
EXIT: THE UPPER-TRIANGULAR PART OF THE INVERSE MATRIX IS
DELIVERED COLUMNWISE IN ARRAY A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX.
PROCEDURES USED:
SEQVEC = CP34016,
SYMMATVEC = CP34018.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: N.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLINV1 CALCULATES THE INVERSE OF A MATRIX, PROVIDED
THAT THE MATRIX HAS BEEN DECOMPOSED BY A SUCCESSFUL CALL OF CHLDEC1
OR CHLDECSOL1;
THE UPPER TRIANGLE OF THE INVERSE MATRIX IS DELIVERED COLUMNWISE
IN THE ONE-DIMENSIONAL ARRAY.
SEE ALSO METHOD AND PERFORMANCE OF CHLINV2 (THIS SECTION).
EXAMPLE OF USE:
THE SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX (THE PASCAL
MATRIX OF ORDER 4) OF THE SYSTEM OF EQUATIONS
X1 + X2 + X3 + X4 = 2
X1 + 2 * X2 + 3 * X3 + 4 * X4 = 4
X1 + 3 * X2 + 6 * X3 + 10 * X4 = 8
X1 + 4 * X2 + 10 * X3 + 20 * X4 = 16
IS STORED IN THE ONE-DIMENSIONAL ARRAY PASCAL1.
THE INVERSE OF THE COEFFICIENT MATRIX AND THE SOLUTION OF THE
LINEAR SYSTEM ARE CALCULATED BY THE FOLLOWING PROGRAM:
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 5
"BEGIN" "COMMENT" TEST CHLDEC1, CHLSOL1 AND CHLINV1;
"INTEGER" I, J, JJ;
"ARRAY" PASCAL1[1:(4 + 1) * 4 // 2], B[1:4], AUX[2:3];
JJ:= 1;
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL1[JJ]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL1[JJ + I - 1]:= "IF" I = J "THEN"
PASCAL1[JJ + I - 2] * 2 "ELSE"
PASCAL1[JJ + I - 2] + PASCAL1[JJ + I - J];
B[J]:= 2 ** J;
JJ:= JJ + J
"END";
AUX[2]:= "-11;
CHLDEC1(PASCAL1, 4, AUX);
"IF" AUX[3] = 4 "THEN"
"BEGIN" CHLSOL1(PASCAL1, 4, B); CHLINV1(PASCAL1, 4) "END"
"ELSE" OUTPUT(61, "(""("MATRIX NOT POSITIVE DEFINITE")", /")");
OUTPUT(61,"("4B")");
OUTPUT(61, "(""("SOLUTION WITH CHLDEC1 AND CHLSOL1:")", /")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
OUTPUT(61, "("4B+D.5D")", B[I]);
OUTPUT(61, "("2/, 4B")");
OUTPUT(61, "(""("INVERSE MATRIX WITH CHLINV1:")", /, 4B")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"IF" J < I "THEN" OUTPUT(61, "("12B")") "ELSE"
OUTPUT(61, "("+ZD.5D3B")", PASCAL1[(J - 1) * J // 2 + I]);
OUTPUT(61, "("/, 4B")")
"END"
"END"
THIS PROGRAM DELIVERS:
SOLUTION WITH CHLDEC1 AND CHLSOL1:
+0.00000 +4.00000 -4.00000 +2.00000
INVERSE MATRIX WITH CHLINV1:
+4.00000 -6.00000 +4.00000 -1.00000
+14.00000 -11.00000 +3.00000
+10.00000 -3.00000
+1.00000
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 6
SUBSECTION: CHLDECINV2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDECINV2(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"CODE" 34402;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY: THE UPPER TRIANGLE OF THE POSITIVE DEFINITE MATRIX
MUST BE GIVEN IN THE UPPER TRIANGLE OF A (THE
ELEMENTS A[I,J], I <= J);
EXIT: THE UPPER-TRIANGULAR PART OF THE INVERSE MATRIX IS
DELIVERED IN THE UPPER TRIANGLE OF A.
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS;
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
PROCEDURES USED:
CHLDEC2 = CP34310,
CHLINV2 = CP34400.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDECINV2 CALCULATES THE INVERSE OF A SYMMETRIC
POSITIVE DEFINITE MATRIX BY CALLING CHLDEC2 AND, IF THIS CALL WAS
SUCCESSFUL, CHLINV2.
THE UPPER TRIANGLE OF THE INVERSE MATRIX IS DELIVERED IN THE UPPER
TRIANGLE OF THE GIVEN ARRAY.
SEE ALSO METHOD AND PERFORMANCE OF CHLINV2 (THIS SECTION) AND
CHLDEC2, SECTION 3.1.1.1.1.2.1.
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 7
EXAMPLE OF USE:
THE SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX (THE PASCAL
MATRIX OF ORDER 4) OF THE SYSTEM OF EQUATIONS
X1 + X2 + X3 + X4 = 2
X1 + 2 * X2 + 3 * X3 + 4 * X4 = 4
X1 + 3 * X2 + 6 * X3 + 10 * X4 = 8
X1 + 4 * X2 + 10 * X3 + 20 * X4 = 16
IS STORED IN THE TWO-DIMENSIONAL ARRAY PASCAL2.
THE DETERMINANT AND THE INVERSE OF THE COEFFICIENT MATRIX AND THE
SOLUTION OF THE LINEAR SYSTEM ARE CALCULATED BY THE FOLLOWING
PROGRAM:
"BEGIN" "COMMENT" TEST CHLDECSOL2, CHLDETERM2 AND CHLDECINV2;
"INTEGER" I, J;
"ARRAY" PASCAL2[1:4,1:4], B[1:4], AUX[2:3];
"REAL" DETERMINANT;
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL2[1,J]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL2[I,J]:= "IF" I = J "THEN" PASCAL2[I-1,J] * 2 "ELSE"
PASCAL2[I,J-1] + PASCAL2[I-1,J];
B[J]:= 2 ** J
"END";
AUX[2]:= "-11;
CHLDECSOL2(PASCAL2, 4, AUX, B);
"IF" AUX[3] = 4 "THEN" DETERMINANT:= CHLDETERM2(PASCAL2, 4)
"ELSE" OUTPUT(61, "(""("MATRIX NOT POSITIVE DEFINITE")", /")");
OUTPUT(61, "("4B")");
OUTPUT(61, "(""("SOLUTION WITH CHLDECSOL2:")", /")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
OUTPUT(61, "("4B+D.5D")", B[I]);
OUTPUT(61, "("//, 4B, "("DETERMINANT WITH CHLDETERM2: ")",
+D.5D, 2/, 4B")", DETERMINANT);
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL2[1,J]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL2[I,J]:= "IF" I = J "THEN" PASCAL2[I-1,J] * 2 "ELSE"
PASCAL2[I,J-1] + PASCAL2[I-1,J]
"END";
CHLDECINV2(PASCAL2, 4, AUX);
OUTPUT(61, "(""("INVERSE MATRIX WITH CHLDECINV2:")", /, 4B")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"IF" J < I "THEN" OUTPUT(61, "("12B")") "ELSE"
OUTPUT(61, "("+ZD.5D3B")", PASCAL2[I,J]);
OUTPUT(61, "("/, 4B")")
"END"
"END"
1SECTION: 3.1.1.1.1.2.4 (DECEMBER 1975) PAGE 8
THIS PROGRAM DELIVERS:
SOLUTION WITH CHLDECSOL2:
+0.00000 +4.00000 -4.00000 +2.00000
DETERMINANT WITH CHLDETERM2: +1.00000
INVERSE MATRIX WITH CHLDECINV2:
+4.00000 -6.00000 +4.00000 -1.00000
+14.00000 -11.00000 +3.00000
+10.00000 -3.00000
+1.00000
SUBSECTION: CHLDECINV1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" CHLDECINV1(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"CODE" 34403;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:(N + 1) * N // 2];
ENTRY: THE UPPER-TRIANGULAR PART OF THE SYMMETRIC POSITIVE
DEFINITE MATRIX MUST BE GIVEN COLUMNWISE IN ARRAY A
(THE (I,J)-TH ELEMENT OF THE MATRIX MUST BE GIVEN IN
A[(J - 1) * J // 2 + I] FOR 1 <= I <= J <= N);
EXIT: THE UPPER-TRIANGULAR PART OF THE INVERSE MATRIX IS
DELIVERED COLUMNWISE IN ARRAY A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:3];
ENTRY: AUX[2]: A RELATIVE TOLERANCE USED TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS; (SEE METHOD
AND PERFORMANCE OF CHLDEC2, SECTION 3.1.1.1.1.2.1);
NORMAL EXIT: AUX[3]:= N;
ABNORMAL EXIT: IF THE DECOMPOSITION CANNOT BE CARRIED OUT
BECAUSE THE MATRIX IS (NUMERICALLY) NOT POSITIVE
DEFINITE, AUX[3]:= K - 1, WHERE K IS THE LAST STAGE
NUMBER.
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 9
PROCEDURES USED:
CHLDEC1 = CP34311,
CHLINV1 = CP34401.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE PROCEDURE CHLDECINV1 CALCULATES THE INVERSE OF A SYMMETRIC
POSITIVE DEFINITE MATRIX BY CALLING CHLDEC1 AND, IF THIS CALL WAS
SUCCESSFUL, CHLINV1.
THE UPPER TRIANGLE OF THE INVERSE MATRIX IS DELIVERED COLUMNWISE IN
THE GIVEN ONE-DIMENSIONAL ARRAY.
SEE ALSO METHOD AND PERFORMANCE OF CHLINV2, (THIS SECTION) AND
CHLDEC1, SECTION 3.1.1.1.1.2.1.
EXAMPLE OF USE:
THE SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX (THE PASCAL
MATRIX OF ORDER 4) OF THE SYSTEM OF EQUATIONS
X1 + X2 + X3 + X4 = 2
X1 + 2 * X2 + 3 * X3 + 4 * X4 = 4
X1 + 3 * X2 + 6 * X3 + 10 * X4 = 8
X1 + 4 * X2 + 10 * X3 + 20 * X4 = 16
IS STORED IN THE ONE-DIMENSIONAL ARRAY PASCAL1.
THE DETERMINANT AND THE INVERSE OF THE COEFFICIENT MATRIX AND THE
SOLUTION OF THE LINEAR SYSTEM ARE CALCULATED BY THE FOLLOWING
PROGRAM:
1SECTION: 3.1.1.1.1.2.4 (DECEMBER 1975) PAGE 10
"BEGIN" "COMMENT" TEST CHLDECSOL1, CHLDETERM1 AND CHLDECINV1;
"INTEGER" I, J, JJ;
"ARRAY" PASCAL1[1:(4 + 1) * 4 // 2], B[1:4], AUX[2:3];
"REAL" DETERMINANT;
JJ:= 1;
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL1[JJ]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL1[JJ + I - 1]:= "IF" I = J "THEN"
PASCAL1[JJ + I - 2] * 2 "ELSE"
PASCAL1[JJ + I - 2] + PASCAL1[JJ + I - J];
B[J]:= 2 ** J;
JJ:= JJ + J
"END";
AUX[2]:= "-11;
CHLDECSOL1(PASCAL1, 4, AUX, B);
"IF" AUX[3] = 4 "THEN" DETERMINANT:= CHLDETERM1(PASCAL1, 4)
"ELSE" OUTPUT(61, "(""("MATRIX NOT POSITIVE DEFINITE")", /")");
OUTPUT(61, "("4B")");
OUTPUT(61, "(""("SOLUTION WITH CHLDECSOL1:")", /")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
OUTPUT(61, "("4B+D.5D")", B[I]);
OUTPUT(61, "("//, 4B, "("DETERMINANT WITH CHLDETERM1: ")",
+D.5D, 2/, 4B")", DETERMINANT);
JJ:= 1;
"FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" PASCAL1[JJ]:= 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" J "DO"
PASCAL1[JJ + I - 1]:= "IF" I = J "THEN"
PASCAL1[JJ + I - 2] * 2 "ELSE"
PASCAL1[JJ + I - 2] + PASCAL1[JJ + I - J];
JJ:= JJ + J
"END";
CHLDECINV1(PASCAL1, 4, AUX);
OUTPUT(61, "(""("INVERSE MATRIX WITH CHLDECINV1:")", /, 4B")");
"FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" 4 "DO"
"IF" J < I "THEN" OUTPUT(61, "("12B")") "ELSE"
OUTPUT(61, "("+ZD.5D3B")", PASCAL1[(J - 1) * J // 2 + I]);
OUTPUT(61, "("/, 4B")")
"END"
"END"
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 11
THIS PROGRAM DELIVERS:
SOLUTION WITH CHLDECSOL1:
+0.00000 +4.00000 -4.00000 +2.00000
DETERMINANT WITH CHLDETERM1: +1.00000
INVERSE MATRIX WITH CHLDECINV1:
+4.00000 -6.00000 +4.00000 -1.00000
+14.00000 -11.00000 +3.00000
+10.00000 -3.00000
+1.00000
0SOURCE TEXT(S) :
"CODE" 34400;
"PROCEDURE" CHLINV2(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"BEGIN" "REAL" R; "INTEGER" I, J, I1;
"ARRAY" U[1:N];
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" R:= 1 / A[I,I]; I1:= I + 1;
DUPVECROW(I1, N, I, U, A);
"FOR" J:= N "STEP" - 1 "UNTIL" I1 "DO" A[I,J]:=
- (TAMVEC(I1, J, J, A, U) + MATVEC(J + 1, N, J, A, U)) * R;
A[I,I]:= (R - MATVEC(I1, N, I, A, U)) * R
"END"
"END" CHLINV2;
"EOP"
"CODE" 34401;
"PROCEDURE" CHLINV1(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"BEGIN" "INTEGER" I, II, I1, J, IJ; "REAL" R;
"ARRAY" U[1:N];
II:= (N + 1) * N // 2;
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" R:= 1 / A[II]; I1:= I + 1; IJ:= II + I;
"FOR" J:= I1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" U[J]:= A[IJ]; IJ:= IJ + J "END";
"FOR" J:= N "STEP" - 1 "UNTIL" I1 "DO"
"BEGIN" IJ:= IJ - J; A[IJ]:= -SYMMATVEC(I1, N, J, A, U) * R
"END";
A[II]:= (R - SEQVEC(I1, N, II + I, 0, A, U)) * R;
II:= II - I
"END"
"END" CHLINV1
1SECTION: 3.1.1.1.1.2.4 (MAY 1974) PAGE 12
;
"EOP"
"CODE" 34402;
"PROCEDURE" CHLDECINV2(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN"
CHLDEC2(A, N, AUX);
"IF" AUX[3] = N "THEN" CHLINV2(A, N)
"END" CHLDECINV2;
"EOP"
"CODE" 34403;
"PROCEDURE" CHLDECINV1(A, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" A, AUX;
"BEGIN"
CHLDEC1(A, N, AUX);
"IF" AUX[3] = N "THEN" CHLINV1(A, N)
"END" CHLDECINV1;
"EOP"
1SECTION : 3.1.1.1.1.3.1 ( DECEMBER 1978 ) PAGE 1
AUTHORS : J.R.BUNCH,L,KAUFMAN,B.N.PARLETT.
CONTRIBUTOR : C.H.CONVENT.
INSTITUTE : UNIVERSITY OF AMSTERDAM.
RECEIVED : 770712.
BRIEF DESCRIPTION :
DECSYM2 CALCULATES THE LDL' DECOMPOSITION OF A SYMMETRIC MATRIX.
THE MATRIX MAY BE INDEFINITE AND/OR SINGULAR;
KEYWORDS :
GENERAL SYMMETRIC MATRIX,
LDL' DECOMPOSITION,
BLOCK DIAGONAL PIVOTING;
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" DECSYM2(A,N,TOL,AUX,P,DETAUX);
"VALUE" N;"INTEGER" N;"REAL" TOL;
"ARRAY" A,DETAUX;"INTEGER" "ARRAY" AUX,P;
"CODE" 34291;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY : THE SYMMETRIC COEFFICIENT MATRIX;
EXIT : THE ELEMENTS OF THE LDL' DECOMPOSITION OF A ARE
STORED IN THE UPPER TRIANGULAR PART OF A. HERE D
IS A BLOCK DIAGONAL MATRIX WITH BLOCKS OF ORDER 1
OR 2. FOR A BLOCK OF ORDER 2 WE ALWAYS HAVE
D[I,I+1]^=0 AND L[I+1,I]=0,SO THAT D AND L' FIT
IN THE UPPER TRIANGULAR PART OF A.
THE STRICTLY LOWER TRIANGULAR PART OF A IS LEFT
UNDISTURBED.
N : <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
1SECTION : 3.1.1.1.1.3.1 ( DECEMBER 1978 ) PAGE 2
TOL : <ARITHMETIC EXPRESSION>;
A RELATIVE TOLERANCE, USED TO CONTROL THE
CALCULATION OF THE BLOCK DIAGONAL ELEMENTS;
AUX : <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" AUX[2:5];
EXIT :
AUX[2] : IF THE MATRIX IS SYMMETRIC THEN 1, OTHER-
WISE 0;IN THE LAST CASE NO DECOMPOSITION
IS PERFORMED;
AUX[3] : IF THE MATRIX IS SYMMETRIC THEN THE
NUMBER OF ITS POSITIVE EIGENVALUES,
OTHERWISE 0. IF AUX[3]=N THEN THE
MATRIX IS POSITIVE DEFINITE;
AUX[4] : IF THE MATRIX IS SYMMETRIC THEN THE
NUMBER OF ITS NEGATIVE EIGENVALUES,
OTHERWISE 0. IF AUX[4]=N THEN THE
MATRIX IS NEGATIVE DEFINITE;
AUX[5] : IF THE MATRIX IS SYMMETRIC THEN THE
NUMBER OF ITS ZERO EIGENVALUES, OTHERWISE
N; SO, IF AUX[5]=0 THEN THE MATRIX IS
SYMMETRIC AND NON-SINGULAR;
P : <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" P[1:N];
EXIT :
A VECTOR RECORDING
1) THE INTERCHANGES PERFORMED ON A DURING THE
COMPUTATION OF THE DECOMPOSITION AND
2) THE BLOCK STRUCTURE OF D.
IF P[I]>0 AND P[I+1]=0 A 2*2 BLOCK HAS BEEN
FOUND I.E. D[I,I+1]^=0 AND L[I+1,I]=0;
DETAUX : <ARRAY IDENTIFIER>;
"ARRAY" DETAUX[1:N];
EXIT :
IF P[I]>0 AND P[I+1]>0 THEN DETAUX[I] EQUALS
THE EXIT-VALUE OF A[I,I].
IF P[I]>0 AND P[I+1]=0 THEN DETAUX[I]=1 AND
DETAUX[I+1] EQUALS THE VALUE OF THE DETERMINANT
OF THE CORRESPONDING 2*2 DIAGONAL BLOCK AS
DETERMINED BY DECSYM2;
PROCEDURES USED :
ELMROW=CP34030.
ICHROW=CP34032.
ICHROWCOL=CP34033.
1SECTION : 3.1.1.1.1.3.1 ( DECEMBER 1978 ) PAGE 3
RUNNING TIME : ROUGHLY PROPORTIONAL TO N**3.
METHOD AND PERFORMANCE :
THE PROCEDURE DECSYM2 COMPUTES THE LDL' DECOMPOSITION OF A
SYMMETRIC MATRIX,ACCORDING TO A METHOD DUE TO BUNCH,KAUFMAN AND
PARLETT (SEE [1],[2]). IT USES BLOCK DIAGONAL PIVOTING.
THE BLOCK DIAGONAL MATRIX D IS DELIVERED IN THE BLOCK DIAGONAL
OF A. IF P[I]>0 AND P[I+1]=0 A 2*2 BLOCK HAS BEEN FOUND AND
FURTHERMORE : L[I+1,I]=0 WHEN D[I,I+1]^=0.
THE STRICTLY UPPER TRIANGULAR PART OF L' IS DELIVERED IN THE
STRICTLY UPPER TRIANGULAR PART OF A. FOR THE INERTIA PROBLEM
IT IS IMPORTANT THAT DECSYM2 CAN ACCEPT SINGULAR MATRICES.
NOTE, HOWEVER, THAT IN ORDER TO FIND THE NUMBER OF ZERO
EIGENVALUES OF SINGULAR MATRICES, THE SINGULAR VALUE
DECOMPOSITION MIGHT BE PREFERRED.
BEFORE THE DECOMPOSITION IS PERFORMED A CHECK IS MADE TO SEE
WHETHER THE MATRIX IS SYMMETRIC. IF THE MATRIX IS ASYMMETRIC
THEN NO DECOMPOSITION IS PERFORMED;
REFERENCES :
1) J.R.BUNCH,L.KAUFMAN.
SOME STABLE METHODS FOR CALCULATING INERTIA AND SOLVING
SYMMETRIC LINEAR SYSTEMS.
MATHEMATICS OF COMPUTATION 31,P 163-180,1977.
2) J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
DECOMPOSITION OF A SYMMETRIC MATRIX.
NUMERISCHE MATHEMATIK 27,P 95-109,1976.
1SECTION : 3.1.1.1.1.3.1 ( DECEMBER 1978 ) PAGE 4
SOURCE TEXT :
"CODE" 34291;
"PROCEDURE" DECSYM2(A,N,TOL,AUX,P,DETAUX);
"VALUE" N;"INTEGER" N;"REAL" TOL;
"ARRAY" A,DETAUX;"INTEGER" "ARRAY" P,AUX;
"BEGIN" "INTEGER" I,J,K,M,IP1,IP2,DUMMY;"BOOLEAN" ONEBYONE,SYM;
"REAL" DET,S,T,ALPHA,LAMBDA,SIGMA,AII,AIP1,AIP1I;
AUX[3]:=AUX[4]:=0;SYM:="TRUE";I:=0;
"FOR" DUMMY:=0 "WHILE" SYM "AND" (I<N) "DO"
"BEGIN" I:=I+1;J:=I;
"FOR" M:=0 "WHILE" SYM "AND" (J<N) "DO"
"BEGIN" J:=J+1;SYM:=SYM "AND" (A[I,J]=A[J,I]) "END";
"END";
"IF" SYM "THEN" AUX[2]:=1
"ELSE" "BEGIN" AUX[2]:=0;"GOTO" ENDDEC "END";
ALPHA:=(1+SQRT(17))/8;P[N]:=N;I:=1;
"FOR" DUMMY:=0 "WHILE" I<N "DO"
"BEGIN" IP1:=I+1;IP2:=I+2;AII:=ABS(A[I,I]);P[I]:=I;
LAMBDA:=ABS(A[I,IP1]);J:=IP1;
"FOR" M:=IP2 "STEP" 1 "UNTIL" N "DO"
"IF" ABS(A[I,M])>LAMBDA "THEN"
"BEGIN" J:=M;LAMBDA:=ABS(A[I,M]) "END";
T:=ALPHA*LAMBDA;ONEBYONE:="TRUE";
"IF" AII<T "THEN"
"BEGIN" SIGMA:=LAMBDA;
"FOR" M:=IP1 "STEP" 1 "UNTIL" J-1 "DO"
"IF" ABS(A[M,J])>SIGMA "THEN" SIGMA:=ABS(A[M,J]);
"FOR" M:=J+1 "STEP" 1 "UNTIL" N "DO"
"IF" ABS(A[J,M])>SIGMA "THEN" SIGMA:=ABS(A[J,M]);
"IF" SIGMA*AII<LAMBDA "THEN"
"BEGIN" "IF" ALPHA*SIGMA<ABS(A[J,J]) "THEN"
"BEGIN" ICHROW(J+1,N,I,J,A);ICHROWCOL(IP1,J-1,I,J,A);
T:=A[I,I];A[I,I]:=A[J,J];A[J,J]:=T;P[I]:=J
"END"
"ELSE"
"BEGIN" "IF" J>IP1 "THEN"
"BEGIN" ICHROW(J+1,N,IP1,J,A);ICHROWCOL(IP2,J-1,IP1,J,A);
T:=A[I,I];A[I,I]:=A[J,J];A[J,J]:=T;
T:=A[I,J];A[I,J]:=A[I,IP1];A[I,IP1]:=T
"END";
DET:=A[I,I]*A[IP1,IP1]-A[I,IP1]**2;AIP1I:=A[I,IP1]/DET;
AII:=A[I,I]/DET;AIP1:=A[IP1,IP1]/DET;P[I]:=J;P[IP1]:=0;
DETAUX[I]:=1;DETAUX[IP1]:=DET; "COMMENT"
1SECTION : 3.1.1.1.1.3.1 ( DECEMBER 1978 ) PAGE 5
;
"FOR" J:=IP2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:=AIP1I*A[IP1,J]-AIP1*A[I,J];
T:=AIP1I*A[I,J]-AII*A[IP1,J];ELMROW(J,N,J,I,A,A,S);
ELMROW(J,N,J,IP1,A,A,T);A[I,J]:=S;A[IP1,J]:=T
"END";
AUX[3]:=AUX[3]+1;AUX[4]:=AUX[4]+1;I:=IP2;
ONEBYONE:="FALSE"
"END"
"END"
"END";
"IF" ONEBYONE "THEN"
"BEGIN" "IF" TOL<ABS(A[I,I]) "THEN"
"BEGIN" AII:=A[I,I];DETAUX[I]:=A[I,I];
"IF" AII>0 "THEN" AUX[3]:=AUX[3]+1 "ELSE" AUX[4]:=AUX[4]+1;
"FOR" J:=IP1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:=-A[I,J]/AII;ELMROW(J,N,J,I,A,A,S);A[I,J]:=S "END"
"END";I:=IP1
"END"
"END" WHILE I;
"IF" I=N "THEN"
"BEGIN" "IF" TOL<ABS(A[N,N]) "THEN"
"BEGIN" "IF" A[N,N]>0 "THEN" AUX[3]:=AUX[3]+1
"ELSE" AUX[4]:=AUX[4]+1
"END";DETAUX[N]:=A[N,N]
"END";
ENDDEC:
AUX[5]:=N-AUX[3]-AUX[4]
"END" DECSYM2;
"EOP"
1SECTION : 3.1.1.1.1.3.2 ( DECEMBER 1978 ) PAGE 1
AUTHORS : J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
CONTRIBUTOR : C.H.CONVENT.
INSTITUTE : UNIVERSITY OF AMSTERDAM.
RECEIVED : 770712.
BRIEF DESCRIPTION :
DETERMSYM2 CALCULATES THE DETERMINANT OF A SYMMETRIC MATRIX.
THE LDL' DECOMPOSITION OF THE MATRIX, AS PRODUCED BY DECSYM2,
SHOULD BE AVAILABLE.
KEYWORDS :
GENERAL SYMMETRIC MATRIX,
LDL' DECOMPOSITION,
BLOCK DIAGONAL PIVOTING;
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"REAL" "PROCEDURE" DETERMSYM2(DETAUX,N,AUX);
"VALUE" N;"INTEGER" N;
"ARRAY" DETAUX;"INTEGER" "ARRAY" AUX;
"CODE" 34294;
DETERMSYM2 : DELIVERS THE CALCULATED VALUE OF THE DETERMINANT OF
THE MATRIX;
THE MEANING OF THE FORMAL PARAMETERS IS :
DETAUX : <ARRAY IDENTIFIER>;
"ARRAY" DETAUX[1:N];
ENTRY : THE ARRAY DETAUX AS PRODUCED BY DECSYM2;
N : <ARITHMETIC EXPRESSION>;
THE ORDER OF THE ARRAY DETAUX
( = THE ORDER OF THE MATRIX );
AUX : <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" AUX[2:5];
ENTRY : THE ARRAY AUX AS PRODUCED BY DECSYM2;
PROCEDURES USED : NONE.
1SECTION : 3.1.1.1.1.3.2 ( DECEMBER 1978 ) PAGE 2
RUNNING TIME : PROPORTIONAL TO N;
METHOD AND PERFORMANCE :
FIRST OF ALL DECSYM2 SHOULD BE CALLED TO PERFORM THE LDL'
DECOMPOSITION OF THE SYMMETRIC MATRIX,ACCORDING TO A METHOD DUE
TO BUNCH,KAUFMAN AND PARLETT (SEE [1],[2]).
IF A 1*1 BLOCK HAS BEEN COMPUTED FOR D THEN DETAUX[I] CONTAINS
THE VALUE OF D[I]. IF A 2*2 BLOCK HAS BEEN COMPUTED FOR D THEN
DETAUX[I]=1 AND DETAUX[I+1] CONTAIN THE VALUE OF THE DETERMINANT
OF THE CORRESPONDING 2*2 BLOCK.
THE COMPUTATION OF THE DETERMINANT IS DONE BY CALCULATING THE
PRODUCT OF THE ELEMENTS OF DETAUX.
REFERENCES :
1) J.R.BUNCH,L.KAUFMAN.
SOME STABLE METHODS FOR CALCULATING INERTIA AND SOLVING
SYMMETRIC LINEAR SYSTEMS.
MATHEMATICS OF COMPUTATION 31,P 163-180,1977.
2) J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
DECOMPOSITION OF A SYMMETRIC MATRIX.
NUMERISCHE MATHEMATIK 27,P 95-109,1976.
EXAMPLE OF USE :
"BEGIN" "COMMENT" EXAMPLE OF USE OF THE PROCEDURE DETERMSYM2;
"INTEGER" I,J;"REAL" TOL,DETERMINANT;
"REAL" "ARRAY" A[1:5,1:5],DETAUX[1:5];
"INTEGER" "ARRAY" AUX[2:5],P[1:5];
A[1,1]:=A[1,2]:=-3;A[1,3]:=-18;A[1,4]:=-30;A[1,5]:=18;
A[2,2]:=-1;A[2,3]:=-4;A[2,4]:=-48;A[2,5]:=8;
A[3,3]:=-6;A[3,4]:=-274;A[3,5]:=6;
A[4,4]:=119;A[4,5]:=19;
A[5,5]:=216;
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
"FOR" J:=I+1 "STEP" 1 "UNTIL" 5 "DO" A[J,I]:=A[I,J];
"COMMENT"
1SECTION : 3.1.1.1.1.3.2 ( DECEMBER 1978 ) PAGE 3
;
OUTPUT(61,"(""("THE COEFFICIENT MATRIX :")",/")");
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
"BEGIN" "FOR" J:=1 "STEP" 1 "UNTIL" 5 "DO"
OUTPUT(61,"("-2ZD,4B")",A[I,J]);
OUTPUT(61,"("/")")
"END";
TOL:="-14;
DECSYM2(A,5,TOL,AUX,P,DETAUX);
"IF" AUX[2]=1 "THEN"
OUTPUT(61,"("/,"("THE MATRIX IS SYMMETRIC")"")")
"ELSE" OUTPUT(61,"("/,"("THE MATRIX IS ASYMMETRIC.THE ")",
"("RESULTS ARE MEANINGLESS")"")");
DETERMINANT:=DETERMSYM2(DETAUX,5,AUX);
OUTPUT(61,"("/,"("THE DETERMINANT OF THE MATRIX : ")",
3Z3D.2D")",DETERMINANT)
"END"
THIS DELIVERS AS RESULT :
THE COEFFICIENT MATRIX :
-3 -3 -18 -30 18
-3 -1 -4 -48 8
-18 -4 -6 -274 6
-30 -48 -274 119 19
18 8 6 19 216
THE MATRIX IS SYMMETRIC.
THE DETERMINANT OF THE MATRIX : 168.00
SOURCE TEXT :
0"CODE" 34294;
"REAL" "PROCEDURE" DETERMSYM2(DETAUX,N,AUX);
"VALUE" N;"INTEGER" N;
"ARRAY" DETAUX;"INTEGER" "ARRAY" AUX;
"BEGIN" "INTEGER" I;"REAL" DET;
"IF" AUX[5]>0 "THEN" DET:=0 "ELSE"
"BEGIN" DET:=1;
"FOR" I:=1 "STEP" 1 "UNTIL" N "DO" DET:=DET*DETAUX[I]
"END";
DETERMSYM2:=DET
"END" DETERMSYM2;
"EOP"
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 1
AUTHORS : J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
CONTRIBUTOR : C.H.CONVENT.
INSTITUTE : UNIVERSITY OF AMSTERDAM.
RECEIVED : 770712.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES :
A) SOLSYM2 SOLVES A SYMMETRIC SYSTEM OF LINEAR EQUATIONS, ASSUMING
THAT THE MATRIX HAS BEEN DECOMPOSED INTO LDL' FORM BY A CALL OF
DECSYM2;
B) DECSOLSYM2 CALCULATES THE LDL' DECOMPOSITION OF A SYMMETRIC
MATRIX; MOREOVER, IF THIS MATRIX IS NON-SINGULAR, THEN IT SOLVES A
CORRESPONDING SYSTEM OF LINEAR EQUATIONS;
KEYWORDS :
GENERAL SYMMETRIC MATRIX,
LDL' DECOMPOSITION,
BLOCK DIAGONAL PIVOTING;
SUBSECTION : SOLSYM2.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" SOLSYM2(A,N,B,P,DETAUX);
"VALUE" N;"INTEGER" N;
"ARRAY" A,B,DETAUX;"INTEGER" "ARRAY" P;
"CODE" 34292;
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 2
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY : THE LDL' DECOMPOSITION OF A AS PRODUCED BY
DECSYM2;
N : <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY : THE RIGHT-HANDSIDE OF A SYSTEM OF LINEAR
EQUATIONS;
EXIT : THE CALCULATED SOLUTION VECTOR;
P : <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" P[1:N];
ENTRY : A VECTOR RECORDING THE INTERCHANGES PERFORMED ON
THE MATRIX BY THE PROCEDURE DECSYM2. P ALSO CONTAINS
INFORMATION ON THE BLOCKSTRUCTURE OF THE MATRIX AS
DECOMPOSED BY DECSYM2;
DETAUX : <ARRAY IDENTIFIER>;
"ARRAY" DETAUX[1:N];
ENTRY : THE ARRAY DETAUX AS PRODUCED BY DECSYM2;
PROCEDURES USED :
MATVEC=CP34011.
ELMVECROW=CP34026.
RUNNING TIME : ROUGHLY PROPORTIONAL TO N**2.
METHOD AND PERFORMANCE :
THE PROCEDURE SOLSYM2 COMPUTES THE SOLUTION OF A SYMMETRIC
SYSTEM OF LINEAR EQUATIONS,ASSUMING THAT THE MATRIX HAS BEEN
DECOMPOSED INTO LDL' FORM BY A CALL OF DECSYM2. B IS OVERWRITTEN
WITH THE SOLUTION VECTOR.
REFERENCES :
1) J.R.BUNCH,L.KAUFMAN.
SOME STABLE METHODS FOR CALCULATING INERTIA AND SOLVING
SYMMETRIC LINEAR SYSTEMS.
MATHEMATICS OF COMPUTATION 31,P 163-180,1977.
2) J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
DECOMPOSITION OF A SYMMETRIC MATRIX.
NUMERISCHE MATHEMATIK 27,P 95-109,1976.
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 3
SUBSECTION : DECSOLSYM2.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" DECSOLSYM2(A,N,B,TOL,AUX);
"VALUE" N;"INTEGER" N;"REAL" TOL;
"ARRAY" A,B;"INTEGER" "ARRAY" AUX;
"CODE" 34293;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:N];
ENTRY : SEE DECSYM2;
EXIT : SEE DECSYM2;
N : <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY : SEE SOLSYM2;
EXIT : THE CALCULATED SOLUTION VECTOR,WHEN A WAS FOUND
TO BE NON-SINGULAR.
B IS LEFT UNDISTURBED OTHERWISE;
TOL : <ARITHMETIC EXPRESSION>;
ENTRY : SEE DECSYM2;
AUX : <ARRAY IDENTIFIER>;
"INTEGER" "ARRAY" AUX[2:5];
EXIT : SEE DECSYM2;
PROCEDURES USED :
DECSYM2=CP34291.
SOLSYM2=CP34292.
RUNNING TIME : ROUGHLY PROPORTIONAL TO N**3.
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 4
METHOD AND PERFORMANCE :
THE PROCEDURE DECSOLSYM2 COMPUTES THE SOLUTION OF A SYMMETRIC
SYSTEM OF LINEAR EQUATIONS. IT DOES SO BY FIRST CALLING THE
PROCEDURE DECSYM2 TO COMPUTE THE LDL' DECOMPOSITION OF THE
SYMMETRIC MATRIX,ACCORDING TO A METHOD DUE TO BUNCH,KAUFMAN AND
PARLETT (SEE [1],[2]).
WHEN THE MATRIX IS FOUND TO BE NON-SINGULAR THE PROCEDURE SOLSYM2
IS CALLED TO COMPUTE THE SOLUTION VECTOR,AND THE LATTER OVERWRITES
B. WHEN THE MATRIX IS FOUND TO BE SINGULAR THE PROCEDURE SOLSYM2
IS NOT CALLED AND B IS LEFT UNDISTURBED.
REFERENCES :
1) J.R.BUNCH,L.KAUFMAN.
SOME STABLE METHODS FOR CALCULATING INERTIA AND SOLVING
SYMMETRIC LINEAR SYSTEMS.
MATHEMATICS OF COMPUTATION 31,P 163-180,1977.
2) J.R.BUNCH,L.KAUFMAN,B.N.PARLETT.
DECOMPOSITION OF A SYMMETRIC MATRIX.
NUMERISCHE MATHEMATIK 27,P 95-109,1976.
EXAMPLES OF USE :
"BEGIN" "COMMENT" EXAMPLE OF USE OF THE PROCEDURE DECSOLSYM2;
"INTEGER" I,J;"REAL" TOL;
"REAL" "ARRAY" A[1:5,1:5],B[1:5];
"INTEGER" "ARRAY" AUX[2:5];
A[1,1]:=A[1,2]:=-3;A[1,3]:=-18;A[1,4]:=-30;A[1,5]:=18;
A[2,2]:=-1;A[2,3]:=-4;A[2,4]:=-48;A[2,5]:=8;
A[3,3]:=-6;A[3,4]:=-274;A[3,5]:=6;
A[4,4]:=119;A[4,5]:=19;
A[5,5]:=216;
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
"FOR" J:=I+1 "STEP" 1 "UNTIL" 5 "DO" A[J,I]:=A[I,J];
OUTPUT(61,"(""("THE COEFFICIENTMATRIX :")",/")");
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
"BEGIN" "FOR" J:=1 "STEP" 1 "UNTIL" 5 "DO"
OUTPUT(61,"("-2ZD,4B")",A[I,J]);
OUTPUT(61,"("/")")
"END"; "COMMENT"
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 5
;
OUTPUT(61,"("/,"("THE RHS-VECTOR :")",/")");
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
"BEGIN" INREAL(70, B[I]);
OUTPUT(61,"("-Z3D,4B")",B[I])
"END";
TOL:="-14;
DECSOLSYM2(A,5,B,TOL,AUX);
OUTPUT(61,"("2/")");
"IF" AUX[2]=1 "THEN"
OUTPUT(61,"(""("THE MATRIX IS SYMMETRIC")",/")")
"ELSE" OUTPUT(61,"(""("THE MATRIX IS ASYMMETRIC.THE ")",
"("RESULTS ARE MEANINGLESS")",/")");
OUTPUT(61,"("/,"("INERTIA : <")",D,"(",")",D,"(",")",D,
"(">")",/")",AUX[3],AUX[4],AUX[5]);
OUTPUT(61,"("/,"("THE COMPUTED SOLUTION :")",/")");
"FOR" I:=1 "STEP" 1 "UNTIL" 5 "DO"
OUTPUT(61,"("-D.5D,4B")",B[I])
"END"
INPUT:
327 291 1290 275 1720
THIS DELIVERS AS RESULT :
THE COEFFICIENT MATRIX :
-3 -3 -18 -30 18
-3 -1 -4 -48 8
-18 -4 -6 -274 6
-30 -48 -274 119 19
18 8 6 19 216
THE RHS-VECTOR :
327 291 1290 275 1720
THE MATRIX IS SYMMETRIC.
INERTIA : <3,2,0>
THE COMPUTED SOLUTION :
-7.00000 -2.00000 -1.00000 -4.00000 9.00000
1SECTION : 3.1.1.1.1.3.3 ( DECEMBER 1978 ) PAGE 6
SOURCE TEXT(S) :
"CODE" 34292;
"PROCEDURE" SOLSYM2(A,N,B,P,DETAUX);
"VALUE" N;"INTEGER" N;
"ARRAY" A,B,DETAUX;"INTEGER" "ARRAY" P;
"BEGIN" "INTEGER" I,II,J,K,IP1,PI,PII,DUMMY;
"REAL" DET,TEMP,SAVE;
I:=1;
"FOR" DUMMY:=0 "WHILE" I<N "DO"
"BEGIN" IP1:=I+1;PI:=P[I];SAVE:=B[PI];
"IF" P[IP1]>0 "THEN"
"BEGIN" B[PI]:=B[I];B[I]:=SAVE/A[I,I];
ELMVECROW(IP1,N,I,B,A,SAVE);I:=IP1
"END"
"ELSE"
"BEGIN" TEMP:=B[I];B[PI]:=B[IP1];DET:=DETAUX[IP1];
B[I]:=(TEMP*A[IP1,IP1]-SAVE*A[I,IP1])/DET;
B[IP1]:=(SAVE*A[I,I]-TEMP*A[I,IP1])/DET;
ELMVECROW(I+2,N,I,B,A,TEMP);ELMVECROW(I+2,N,IP1,B,A,SAVE);
I:=I+2
"END"
"END" WHILE I;
"IF" I=N "THEN" "BEGIN" B[I]:=B[I]/A[I,I];I:=N-1 "END"
"ELSE" I:=N-2;
"FOR" DUMMY:=0 "WHILE" I>0 "DO"
"BEGIN" "IF" P[I]=0 "THEN" II:=I-1 "ELSE" II:=I;
"FOR" K:=II "STEP" 1 "UNTIL" I "DO"
"BEGIN" SAVE:=B[K];SAVE:=SAVE+MATVEC(I+1,N,K,A,B);
B[K]:=SAVE
"END";
PII:=P[II];B[I]:=B[PII];B[PII]:=SAVE;I:=II-1
"END" WHILE I
"END" SOLSYM2;
"EOP"
"CODE" 34293;
"PROCEDURE" DECSOLSYM2(A,N,B,TOL,AUX);
"VALUE" N;"INTEGER" N;"REAL" TOL;
"ARRAY" A,B;"INTEGER" "ARRAY" AUX;
"BEGIN" "REAL" "ARRAY" DETAUX[1:N];"INTEGER" "ARRAY" P[1:N];
DECSYM2(A,N,TOL,AUX,P,DETAUX);
"IF" AUX[5]=0 "THEN" SOLSYM2(A,N,B,P,DETAUX)
"END" DECSOLSYM2;
"EOP"
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 1
AUTHOR : T.J. DEKKER.
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 731015.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES :
A) LSQORTDEC, FOR THE HOUSEHOLDER TRIANGULARIZATION WITH COLUMN
INTERCHANGES OF THE COEFFICIENT MATRIX OF A LINEAR LEAST SQUARES
PROBLEM;
B) LSQDGLINV, FOR THE CALCULATION OF THE DIAGONAL ELEMENTS OF THE
INVERSE OF M'M, WHERE M IS THE COEFFICIENT MATRIX OF A LINEAR
LEAST SQUARES PROBLEM.
KEY WORDS :
LINEAR LEAST SQUARES PROBLEM,
HOUSEHOLDER TRIANGULARIZATION.
SUBSECTION : LSQORTDEC.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" LSQORTDEC(A, N, M, AUX, AID, CI); "VALUE" N, M;
"INTEGER" N, M; "INTEGER""ARRAY" CI; "ARRAY" A, AUX, AID;
"CODE" 34134;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1 : N,1 : M];
ENTRY :THE COEFFICIENT MATRIX OF THE
LINEAR LEAST SQUARES PROBLEM;
EXIT : IN THE UPPER TRIANGLE OF A (THE ELEMENTS
A[I,J] WITH I < J) THE SUPERDIAGONAL
ELEMENTS OF THE UPPER-TRIANGULAR MATRIX, PRODUCED BY
THE HOUSEHOLDER TRANSFORMATION; IN THE OTHER PART OF
THE COLUMNS OF A THE SIGNIFICANT ELEMENTS OF THE
GENERATING VECTORS OF THE HOUSEHOLDER MATRICES USED
FOR THE HOUSEHOLDER TRIANGULARIZATION;
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 2
N : <ARITHMETIC EXPRESSION>;
NUMBER OF ROWS OF THE MATRIX;
M : <ARITHMETIC EXPRESSION>;
NUMBER OF COLUMNS OF THE MATRIX (N >= M);
AUX : <ARRAY IDENTIFIER>;
"ARRAY" AUX[2 : 5];
ENTRY : AUX[2] CONTAINS A RELATIVE TOLERANCE USED FOR
CALCULATING THE DIAGONAL ELEMENTS OF THE
UPPER-TRIANGULAR MATRIX;
EXIT :
AUX[3] DELIVERS THE NUMBER OF THE DIAGONAL ELEMENTS OF
THE UPPER-TRIANGULAR MATRIX WHICH HAVE BEEN FOUND NOT
NEGLIGIBLE;
NORMAL EXIT AUX[3] = M;
AUX[5] := THE MAXIMUM OF THE EUCLIDEAN NORMS OF THE
COLUMNS OF THE GIVEN MATRIX;
AID : <ARRAY IDENTIFIER>;
"ARRAY" AID[1 : M];
NORMAL EXIT (AUX[3] = M) : AID CONTAINS THE DIAGONAL
ELEMENTS OF THE UPPER-TRIANGULAR MATRIX PRODUCED BY THE
HOUSEHOLDER TRIANGULARIZATION;
CI : <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1 : M];
EXIT : CI CONTAINS THE PIVOTAL INDICES OF THE
INTERCHANGES OF THE COLUMNS OF THE GIVEN MATRIX.
PROCEDURES USED :
TAMMAT = CP34014,
ELMCOL = CP34023,
ICHCOL = CP34031.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : AN ARRAY OF M ELEMENTS IS DECLARED.
RUNNING TIME:
(C1 * M + C2) * M * (N - M / 3);
THE CONSTANTS C1 AND C2 DEPEND ON THE
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 3
METHOD AND PERFORMANCE :
THE PROCEDURE LSQORTDEC IS A MODIFICATION OF THE PROCEDURE LSQDEC
DUE TO T.J. DEKKER (SEE REF [1], WHERE A DERIVATION IS GIVEN OF
A SET OF PROCEDURES BY
P. BUSINGER AND G.H. GOLUB, SEE REF [2]). THE METHOD IS
HOUSEHOLDER TRIANGULARIZATION WITH COLUMN INTERCHANGES.
LET M DENOTE THE GIVEN MATRIX. LSQORTDEC PRODUCES AN N-TH ORDER
ORTHOGONAL MATRIX Q AND AN N * M UPPER-TRIANGULAR MATRIX R SUCH
THAT R EQUALS QM WITH PERMUTED COLUMNS, Q IS THE PRODUCT OF
AT MOST M HOUSEHOLDER MATRICES WHICH ARE REPRESENTED BY THEIR
GENERATING VECTORS. M IS REDUCED TO R IN AT MOST M STAGES : AT
THE K-TH STAGE THE K-TH COLUMN OF THE (ALREADY MODIFIED) MATRIX IS
INTERCHANGED WITH THE COLUMN OF MAXIMUM EUCLIDEAN NORM (THE
PIVOTAL COLUMN); THEN THE MATRIX IS MULTIPLIED WITH A HOUSEHOLDER
MATRIX SUCH, THAT THE SUBDIAGONAL ELEMENTS OF THE K-TH COLUMN
BECOME ZERO, WHILE THE FIRST K - 1 COLUMNS REMAIN UNCHANGED.
THE PROCESS TERMINATES PREMATURELY, IF AT SOME STAGE THE
EUCLIDEAN NORM OF THE PIVOTAL COLUMN IS LESS THAN SOME TOLERANCE,
VIZ. A GIVEN TOLERANCE (AUX[2]) TIMES THE MAXIMUM OF THE
EUCLIDEAN NORMS OF THE COLUMNS OF THE GIVEN MATRIX.
LSQORTDEC DELIVERS THE SIGNIFICANT ELEMENTS OF THE GENERATING
VECTOR OF THE K-TH HOUSEHOLDER MATRIX (THE FIRST K - 1
ELEMENTS OF THIS VECTOR BEING ZERO) IN THE LOWER TRIANGLE PART OF
THE K-TH COLUMN OF THE ARRAY A (A[I,K] FOR I >= K).
OF THE RESULTING UPPER-TRIANGULAR MATRIX THE
DIAGONAL ELEMENTS ARE DELIVERED SEPARATELY IN AN ARRAY AID, AND THE
REMAINING ELEMENTS IN THE SUPER-TRIANGULAR PART OF THE ARRAY A.
FOR THE SOLUTION OF LEAST SQUARES PROBLEMS, ONLY CALLS WITH
N >= M ARE USEFUL.
REFERENCES :
[1] DEKKER, T.J. :
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MC TRACT 22, 1968, MATHEMATISCH CENTRUM, AMSTERDAM.
[2] BUSINGER, P. AND G.H. GOLUB :
LINEAR LEAST SQUARES SOLUTION BY HOUSEHOLDER TRANSFORMATIONS,
NUM. MATH. 7 (1965), PP. 269 - 276.
EXAMPLE OF USE :
SEE EXAMPLE OF USE OF LSQSOL.
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 4
SUBSECTION : LSQDGLINV.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" LSQDGLINV(A, M, AID, CI, DIAG); "VALUE" M; "INTEGER" M;
"INTEGER""ARRAY" CI; "ARRAY" A, AID, DIAG;
"CODE" 34132;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, M, AID, CI :
SEE 'CALLING SEQUENCE' OF LSQORTDEC (THIS SECTION);
THE CONTENTS OF A, AID AND CI SHOULD BE PRODUCED BY A
SUCCESSFUL CALL OF LSQORTDEC (AUX[3] = M) .
DIAG : <ARRAY IDENTIFIER>; "ARRAY" DIAG[1 : M];
EXIT : THE DIAGONAL ELEMENTS OF THE INVERSE OF M'M
WHERE M IS THE MATRIX OF THE LINEAR LEAST SQUARES
PROBLEM.
PROCEDURES USED :
VECVEC = CP34010,
TAMVEC = CP34012.
RUNNING TIME :
(C3 * M + C4) * M * M;
THE CONSTANTS C3 AND C4 DEPEND ON THE ARITHMETIC
OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
LSQDGLINV SHOULD BE CALLED AFTER A SUCCESSFUL CALL OF LSQORTDEC,
I.E. IF AUX[3] = M. LSQDGLINV CALCULATES THE DIAGONAL ELEMENTS
OF THE INVERSE OF M'M, WHERE M IS THE MATRIX OF A LINEAR
LEAST SQUARES PROBLEM.
THESE VALUES CAN BE USED FOR THE COMPUTATION OF THE STANDARD
DEVIATIONS OF LEAST SQUARES SOLUTIONS.
EXAMPLE OF USE :
SEE EXAMPLE OF USE OF LSQSOL.
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 5
SOURCE TEXT(S) :
"CODE" 34134;
"PROCEDURE" LSQORTDEC(A, N, M, AUX, AID, CI); "VALUE" N, M;
"INTEGER" N, M; "ARRAY" A, AUX, AID; "INTEGER" "ARRAY" CI;
"BEGIN" "INTEGER" J, K, KPIV;
"REAL" BETA, SIGMA, NORM, W, EPS, AKK, AIDK;
"ARRAY" SUM[1:M];
NORM:= 0; AUX[3]:= M;
"FOR" K:= 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" W:= SUM[K]:= TAMMAT(1, N, K, K, A, A);
"IF" W > NORM "THEN" NORM:= W
"END";
W:= AUX[5]:= SQRT(NORM); EPS:= AUX[2] * W;
"FOR" K:= 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" SIGMA:= SUM[K]; KPIV:= K;
"FOR" J:= K + 1 "STEP" 1 "UNTIL" M "DO"
"IF" SUM[J] > SIGMA "THEN"
"BEGIN" SIGMA:= SUM[J]; KPIV:= J "END";
"IF" KPIV ^= K "THEN"
"BEGIN" SUM[KPIV]:= SUM[K]; ICHCOL(1, N, K, KPIV, A) "END";
CI[K]:= KPIV; AKK:= A[K,K];
SIGMA:= TAMMAT(K, N, K, K, A, A); W:= SQRT(SIGMA);
AIDK:= AID[K]:= "IF" AKK < 0 "THEN" W "ELSE" - W;
"IF" W < EPS "THEN"
"BEGIN" AUX[3]:= K - 1; "GO TO" ENDDEC "END";
BETA:= 1 / (SIGMA - AKK * AIDK); A[K,K]:= AKK - AIDK;
"FOR" J:= K + 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" ELMCOL(K, N, J, K, A, A, - BETA * TAMMAT(K, N,
K, J, A, A)); SUM[J]:= SUM[J] - A[K,J] ** 2
"END"
"END" FOR K;
ENDDEC:
"END" LSQORTDEC
1SECTION : 3.1.1.2.1.1 (MAY 1974) PAGE 6
;
"EOP"
"CODE" 34132;
"PROCEDURE" LSQDGLINV(A, M, AID, CI, DIAG); "VALUE" M; "INTEGER" M;
"ARRAY" A, AID, DIAG; "INTEGER" "ARRAY" CI;
"BEGIN" "INTEGER" J, K, CIK;
"REAL" W;
"FOR" K:= 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" DIAG[K]:= 1 / AID[K];
"FOR" J:= K + 1 "STEP" 1 "UNTIL" M "DO"
DIAG[J]:= - TAMVEC(K, J - 1, J, A, DIAG) / AID[J];
DIAG[K]:= VECVEC(K, M, 0, DIAG, DIAG)
"END";
"FOR" K:= M "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" CIK:= CI[K]; "IF" CIK ^= K "THEN"
"BEGIN" W:= DIAG[K]; DIAG[K]:= DIAG[CIK]; DIAG[CIK]:= W
"END"
"END"
"END" LSQDGLINV;
"EOP"
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 1
AUTHOR : T.J. DEKKER.
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 731015.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES :
A) LSQSOL, FOR THE SOLUTION OF A LINEAR LEAST SQUARES PROBLEM IF
THE COEFFICIENT MATRIX HAS BEEN DECOMPOSED BY LSQORTDEC
(SECTION 3.1.1.2.1.1.);
B) LSQORTDECSOL, FOR THE SOLUTION OF A LINEAR LEAST SQUARES PROBLEM
BY HOUSEHOLDER TRIANGULARIZATION WITH COLUMN INTERCHANGES AND FOR
THE CALCULATION OF THE DIAGONAL OF THE INVERSE OF M'M, WHERE
M IS THE COEFFICIENT MATRIX.
KEY WORDS :
LINEAR LEAST SQUARES PROBLEM,
HOUSEHOLDER TRIANGULARIZATION.
SUBSECTION : LSQSOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" LSQSOL(A, N, M, AID, CI, B); "VALUE" N, M;
"INTEGER" N, M; "INTEGER""ARRAY" CI; "ARRAY" A, AID, B;
"CODE" 34131;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, M, AID, CI : SEE 'CALLING SEQUENCE' OF LSQORTDEC
(SECTION 3.1.1.2.1.1.); THE CONTENTS OF THE ARRAYS
A, AID AND CI SHOULD BE PRODUCED BY A SUCCESSFUL CALL
OF LSQORTDEC, I.E. IF AUX[3] = M;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1 : N];
ENTRY : B CONTAINS THE RIGHT HAND SIDE OF A LINEAR
LEAST SQUARES PROBLEM;
EXIT : B[1 : M] CONTAINS THE SOLUTION OF THE PROBLEM;
B[M + 1 : N] CONTAINS A VECTOR WITH EUCLIDEAN
LENGTH EQUAL TO THE EUCLIDEAN LENGTH OF THE RESIDUAL
VECTOR.
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 2
PROCEDURES USED :
MATVEC = CP34011,
TAMVEC = CP34012,
ELMVECCOL = CP34021.
RUNNING TIME:
(C5 * M + C6) * N;
THE CONSTANTS C5 AND C6 DEPEND UPON THE ELEMENTARY
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
LSQSOL SHOULD BE CALLED AFTER A SUCCESSFUL CALL OF LSQORTDEC
(SECTION 3.1.1.2.1.1.), I.E. IF AUX[3] = M. LSQSOL YIELDS
THE LEAST SQUARES SOLUTION OF THE OVERDETERMINED SYSTEM WITH THE
DECOMPOSED COEFFICIENT MATRIX IN ARRAY A AND THE RIGHT HAND SIDE IN
ARRAY B.
FIRST THE ORTHOGONAL TRANSFORMATION WITH THE HOUSEHOLDER MATRICES
IS PERFORMED ON THE RIGHT HAND SIDE. NEXT THE SYSTEM OF THE FIRST M
EQUATIONS AND WITH AN UPPER-TRIANGULAR COEFFICIENT MATRIX IS SOLVED
BY BACK SUBSTITUTION, YIELDING A SOLUTION WITH M PERMUTED
COMPONENTS DUE TO THE COLUMN INTERCHANGES OF THE TRIANGULARIZATION.
FINALLY THE ORDER OF THE M COMPONENTS IS RESTORED. SEE ALSO METHOD
AND PERFORMANCE OF LSQORTDEC (SECTION 3.1.1.2.1.1.).
THE LEAST SQUARES SOLUTIONS OF SEVERAL OVERDETERMINED SYSTEMS WITH
THE SAME COEFFICIENT MATRIX CAN BE OBTAINED BY SUCCESSIVE CALLS OF
LSQSOL WITH DIFFERENT RIGHT HAND SIDES.
EXAMPLE OF USE :
THE NEXT PROGRAM SOLVES THE SYSTEM
- 2 * X1 + X2 = 0
- X1 + X2 = 1
X1 + X2 = 2
2 * X1 + X2 = 2
X1 + 2 * X2 = 3
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 3
"BEGIN""COMMENT" 730912, TEST LSQORTDEC, LSQSOL, LSQDGLINV;
"ARRAY" A, C[1 : 5,1 : 2], B, X[1 : 5], DIAG, AID[1 : 2],
AUX[2 : 5];
"INTEGER""ARRAY" PIV[1 : 2];
"INTEGER" I, J;
"REAL" H;
"REAL""PROCEDURE" SUM(I, A, B, X); "VALUE" A, B;
"INTEGER" I, A, B; "REAL" X;
"BEGIN""REAL" S; S:= 0; "FOR" I:= A "STEP" 1 "UNTIL" B "DO"
S:= S + X; SUM:= S
"END" SUM;
AUX[2]:= "-12; I:= J:= 1;
"FOR" H:= - 2, - 1, 1, 2, 1, 1, 1, 1, 1, 2 "DO"
"BEGIN" A[I,J]:= C[I,J]:= H; "IF" I < 5 "THEN" I:= I + 1 "ELSE"
"BEGIN" I:= 1; J:= J + 1 "END"
"END";
"FOR" H:= 0, 1, 2, 2, 3 "DO"
"BEGIN" B[I]:= X[I]:= H; I:= I + 1 "END";
LSQORTDEC(A, 5, 2, AUX, AID, PIV);
"IF" AUX[3] = 2 "THEN"
"BEGIN" LSQSOL(A, 5, 2, AID, PIV, X);
LSQDGLINV(A, 2, AID, PIV, DIAG);
OUTPUT(61, "("/, "("AUX[2, 3, 5] = ")" +.4D"+DD5B, 3ZD5B,
+.4D"+DD/, "("LSQ SOLUTION :")", 2(2B+.8D"+DD), /
"("RESIDUE (DELIVERED) :")" +.8D"+DD/,
"("RESIDUE (CHECKED) :")" +.8D"+DD/,
"("DIAGONAL OF INVERSE M'M :")", 2(2B+.8D"+DD)")",
AUX[2], AUX[3], AUX[5], X[1], X[2],
SQRT(VECVEC(3, 5, 0, X, X)),
SQRT(SUM(I, 1, 5, (B[I] - C[I,1] * X[1] - C[I,2] * X[2])
** 2)), DIAG[1], DIAG[2])
"END"
"END"
DELIVERS :
AUX[2, 3, 5] = +.1000"-11 2 +.3317"+01
LSQ SOLUTION : +.50000000"+00 +.12500000"+01
RESIDUE (DELIVERED) :+.50000000"+00
RESIDUE (CHECKED) :+.50000000"+00
DIAGONAL OF INVERSE M'M : +.95238095"-01 +.13095238"+00
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 4
SUBSECTION : LSQORTDECSOL.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" LSQORTDECSOL(A, N, M, AUX, DIAG, B); "VALUE" N, M;
"INTEGER" N, M; "ARRAY" A, AUX, DIAG, B;
"CODE" 34135;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1 : N,1 : M];
ENTRY : A CONTAINS THE COEFFICIENT MATRIX OF THE
LINEAR LEAST SQUARES PROBLEM;
EXIT : IN THE UPPER TRIANGLE OF A (THE ELEMENTS
A[I,J] WITH I < J) THE SUPERDIAGONAL
ELEMENTS OF THE UPPER-TRIANGULAR MATRIX, PRODUCED BY
THE HOUSEHOLDER TRANSFORMATION; IN THE OTHER PART OF
THE COLUMNS OF A THE SIGNIFICANT ELEMENTS OF THE
GENERATING VECTORS OF THE HOUSEHOLDER MATRICES USED
FOR THE HOUSEHOLDER TRIANGULARIZATION;
N : <ARITHMETIC EXPRESSION>;
NUMBER OF ROWS OF THE MATRIX;
M : <ARITHMETIC EXPRESSION>;
NUMBER OF COLUMNS OF THE MATRIX (N >= M);
AUX : <ARRAY IDENTIFIER>;
"ARRAY" AUX[2 : 5];
ENTRY : AUX[2] CONTAINS A RELATIVE TOLERANCE USED FOR
CALCULATING THE DIAGONAL ELEMENTS OF THE
UPPER-TRIANGULAR MATRIX;
EXIT :
AUX[3] DELIVERS THE NUMBER OF THE DIAGONAL ELEMENTS OF
THE UPPER-TRIANGULAR MATRIX WHICH ARE FOUND NOT
NEGLIGIBLE; NORMAL EXIT AUX[3] = M;
AUX[5] := THE MAXIMUM OF THE EUCLIDEAN NORMS OF THE
COLUMNS OF THE GIVEN MATRIX;
DIAG : <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1 : M];
EXIT : THE DIAGONAL ELEMENTS OF THE INVERSE OF M'M
WHERE M IS THE MATRIX OF THE LINEAR LEAST SQUARES
PROBLEM;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1 : N];
ENTRY : B CONTAINS THE RIGHT HAND SIDE OF A LINEAR
LEAST SQUARES PROBLEM;
EXIT : B[1 : M] CONTAINS THE SOLUTION OF THE PROBLEM;
B[M + 1 : N] CONTAINS A VECTOR WITH EUCLIDEAN LENGTH
EQUAL TO THE EUCLIDEAN LENGTH OF THE RESIDUAL VECTOR.
PROCEDURES USED :
LSQORTDEC = CP34134,
LSQDGLINV = CP34132,
LSQSOL = CP34131.
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 5
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : AN INTEGER ARRAY AND A REAL ARRAY,
BOTH OF M ELEMENTS, ARE DECLARED.
RUNNING TIME : ROUGHLY PROPORTIONAL TO N * (M ** 2).
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
LSQORTDECSOL SOLVES AN OVERDETERMINED SYSTEM OF N LINEAR EQUATIONS
IN M UNKNOWNS BY CALLING LSQORTDEC AND, IF THIS CALL WAS SUCCESSFUL
LSQDGLINV AND LSQSOL. LSQORTDECSOL DELIVERS THE LEAST SQUARES
SOLUTION AND THE DIAGONAL OF THE INVERSE OF M'M, WHERE M IS
THE COEFFICIENT MATRIX OF THE SYSTEM. SEE SECTION 3.1.1.2.1.1.,
AND LSQSOL (THIS SECTION).
EXAMPLE OF USE :
THE PROGRAM
"BEGIN""COMMENT" 730914, TEST LSQORTDECSOL;
"ARRAY" A, C[1 : 5,1 : 2], B, X[1 : 5], DIAG[1 : 2],
AUX[2 : 5];
"INTEGER" I, J;
"REAL" H;
"REAL""PROCEDURE" SUM(I, A, B, X); "VALUE" A, B;
"INTEGER" I, A, B; "REAL" X;
"BEGIN""REAL" S; S:= 0; "FOR" I:= A "STEP" 1 "UNTIL" B "DO"
S:= S + X; SUM:= S
"END" SUM;
AUX[2]:= "-12; I:= J:= 1;
"FOR" H:= - 2, - 1, 1, 2, 1, 1, 1, 1, 1, 2 "DO"
"BEGIN" A[I,J]:= C[I,J]:= H; "IF" I < 5 "THEN" I:= I + 1 "ELSE"
"BEGIN" I:= 1; J:= J + 1 "END"
"END";
"FOR" H:= 0, 1, 2, 2, 3 "DO"
"BEGIN" B[I]:= X[I]:= H; I:= I + 1 "END";
LSQORTDECSOL(A, 5, 2, AUX, DIAG, X);
"IF" AUX[3] = 2 "THEN"
OUTPUT(61, "("/, "("AUX[2, 3, 5] = ")" +.4D"+DD5B, 3ZD5B,
+.4D"+DD/, "("LSQ SOLUTION :")", 2(2B+.8D"+DD), /
"("RESIDUE (DELIVERED) :")" +.8D"+DD/,
"("RESIDUE (CHECKED) :")" +.8D"+DD/,
"("DIAGONAL OF INVERSE M'M :")", 2(2B+.8D"+DD)")",
AUX[2], AUX[3], AUX[5], X[1], X[2],
SQRT(VECVEC(3, 5, 0, X, X)),
SQRT(SUM(I, 1, 5, (B[I] - C[I,1] * X[1] - C[I,2] * X[2])
** 2)), DIAG[1], DIAG[2])
"END"
1SECTION : 3.1.1.2.1.2 (MAY 1974) PAGE 6
WHICH SOLVES THE PROBLEM OF THE EXAMPLE OF USE OF LSQSOL,
DELIVERS :
AUX[2, 3, 5] = +.1000"-11 2 +.3317"+01
LSQ SOLUTION : +.50000000"+00 +.12500000"+01
RESIDUE (DELIVERED) :+.50000000"+00
RESIDUE (CHECKED) :+.50000000"+00
DIAGONAL OF INVERSE M'M : +.95238095"-01 +.13095238"+00
SOURCE TEXT(S) :
0"CODE" 34131;
"PROCEDURE" LSQSOL(A, N, M, AID, CI, B); "VALUE" N, M;
"INTEGER" N, M; "ARRAY" A, AID, B; "INTEGER" "ARRAY" CI;
"BEGIN" "INTEGER" K, CIK;
"REAL" W;
"FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" ELMVECCOL(K, N, K, B, A,
TAMVEC(K, N, K, A, B) / (AID[K] * A[K,K]));
"FOR" K:= M "STEP" - 1 "UNTIL" 1 "DO" B[K]:= (B[K] - MATVEC
(K + 1, M, K, A, B)) / AID[K];
"FOR" K:= M "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" CIK:= CI[K]; "IF" CIK ^= K "THEN"
"BEGIN" W:= B[K]; B[K]:= B[CIK]; B[CIK]:= W "END"
"END"
"END" LSQSOL;
"EOP"
0"CODE" 34135;
"PROCEDURE" LSQORTDECSOL(A, N, M, AUX, DIAG, B); "VALUE" N, M;
"INTEGER" N, M; "ARRAY" A, AUX, DIAG, B;
"BEGIN" "ARRAY" AID[1:M];
"INTEGER" "ARRAY" CI[1:M];
LSQORTDEC(A, N, M, AUX, AID, CI);
"IF" AUX[3] = M "THEN"
"BEGIN" LSQDGLINV(A, M, AID, CI, DIAG);
LSQSOL(A, N, M, AID, CI, B)
"END"
"END" LSQORTDECSOL;
"EOP"
1SECTION : 3.1.1.2.1.3 (OCTOBER 1974) PAGE 1
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 740617.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS ONE PROCEDURE,
LSQINV, FOR THE CALCULATION OF THE INVERSE OF THE MATRIX S'S,
WHERE S IS THE COEFFICIENT MATRIX OF A LINEAR LEAST SQUARES
PROBLEM.
KEYWORDS :
INVERSE MATRIX,
LINEAR LEAST SQUARES PROBLEM.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" LSQINV(A, M, AID, C); "VALUE" M; "INTEGER" M;
"ARRAY" A, AID; "INTEGER""ARRAY" C;
"CODE" 34136;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1 : M, 1 : M];
ENTRY : IN THE UPPER TRIANGLE OF A (THE ELEMENTS
A[I,J] WITH 1 <= I < J <= M) THE SUPERDIAGONAL
ELEMENTS SHOULD BE GIVEN OF THE UPPERTRIANGULAR MATRIX
THAT IS PRODUCED BY THE HOUSEHOLDER TRIANGULARIZATION
IN A CALL OF THE PROCEDURE LSQORTDEC (SECTION
3.1.1.2.1.1.) WITH A NORMAL EXIT (AUX[3] = M).
SEE ALSO THE MEANING OF THE PARAMETER AID;
EXIT : THE UPPER TRIANGLE OF THE (SYMMETRIC) INVERSE
MATRIX IS DELIVERED IN THE UPPERTRIANGULAR ELEMENTS OF
THE ARRAY A (A[I,J] FOR 1 <= I <= J <= M);
M : <ARITHMETIC EXPRESSION>;
NUMBER OF COLUMNS OF THE MATRIX OF THE LINEAR LEAST
SQUARES PROBLEM;
AID : <ARRAY IDENTIFIER>;
"ARRAY" AID[1 : M];
ENTRY : AID CONTAINS THE DIAGONAL ELEMENTS OF THE
UPPERTRIANGULAR MATRIX THAT IS PRODUCED BY LSQORTDEC;
C : <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" C[1 : M];
ENTRY : C CONTAINS THE PIVOTAL INDICES AS PRODUCED BY
A CALL OF LSQORTDEC.
1SECTION : 3.1.1.2.1.3 (OCTOBER 1974) PAGE 2
PROCEDURES USED :
CHLINV2 = CP34400,
ICHCOL = CP34031,
ICHROW = CP34032,
ICHROWCOL = CP34033.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : A REAL ARRAY OF M ELEMENTS IS
DECLARED (IN THE CALL OF CHLINV2).
RUNNING TIME : PROPORTIONAL TO M CUBED.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
LSQINV SHOULD BE CALLED AFTER A SUCCESSFUL CALL OF LSQORTDEC
(SECTION 3.1.1.2.1.1.). LSQINV CAN BE USED FOR THE CALCULATION OF
THE COVARIANCE MATRIX OF A LINEAR LEAST SQUARES PROBLEM.
LET S BE THE MATRIX OF THE LEAST SQUARES SYSTEM WITH PERMUTED
COLUMNS AND Q * R THE HOUSEHOLDER DECOMPOSITION OF S. THEN THE
INVERSE OF S'S ALSO IS THE INVERSE OF R'R. SINCE R IS THE
CHOLESKY MATRIX OF R'R, THE INVERSE MATRIX IS COMPUTED IN A CALL
OF CHLINV2 (SECTION 3.1.1.1.1.2.4.). AFTERWARDS THE COVARIANCE
MATRIX IS OBTAINED BY INTERCHANGES OF THE COLUMNS AND ROWS OF THE
INVERSE MATRIX.
EXAMPLE OF USE :
THE FOLLOWING PROGRAM COMPUTES THE INVERSE T OF S'S, WHERE S IS
THE COEFFICIENT MATRIX OF THE SYSTEM IN THE EXAMPLE OF USE OF
LSQORTDEC AND LSQDGLINV (SECTION 3.1.1.2.1.1.). THE DIAGONAL OF S
CAN BE COMPARED WITH THE RESULT OF LSQDGLINV. TO CHECK THE ANSWERS
S' * (S * T) IS PRINTED.
1SECTION : 3.1.1.2.1.3 (OCTOBER 1974) PAGE 3
"BEGIN""COMMENT" JKOK, 740530, EXAMPLE OF USE OF LSQORTDEC AND
LSQINV;
"ARRAY" A, C[1 : 5, 1 : 2], AID[1 : 2], T[1 : 2, 1 : 2],
AUX[2 : 5];
"INTEGER""ARRAY" PIV[1 : 2];
"INTEGER" I, J; "REAL" H;
AUX[2]:= "-12; I:= J:= 1;
"FOR" H:= - 2, - 1, 1, 2, 1, 1, 1, 1, 1, 2 "DO"
"BEGIN" A[I,J]:= C[I,J]:= H; "IF" I < 5 "THEN" I:= I + 1 "ELSE"
"BEGIN" I:= 1; J:= J + 1 "END"
"END";
LSQORTDEC(A, 5, 2, AUX, AID, PIV); "IF" AUX[3] = 2 "THEN"
"BEGIN" LSQINV(A, 2, AID, PIV);
T[1,1]:= A[1,1]; T[2,2]:= A[2,2]; T[2,1]:= T[1,2]:= A[1,2];
"FOR" J:= 1, 2 "DO""FOR" I:= 1 "STEP" 1 "UNTIL" 5 "DO"
A[I,J]:= MATMAT(1, 2, I, J, C, T);
OUTPUT(61, "("/4B, "(" AUX[2, 3, 5] = ")",
-.4D"+DD5B, 3ZD5B, +.4D"+DD/,
2(/4B, 30S, /, 2(/4B, 2(2B+.8D"+DD)), /) ")",
AUX[2], AUX[3], AUX[5],
"(" INVERSE :")", T,
"(" CHECK : S' * (S * T) :")",
TAMMAT(1, 5, 1, 1, C, A),
TAMMAT(1, 5, 1, 2, C, A),
TAMMAT(1, 5, 2, 1, C, A),
TAMMAT(1, 5, 2, 2, C, A) )
"END"
"END"
OUTPUT :
AUX[2, 3, 5] = .1000"-11 2 +.3317"+01
INVERSE :
+.95238095"-01 -.23809524"-01
-.23809524"-01 +.13095238"+00
CHECK : S' * (S * T) :
+.10000000"+01 +.17763568"-14
+.00000000"+00 +.10000000"+01
1SECTION : 3.1.1.2.1.3 (OCTOBER 1974) PAGE 4
SOURCE TEXT(S):
"CODE" 34136;
"PROCEDURE" LSQINV(A, M, AID, C); "VALUE" M; "INTEGER" M;
"ARRAY" A, AID; "INTEGER""ARRAY" C;
"BEGIN""INTEGER" I, CI;
"REAL" W;
"FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" A[I,I]:= AID[I];
CHLINV2(A, M);
"FOR" I:= M "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" CI:= C[I]; "IF" CI ^= I "THEN"
"BEGIN" ICHCOL(1, I - 1, I, CI, A); ICHROWCOL(I + 1, CI - 1,
I, CI, A); ICHROW(CI + 1, M, I, CI, A);
W:= A[I,I]; A[I,I]:= A[CI,CI]; A[CI,CI]:= W
"END"
"END"
"END" LSQINV;
"EOP"
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 1
AUTHORS: A.BJOERCK AND G.H.GOLUB.
CONTRIBUTOR: J.KOOPMAN.
INSTITUTE: UNIVERSITY OF AMSTERDAM.
RECEIVED: 780701 .
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES FOR SOLVING A LINEAR LEAST
SQUARES PROBLEM SUBJECT TO LINEAR CONSTRAINTS:
LSQDECOMP , FOR THE QR-DECOMPOSITION OF A LEAST SQUARES MATRIX,
WHERE THIS MATRIX ALSO CONTAINS THE COEFFICIENTS OF
THE LINEAR CONSTRAINTS (LINEAR EQUATIONS);
LSQREFSOL , FOR THE SOLUTION OF THIS LEAST SQUARES PROBLEM, IF
THE MATRIX HAS BEEN DECOMPOSED BY LSQDECOMP.
KEYWORDS:
LEAST SQUARES,
LINEAR CONSTRAINTS,
HOUSEHOLDER TRIANGULARIZATION,
ITERATIVE REFINEMENT.
SUBSECTION: LSQDECOMP.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LSQDECOMP(A, N, M, N1, AUX, AID, CI);
"VALUE"N,M,N1;"INTEGER"N,M,N1;"ARRAY" A,AUX,AID;
"INTEGER""ARRAY" CI;
"CODE" 34137;
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 2
THE MEANING OF THE FORMAL PARAMETERS IS:
A :<ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:M];
ENTRY: THE ORIGINAL LEAST SQUARES MATRIX, WHERE THE FIRST
N1 ROWS SHOULD FORM THE CONSTRAINT MATRIX (I.E. THE
FIRST N1 EQUATIONS ARE TO BE STRICTLY SATISFIED);
EXIT : IN THE UPPER TRIANGLE OF A (THE ELEMENTS A[I,J] WITH
I<J) THE SUPERDIAGONAL PART OF THE UPPER TRIANGULAR
MATRIX, PRODUCED BY HOUSEHOLDER TRANSFORMATIONS; IN
THE OTHER PART OF THE COLUMNS OF A THE SIGNIFICANT
ELEMENTS OF THE GENERATING VECTORS OF THE HOUSEHOLDER
MATRICES USED FOR THE TRIANGULARIZATION;
N :<ARITHMETIC EXPRESSION>;
NUMBER OF ROWS OF THE MATRIX;
M :<ARITHMETIC EXPRESSION>;
NUMBER OF COLUMNS OF THE MATRIX;
N1 :<ARITHMETIC EXPRESSION>;
NUMBER OF LINEAR CONSTRAINTS, I.E. THE FIRST N1 ROWS OF A
SET UP A SYSTEM OF N1 LINEAR EQUATIONS THAT MUST BE
STRICTLY SATISFIED (OF COURSE, IF THERE ARE NO CON-
STRAINTS, N1 MUST BE CHOSEN ZERO);
AUX :<ARRAY IDENTIFIER>;
"ARRAY" AUX[2:7];
ENTRY: AUX[2] CONTAINS A RELATIVE TOLERANCE FOR CALCULATING
THE DIAGONAL ELEMENTS OF THE UPPER TRIANGULAR MATRIX;
EXIT: AUX[3] CONTAINS THE NUMBER OF DIAGONAL ELEMENTS WHICH
ARE NOT NEGLIGIBLE, NORMAL EXIT AUX[3]=M;
AID :<ARRAY IDENTIFIER>;
"ARRAY" AID[1:M];
NORMAL EXIT (AUX[3]=M): THE DIAGONAL ELEMENTS OF THE UPPER
TRIANGULAR MATRIX PRODUCED BY THE HOUSEHOLDER TRANSFORMATION
CI :<ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:M];
EXIT: THE PIVOTAL INDICES OF THE INTERCHANGES OF THE COLUMNS
OF THE GIVEN MATRIX;
PROCEDURES USED:
MATMAT = CP34013.
TAMMAT = CP34014.
ELMCOL = CP34023.
ICHCOL = CP34031.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO N*M**2-M**3/3.
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 3
METHOD AND PERFORMANCE:
LET A DENOTE THE GIVEN MATRIX. LSQDECOMP PRODUCES AN N-TH ORDER
ORTHOGONAL MATRIX Q AND AN N*M UPPER TRIANGULAR MATRIX R SUCH THAT
R EQUALS QA WITH PERMUTED COLUMNS.
THE ORTHOGONAL MATRIX Q IS FORMED AS A PRODUCT OF AT MOST M TRANS-
FORMATIONS OF THE FORM (I-BETA U U'). THESE HOUSEHOLDER MATRICES
REDUCE A TO THE MATRIX R: AT THE K-TH STAGE THE K-TH COLUMN OF THE
(ALREADY MODIFIED) MATRIX IS INTERCHANGED WITH THE COLUMN OF
MAXIMUM EUCLIDEAN NORM.THESE INTERCHANGES ARE RECORDED IN THE ARRAY
CI. PREMATURE TERMINATION OCCURS IF AT SOME STAGE THE EUCLIDEAN
NORM OF THE PIVOTAL COLUMN IS LESS THAN SOME TOLERANCE (AUX[2])
TIMES THE MAXIMUM OF THE EUCLIDEAN NORMS OF THE COLUMNS OF THE
MATRIX.
LSQDECOMP DELIVERS THE UPPER TRIANGULAR MATRIX, WHERE THE DIAGONAL
ELEMENTS ARE STORED IN THE ARRAY AID AND THE OFF-DIAGONAL ELEMENTS
IN THE UPPER TRIANGULAR PART OF A. THE SIGNIFICANT ELEMENTS OF THE
GENERATING VECTORS OF THE HOUSEHOLDER TRANSFORMATIONS ARE STORED
IN THE COLUMNS OF A, I.E. ON AND BELOW THE LEADING DIAGONAL OF
A. IT SHOULD BE NOTED THAT FOR THE SOLUTION OF LEAST SQUARES
PROBLEMS, ONLY CALLS WITH N>=M ARE USEFUL.
REFERENCES:
A.BJOERCK AND G.H.GOLUB:
ITERATIVE REFINEMENT OF LEAST SQUARES SOLUTIONS BY HOUSEHOLDER
TRANSFORMATION, BIT 7 (1967), PP. 322-337
SUBSECTION: LSQREFSOL.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" LSQREFSOL(A, QR, N, M, N1, AUX,AID,CI,B,LDX,X,RES);
"VALUE"N,M,N1;"INTEGER"N,M,N1;"INTEGER""ARRAY"CI;"REAL" LDX;
"ARRAY"A,QR,AUX,AID,B,X,RES;
"CODE" 34138;
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 4
THE MEANING OF THE FORMAL PARAMETERS IS:
A :<ARRAY IDENTIFIER>;
"ARRAY" A[1:N,1:M];
THE ORIGINAL LEAST SQUARES MATRIX, WHERE THE FIRST N1
ROWS SET UP A SYSTEM OF LINEAR EQUATIONS THAT MUST BE
STRICTLY SATISFIED;
QR :<ARRAY IDENTIFIER>;
"ARRAY" QR[1:N,1:M];
THE QR-DECOMPOSITION OF THE ORIGINAL LEAST SQUARES
MATRIX AS DELIVERED BY A SUCCESSFUL CALL OF LSQDECOMP;
N :<ARITHMETIC EXPRESSION>;
NUMBER OF ROWS OF THE MATRICES A AND QR;
M :<ARITHMETIC EXPRESSION>;
NUMBER OF COLUMNS OF THE MATRICES A AND QR;
N1 :<ARITHMETIC EXPRESSION>;
NUMBER OF LINEAR CONSTRAINTS;
AUX :<ARRAY IDENTIFIER>;
"ARRAY" AUX[2:7];
ENTRY: AUX[2] CONTAINS A RELATIVE TOLERANCE AS A
CRITERION TO STOP ITERATIVE REFINING: IF THE
EUCLIDEAN NORM OF THE CORRECTION IS SMALLER
THAN AUX[2] TIMES THE CURRENT APPROXIMATION OF
THE SOLUTION, THE ITERATIVE REFINING IS STOPPED;
AUX[6]: MAXIMUM NUMBER OF ITERATIONS ALLOWED
(USUALLY AUX[6]=5 WILL BE SUFFICIENT);
EXIT : AUX[7]: THE NUMBER OF ITERATIONS PERFORMED;
AID :<ARRAY IDENTIFIER>;
"ARRAY" AID[1:M];
THE DIAGONAL ELEMENTS OF THE UPPER TRIANGULAR MATRIX
AS DELIVERED BY A SUCCESSFUL CALL OF LSQDECOMP;
CI :<ARRAY IDENTIFIER>;
"INTEGER""ARRAY" CI[1:M];
THE PIVOTAL INDICES AS PRODUCED BY LSQDECOMP;
B :<ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
THE RIGHT-HAND SIDE OF THE LEAST SQUARES PROBLEM;
FIRST N1 ELEMENTS FORM THE RIGHT HAND SIDES OF THE
CONSTRAINTS;
LDX :<REAL VARIABLE>;
THE EUCLIDEAN NORM OF THE LAST CORRECTION OF THE
SOLUTION;
X :<ARRAY IDENTIFIER>;
"ARRAY" X[1:M];
EXIT: THE SOLUTION VECTOR;
RES :<ARRAY IDENTIFIER>;
"ARRAY" RES[1:N];
EXIT: THE RESIDUAL VECTOR CORRESPONDING TO THE SOLUTION;
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 5
PROCEDURES USED:
VECVEC = CP34010.
MATVEC = CP34011.
TAMVEC = CP34012.
ELMVECCOL = CP34021.
ICHCOL = CP34031.
LNG SUB = CP31106.
LNGMATVEC = CP34411.
LNGTAMVEC = CP34412.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO C1*M*N-C2*M**2, WHERE C1 AND C2
ARE CONSTANTS.
METHOD AND PERFORMANCE:
LSQREFSOL SHOULD BE CALLED AFTER A SUCCESSFUL CALL OF LSQDECOMP
(I.E. AUX[3]=M). LSQREFSOL YIELDS THE LEAST SQUARES SOLUTION OF
THE OVERDETERMINED SYSTEM WITH THE DECOMPOSED COEFFICIENT MATRIX
IN THE ARRAY A AND THE RIGHT-HAND SIDE IN ARRAY B. THE ORIGINAL
LEAST SQUARES MATRIX ALSO CONTAINS THE LINEAR CONSTRAINTS (THE
FIRST N1 ROWS OF THIS MATRIX SET UP A SYSTEM THAT MUST BE STRICT-
LY SATISFIED). FIRST, THE ORTHOGONAL TRANSFORMATION WITH THE
HOUSEHOLDER MATRICES IS PERFORMED ON THE RIGHT-HAND SIDE. NEXT
THE SYSTEM IS SOLVED BY MEANS OF A BACK SUBSTITUTION. IN THIS
WAY THE FIRST APPROXIMATION OF THE SOLUTION (WITH PERMUTED
COLUMNS) IS OBTAINED.
AFTER THIS AN ITERATIVE PROCESS REFINES THE APPROXIMATION
UNTIL THE EUCLIDEAN NORM OF THE CORRECTION VECTOR IS NEGLIGIBLY
SMALL COMPARED TO THE APPROXIMATION OR UNTIL THE MAXIMUM NUMBER
OF ITERATIONS IS REACHED. FOR A MORE DETAILED DESCRIPTION OF
THE ITERATIVE IMPROVEMENT, SEE REF[1].
AFTER THE ITERATIVE PROCESS AN APPROXIMATION TO THE SOLUTION IS
FOUND. HOWEVER, THE ORDER OF THE COMPONENTS POSSIBLY IS NOT
CORRECT. THEREFORE THIS ORDER IS RESTORED AT THE END OF THE
PROCEDURE (SEE ALSO METHOD AND PERFORMANCE OF LSQDECOMP).
THE LEAST SQUARES SOLUTIONS OF SEVERAL OVERDETERMINED SYSTEMS
WITH THE SAME CONSTRAINTS AND COEFFICIENT MATRIX CAN BE SOLVED
BY SUCCESSIVE CALLS OF LSQREFSOL WITH DIFFERENT RIGHT-HAND
SIDES.
REFERENCES:
[1] A.BJOERCK AND G.H.GOLUB:
ITERATIVE REFINEMENT OF LEAST SQUARES SOLUTIONS BY HOUSEHOLDER
TRANSFORMATION, BIT 7 (1967), PP. 322-337.
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 6
EXAMPLE OF USE:
THE FOLLOWING PROGRAM SOLVES THE PROBLEM:
MINIMIZE //B - A*X//
UNDER X1 + 1000*X2 + 5*X3 = 2016 ,WHERE
A IS THE MATRIX
1 0 8
0 3 2
1 2 "-5
0 0 0
AND THE VECTOR B = (25, 12, 5.00003, 1)' , X = (X1, X2, X3)' ;
"BEGIN" "INTEGER" N,M,N1,I,J;
N := 5; M := 3; N1 := 1;
"BEGIN""INTEGER""ARRAY" CI[1:M];
"ARRAY" AUX[2:7],QR,A[1:N,1:M],B,RES[1:N],AID,X[1:M];
"REAL" LDX;
A[1,1] := 1; A[1,2] := 1000; A[1,3] := 5;
A[2,1] := 1; A[2,2] := 0; A[2,3] := 8;
A[3,1] := 0; A[3,2] := 3; A[3,3] := 2;
A[4,1] := 1; A[4,2] := 2; A[4,3] := "-5;
A[5,1]:=A[5,2]:=A[5,3]:=0;
B[1] := 2016; B[2] :=25; B[3] :=12; B[4] := 5.00003;
B[5] := 1; AUX[2] := "-14; AUX[6] := 5;
"FOR" I:=1 "STEP" 1 "UNTIL" N "DO"
"FOR" J:=1 "STEP" 1 "UNTIL" M "DO" QR[I,J] := A[I,J];
LSQDECOMP(QR,N,M,N1,AUX,AID,CI);
LSQREFSOL(A,QR,N,M,N1,AUX,AID,CI,B,LDX,X,RES);
OUTPUT(61,"(""("THE SOLUTION VECTOR: ")",//")");
"FOR" I:=1 "STEP" 1 "UNTIL" M "DO" OUTPUT(61,"("/")",X[I]);
OUTPUT(61,"("//,"("THE RESIDUAL VECTOR: ")",//")");
"FOR" J:=N1+1 "STEP" 1 "UNTIL" N "DO" OUTPUT(61,"("/")",RES[J]);
OUTPUT(61,"("///,"("NUMBER OF ITERATIONS: ")",D,//,
"("NORM LAST CORRECTION OF X: ")",N")",AUX[7],LDX)
"END"
"END"
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 7
DELIVERS:
THE SOLUTION VECTOR:
+1.0000000000000"+000
+2.0000000000000"+000
+3.0000000000000"+000
THE RESIDUAL VECTOR:
-5.2734444477081"-016
+2.1280091641666"-015
+5.3479806840033"-016
+1.0000000000000"+000
NUMBER OF ITERATIONS: 2
NORM LAST CORRECTION OF X: +2.1657844626990"-015
SOURCE TEXT(S):
0"CODE" 34137;
"PROCEDURE" LSQDECOMP( A, N ,M ,N1 ,AUX ,AID ,CI );
"VALUE" N , M ,N1;"INTEGER" N,M,N1;"ARRAY" A,AUX, AID;
"INTEGER""ARRAY" CI;
"BEGIN""INTEGER"J,K,KPIV,NR,S;"BOOLEAN" FSUM;
"REAL" BETA,SIGMA,NORM,AIDK,AKK,W,EPS;
"ARRAY" SUM[1:M];
NORM:=0 ; AUX[3]:=M;NR:=N1;FSUM:="TRUE";
"FOR" K:=1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" "IF" K=N1+1 "THEN" "BEGIN" FSUM:="TRUE"; NR:=N "END";
"IF" FSUM "THEN"
"FOR" J:=K "STEP" 1 "UNTIL" M "DO"
"BEGIN" W:=SUM[J]:= TAMMAT(K ,NR ,J ,J ,A ,A);
"IF" W>NORM "THEN" NORM:=W
"END"; FSUM:="FALSE";EPS:=AUX[2]*SQRT(NORM);
SIGMA:=SUM[K]; KPIV:=K;
"FOR" J:=K+1 "STEP" 1 "UNTIL" M "DO"
"IF" SUM[J]>SIGMA "THEN"
"BEGIN" SIGMA:=SUM[J]; KPIV:=J "END";
"COMMENT"
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 8
;
"IF" KPIV^=K "THEN"
"BEGIN" SUM[KPIV]:=SUM[K]; ICHCOL( 1 ,N ,K ,KPIV ,A) "END";
CI[K]:=KPIV; AKK:=A[K,K];
SIGMA:=TAMMAT(K ,NR ,K ,K ,A ,A); W:=SQRT(SIGMA);
AIDK:=AID[K]:="IF" AKK<0 "THEN" W "ELSE" -W;
"IF" W<EPS "THEN"
"BEGIN" AUX[3]:=K-1;"GOTO" ENDDEC "END";
BETA:= 1/(SIGMA-AKK*AIDK); A[K,K]:=AKK-AIDK;
"FOR" J:=K+1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" ELMCOL(K ,NR ,J ,K ,A ,A ,-BETA*TAMMAT(K ,NR ,
K ,J ,A ,A)); SUM[J]:=SUM[J]-A[K,J]**2
"END";
"IF" K=N1 "THEN"
"FOR" J:=N1+1 "STEP" 1 "UNTIL" N "DO"
"FOR" S:=1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" NR:="IF" S>N1 "THEN" N1 "ELSE" S-1;
W:=A[J,S]-MATMAT(1 ,NR , J ,S ,A ,A);
A[J,S]:="IF" S>N1 "THEN" W "ELSE" W/AID[S]
"END"
"END" FOR K;
ENDDEC:
"END" LSQDECOMP;
"EOP"
"CODE" 34138;
"PROCEDURE" LSQREFSOL(A, QR, N, M, N1, AUX, AID, CI, B,LDX,X,RES);
"VALUE"N,M,N1;"INTEGER"N,M,N1;"INTEGER""ARRAY"CI;"REAL"LDX;
"ARRAY"QR,A,AID,AUX,X,RES,B;
"BEGIN""INTEGER"I,J,K,S;
"REAL"C1,NEXVE,NDX,NDR,D,DD,OP,OPL,CORRNORM;
"ARRAY"F[1:N],G[1:M];
"PROCEDURE" HOUSEHOLDER(P, Q, R, E);
"VALUE" P,Q,R,E;"INTEGER" P,Q,R,E;
"BEGIN" "FOR" S:=P "STEP" Q "UNTIL" R "DO"
ELMVECCOL(S, E, S, F, QR,TAMVEC(S, E, S,QR, F)/(QR[S,S]*
AID[S]))
"END";
"FOR" J:=1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" S:=CI[J];"IF" S^=J"THEN" ICHCOL(1,N,J,S,A) "END";
"FOR"J:=1"STEP"1"UNTIL"M"DO"X[J]:=G[J]:=0;
"FOR"I:=1"STEP"1"UNTIL"N"DO"
"BEGIN"RES[I]:=0;F[I]:=B[I]"END";
"COMMENT"
1SECTION : 3.1.1.2.1.4 (DECEMBER 1978) PAGE 9
;
"FOR"K:=0,1,K+1
"WHILE" (CORRNORM>AUX[2]*NEXVE & K<=AUX[6])
"DO"
"BEGIN"NDX:=NDR:=0;
"IF"K^=0"THEN"
"BEGIN""FOR"I:=1"STEP"1"UNTIL"N"DO"RES[I]:=RES[I]+F[I];
"FOR"S:=1"STEP"1"UNTIL"M"DO"
"BEGIN" X[S]:=X[S]+G[S];
LNGTAMVEC(1,N,S,A,RES,0,0,D,DD);
G[S]:=(-D-TAMVEC(1,S-1,S,QR,G))/AID[S]
"END";
"FOR"I:=1"STEP"1"UNTIL"N"DO"
"BEGIN" LNGMATVEC(1, M, I, A, X,
"IF" I>N1 "THEN" RES[I] "ELSE" 0, 0, D, DD);
LNG SUB(B[I],0,D,DD,OP,OPL);
F[I]:=OP
"END"
"END";
NEXVE:=SQRT(VECVEC(1,M,0,X,X)+VECVEC(1,N,0,RES,RES));
HOUSEHOLDER(1, 1, N1, N1);
"FOR" I:=N1+1 "STEP" 1 "UNTIL" N "DO"
F[I]:=F[I]-MATVEC(1,N1, I, QR, F);
HOUSEHOLDER(N1+1, 1, M, N);
"FOR" I:=1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" C1:=F[I];F[I]:=G[I];
G[I]:="IF" I>N1 "THEN" C1-G[I] "ELSE" C1
"END";
"FOR"S:=M"STEP"-1"UNTIL"1"DO"
"BEGIN"G[S]:=(G[S]-MATVEC(S+1,M,S,QR,G))/AID[S];
NDX:=NDX+G[S]**2
"END";
HOUSEHOLDER(M, -1, N1+1, N);
"FOR" S:=1 "STEP" 1 "UNTIL" N1 "DO"
F[S]:=F[S]-TAMVEC(N1+1, N, S, QR, F);
HOUSEHOLDER(N1, -1, 1, N1);
AUX[7]:=K;
"FOR"I:=1"STEP"1"UNTIL"N"DO"NDR:=NDR+F[I]**2;
CORRNORM:=SQRT(NDX+NDR)
"END"FOR K;
LDX:=SQRT(NDX);
"FOR" S:=M "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" J:=CI[S];"IF" J^=S "THEN"
"BEGIN" C1:=X[J];X[J]:=X[S];X[S]:=C1;ICHCOL(1,N,J,S,A)"END"
"END"
"END" LSQREFSOL;
"EOP"
1SECTION : 3.1.1.3.1.1 (FEBRUARY 1979) PAGE 1
AUTHOR : D.T.WINTER
INSTITUTE : MATHEMATICAL CENTRE
RECEIVED : 731217
BRIEF DESCRIPTION :
THIS SECTION CONTAINS 2 PROCEDURES FOR THE SOLUTION OF AN
OVERDETERMINED SYSTEM OF LINEAR EQUATIONS:
SOLSVDOVR SOLVES AN OVERDETERMINED SYSTEM OF LINEAR EQUATIONS ,
MULTIPLYING THE RIGHT-HAND SIDE BY THE PSEUDO-INVERSE OF THE GIVEN
MATRIX; THE SINGULAR VALUES DECOMPOSITION SHOULD BE AVAILABLE.
SOLOVR CALCULATES THE SINGULAR VALUES DECOMPOSITION AND SOLVES AN
OVERDETERMINED SYSTEM OF LINEAR EQUATIONS.
KEYWORDS :
BEST LEAST-SQUARES SOLUTION
SINGULAR VALUES
PSEUDO-INVERSE
SUBSECTION : SOLSVDOVR
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" SOLSVDOVR(U, VAL, V, M, N, X, EM);
"VALUE" M,N; "INTEGER" M,N; "ARRAY" U, VAL, V, X, EM; "CODE" 34280;
THE MEANING OF THE FORMAL PARAMETERS IS :
U: <ARRAY IDENTIFIER>;
"ARRAY" U[1:M,1:N];
ENTRY:THE MATRIX U IN THE SINGULAR VALUES DECOMPOSITION U*S*V'.
VAL: <ARRAY IDENTIFIER>;
"ARRAY" VAL[1:N];
ENTRY:THE SINGULAR VALUES.
V: <ARRAY IDENTIFIER>;
"ARRAY" V[1:N,1:N];
ENTRY:THE MATRIX V IN THE SINGULAR VALUES DECOMPOSITION.
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF UNKNOWNS.
M: <ARITHMETIC EXPRESSION>;
THE LENGTH OF THE RIGHT-HAND SIDE VECTOR, N SHOULD SATISFY
N <= M.
X: <ARRAY IDENTIFIER>;
"ARRAY" X[1:M];
ENTRY: THE RIGHT-HAND SIDE VECTOR;
EXIT: THE SOLUTION VECTOR IN X[1:N].
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[6:6];
ENTRY: EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE.
1SECTION : 3.1.1.3.1.1 (FEBRUARY 1979) PAGE 2
PROCEDURES USED :
MATVEC = CP34011
TAMVEC = CP34012
REQUIRED CENTRAL MEMORY : AN AUXILIARY ARRAY OF N REALS IS DECLARED.
RUNNING TIME : ROUGHLY PROPORTIONAL TO (M + N) * N
METHOD AND PERFORMANCE :
THE SOLUTION IS FOUND IN THREE STEPS :
1. U' * X = X1 IS CALCULATED,
2. VAL+ * X1 = X2 IS CALCULATED, HERE VAL+ DENOTES THE DIAGONAL
MATRIX OBTAINED FROM VAL BY SETTING VAL+[I,I] = 1/VAL[I] IF
VAL[I] GREATER THAN OR EQUAL TO EM[6], AND 0 OTHERWISE,
3. THE SOLUTION V * X2 IS CALCULATED.
SUBSECTION : SOLOVR
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"INTEGER" "PROCEDURE" SOLOVR(A, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, X, EM; "CODE" 34281;
SOLOVR:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. ZERO IF ALL
SINGULAR VALUES ARE CALCULATED.
THE MEANING OF THE FORMAL PAREMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:M,1:N];
ENTRY: THE MATRIX OF THE SYSTEM;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF A;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF A, N <= M;
X: <ARRAY IDENTIFIER>;
"ARRAY" X[1:M];
ENTRY: THE RIGHT-HAND SIDE VECTOR;
EXIT: THE SOLUTION VECTOR IN X[1:N];
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[0:7];
ENTRY: EM[0]: THE MACHINE PRECISION;
EM[2]: THE RELATIVE PRECISION OF THE SINGULAR VALUES;
EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED IN
THE SINGULAR VALUES DECOMPOSITION;
EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE;
EXIT:EM[1]: THE INFINITY NORM OF THE MATRIX;
EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT;
EM[5]: THE NUMBER OF ITERATIONS PERFORMED IN THE SINGULAR
VALUES DECOMPOSITION;
EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF
SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6].
1SECTION : 3.1.1.3.1.1 (FEBRUARY 1979) PAGE 3
PROCEDURES USED :
QRISNGVALDEC = CP34273
SOLSVDOVR = CP34280
REQUIRED CENTRAL MEMORY :
AUXILIARY ARRAYS ARE DECLARED TO A TOTAL OF (N + 2) * N REALS
RUNNING TIME : ROUGHLY PROPORTIONAL TO (M + N) * N * N
METHOD AND PERFORMANCE :
THE SOLUTION IS FOUND IN TWO STEPS :
1. THE SINGULAR VALUES DECOMPOSITION IS CALCULATED BY MEANS OF THE
PROCEDURE QRISNGVALDEC (SECTION 3.5.1.2);
2. THE SOLUTION IS CALCULATED BY MEANS OF THE PROCEDURE SOLSVDOVR,
(THIS SECTION);
REFERENCES :
WILKINSON, J.H. AND C.REINSCH
HANDBOOK OF AUTOMATIC COMPUTATION, VOL. 2 (CONTRIBUTION I-10)
LINEAR ALGEBRA
HEIDELBERG (1971)
EXAMPLE OF USE :
FIRST A PROGRAM IS GIVEN, AND THEN THE RESULTS OF THIS PROGRAM :
"BEGIN" "ARRAY" A[1:8,1:5], B[1:8], EM[0:7];
"INTEGER" I;
A[1,1]:=22; A[1,2]:= A[2,3]:=10; A[1,3]:= A[7,1]:= A[8,5]:=2;
A[1,4]:= A[3,5]:=3; A[1,5]:= A[2,2]:=7; A[2,1]:=14; A[2,5]:=8;
A[2,4]:= A[8,3]:=0; A[3,1]:= A[3,3]:= A[6,5]:=-1; A[3,2]:=13;
A[3,4]:=-11; A[4,1]:=-3; A[4,2]:= A[4,4]:= A[5,4]:= A[8,4]:=-2;
A[4,3]:=13; A[4,5]:= A[5,5]:= A[8,1]:=4; A[5,1]:= A[6,1]:=9;
A[5,2]:=8; A[5,3]:= A[6,2]:= A[7,5]:=1; A[6,3]:=-7;
A[6,4]:= A[7,4]:= A[8,2]:=5; A[7,2]:=-6; A[7,3]:=6;
B[1]:=-1; B[2]:=2; B[3]:= B[7]:=1; B[4]:=4; B[5]:= B[8]:=0;
B[6]:=-3; EM[0]:="-14; EM[2]:="-12; EM[4]:=80; EM[6]:="-10;
I:= SOLOVR(A, 8, 5, B, EM);
OUTPUT(61, "("4B, "("NUMBER SINGULAR VALUES NOT FOUND : ")",
3ZD,/, 4B, "("NORM : ")", N,/, 4B, "("MAX NEGL SUBD ELEM : ")",
N,/, 4B, "("NUMBER ITERATIONS : ")", 3ZD, /, 4B, "("RANK : ")",
3ZD, /")", I, EM[1], EM[3], EM[5], EM[7]);
OUTPUT(61, "("/, 4B, "("SOLUTION VECTOR")",/,/, 5(4B, N, /)")",
B[1], B[2], B[3], B[4], B[5])
"END"
1SECTION : 3.1.1.3.1.1 (FEBRUARY 1979) PAGE 4
NUMBER SINGULAR VALUES NOT FOUND : 0
NORM : +4.4000000000000"+001
MAX NEGL SUBD ELEM : +4.3977072741076"-014
NUMBER ITERATIONS : 6
RANK : 3
SOLUTION VECTOR
-8.3333333333334"-002
+1.0989227456287"-015
+2.5000000000000"-001
-8.3333333333332"-002
+8.3333333333334"-002
"EOP"
SOURCE TEXT(S):
0"CODE" 34280;
"PROCEDURE" SOLSVDOVR(U, VAL, V, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V, X, EM;
"BEGIN" "INTEGER" I;
"REAL" MIN;
"ARRAY" X1[1:N];
MIN:= EM[6];
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
X1[I]:= "IF" VAL[I] <= MIN "THEN" 0 "ELSE" TAMVEC(1, M, I, U, X) /
VAL[I];
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
X[I]:= MATVEC(1, N, I, V, X1)
"END" SOLSVDOVR;
"EOP"
0"CODE" 34281;
"INTEGER" "PROCEDURE" SOLOVR(A, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, X, EM;
"BEGIN" "INTEGER" I;
"ARRAY" VAL[1:N], V[1:N,1:N];
SOLOVR:= I:= QRISNGVALDEC(A, M, N, VAL, V, EM);
"IF" I = 0 "THEN" SOLSVDOVR(A, VAL, V, M, N, X, EM)
"END" SOLOVR;
"EOP"
1SECTION 3.1.1.3.1.2 (DECEMBER 1975) PAGE 1
AUTHOR : D.T.WINTER
INSTITUTE : MATHEMATICAL CENTRE
RECEIVED : 731217
BRIEF DESCRIPTION :
THIS SECTION CONTAINS 2 PROCEDURES FOR THE SOLUTION OF AN
UNDERDETERMINED SYSTEM OF LINEAR EQUATIONS:
SOLUND EXPECTS AS INPUT THE MATRIX OF THE SYSTEM OF EQUATIONS,
CALCULATES THE SINGULAR VALUES DECOMPOSITION BY MEANS OF THE
PROCEDURE QRISNGVALDEC, AND SOLVES THE SYSTEM BY MEANS OF THE
PROCEDURE SOLSVDUND.
SOLSVDUND ASSUMES THAT THE MATRIX IS
ALREADY DECOMPOSED AND SOLVES THE SYSTEM OF EQUATIONS, MULTIPLYING
THE RIGHT-HAND SIDE BY THE PSEUDO-INVERSE OF THE GIVEN MATRIX.
KEYWORDS :
BEST LEAST-SQUARES SOLUTION
SINGULAR VALUES
PSEUDO-INVERSE
1SECTION 3.1.1.3.1.2 (DECEMBER 1975) PAGE 2
SUBSECTION : SOLSVDUND
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" SOLSVDUND(U, VAL, V, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V, X, EM;
"CODE" 34282;
THE MEANING OF THE FORMAL PARAMETERS IS :
U: <ARRAY IDENTIFIER>;
"ARRAY" U[1:M,1:N];
ENTRY:THE MATRIX U IN THE SINGULAR VALUES DECOMPOSITION V*S*U'.
VAL: <ARRAY IDENTIFIER>;
"ARRAY" VAL[1:N];
ENTRY:THE SINGULAR VALUES;
V: <ARRAY IDENTIFIER>;
"ARRAY" V[1:N,1:N];
ENTRY:THE MATRIX V IN THE SINGULAR VALUES DECOMPOSITION.
N: <ARITHMETIC EXPRESSION>;
THE LENGTH OF THE RIGHT-HAND SIDE VECTOR;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF UNKNOWNS, N SHOULD SATISFY N <= M;
X: <ARRAY IDENTIFIER>;
"ARRAY" X[1:M];
ENTRY: THE RIGHT-HAND SIDE VECTOR IN X[1:N];
EXIT: THE SOLUTION VECTOR IN X[1:M];
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[6:6];
ENTRY: EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE.
PROCEDURES USED :
MATVEC = CP34011
TAMVEC = CP34012
REQUIRED CENTRAL MEMORY : AN AUXILIARY ARRAY OF N REALS IS DECLARED
RUNNING TIME : ROUGHLY PROPORTIONAL TO (M + N) * N
METHOD AND PERFORMANCE :
THE SOLUTION IS FOUND IN THREE STEPS :
1. V' * X = X1 IS CALCULATED,
2. VAL+ * X1 = X2 IS CALCULATED, HERE VAL+ DENOTES THE DIAGONAL
MATRIX OBTAINED FROM VAL BY SETTING VAL+[I,I] = 1/VAL[I] IF
VAL[I] GREATER THAN OR EQUAL TO EM[6], AND 0 OTHERWISE,
3. THE SOLUTION U * X2 IS CALCULATED.
LANGUAGE : ALGOL 60
1SECTION 3.1.1.3.1.2 (DECEMBER 1975) PAGE 3
SUBSECTION : SOLUND
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"INTEGER" "PROCEDURE" SOLUND(A, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, X, EM;
"CODE" 34283;
SOLUND:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. ZERO IF ALL
SINGULAR VALUES ARE CALCULATED.
THE MEANING OF THE FORMAL PAREMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:M,1:N];
ENTRY: THE TRANSPOSE OF THE MATRIX;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF A;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF A, N <= M;
X: <ARRAY IDENTIFIER>;
"ARRAY" X[1:M];
ENTRY: THE RIGHT-HAND SIDE VECTOR IN X[1:N];
EXIT: THE SOLUTION VECTOR.
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[0:7];
ENTRY: EM[0]: THE MACHINE PRECISION;
EM[2]: THE RELATIVE PRECISION FOR THE SINGULAR VALUES;
EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED IN
THE SINGULAR VALUES DECOMPOSITION;
EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE;
EXIT: EM[1]: THE INFINITY NORM OF THE MATRIX;
EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT;
EM[5]: THE NUMBER OF ITERATIONS PERFORMED IN THE SINGULAR
VALUES DECOMPOSITION;
EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF
SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6].
PROCEDURES USED :
QRISNGVALDEC = CP34273
SOLSVDUND = CP34282
REQUIRED CENTRAL MEMORY :
AUXILIARY ARRAYS ARE DECLARED TO A TOTAL OF (N + 1) * N REALS
METHOD AND PERFORMANCE :
THE SOLUTION IS FOUND IN TWO STEPS :
1. THE SINGULAR VALUES DECOMPOSITION IS CALCULATED BY MEANS OF THE
PROCEDURE QRISNGVALDEC;
2. THE SOLUTION IS CALCULATED BY MEANS OF THE PROCEDURE SOLSVDUND.
1SECTION 3.1.1.3.1.2 (MAY 1974) PAGE 4
RUNNING TIME : ROUGHLY PROPORTIONAL TO (M + N) * N * N
LANGUAGE : ALGOL-60
REFERENCES :
WILKINSON, J.H. AND C.REINSCH
HANDBOOK OF AUTOMATIC COMPUTATION, VOL. 2
LINEAR ALGEBRA
HEIDELBERG (1971)
EXAMPLE OF USE :
FIRST WE GIVE A PROGRAM, AND THAN THE RESULTS OF THIS PROGRAM :
"BEGIN" "ARRAY" A[1:8,1:5], B[1:8], EM[0:7];
"INTEGER" I;
A[1,1]:=22; A[1,2]:= A[2,3]:=10; A[1,3]:= A[7,1]:= A[8,5]:=2;
A[1,4]:= A[3,5]:=3; A[1,5]:= A[2,2]:=7; A[2,1]:=14; A[2,5]:=8;
A[2,4]:= A[8,3]:=0; A[3,1]:= A[3,3]:= A[6,5]:=-1; A[3,2]:=13;
A[3,4]:=-11; A[4,1]:=-3; A[4,2]:= A[4,4]:= A[5,4]:= A[8,4]:=-2;
A[4,3]:=13; A[4,5]:= A[5,5]:= A[8,1]:=4; A[5,1]:= A[6,1]:=9;
A[5,2]:=8; A[5,3]:= A[6,2]:= A[7,5]:=1; A[6,3]:=-7;
A[6,4]:= A[7,4]:= A[8,2]:=5; A[7,2]:=-6; A[7,3]:=6;
B[1]:=-1; B[2]:=2; B[3]:=1; B[4]:=4; B[5]:=0;
EM[0]:="-14; EM[2]:="-12; EM[4]:=80; EM[6]:="-10;
I:= SOLUND(A, 8, 5, B, EM);
OUTPUT(61, "("4B, "("NUMBER SINGULAR VALUES NOT FOUND : ")",
3ZD,/, 4B, "("NORM : ")",N, /, 4B, "("MAX NEGL SUBD ELEM : ")",
N,/, 4B, "("NUMBER ITERATIONS : ")", 3ZD, /, 4B, "("RANK : ")",
3ZD, /")", I, EM[1], EM[3], EM[5], EM[7]);
OUTPUT(61, "("/, 4B, "("SOLUTION VECTOR")", /,/, 8(4B, N,/)")",
B[1], B[2], B[3], B[4], B[5], B[6], B[7], B[8])
"END"
NUMBER SINGULAR VALUES NOT FOUND : 0
NORM : +4.4000000000000"+001
MAX NEGL SUBD ELEM : +4.3977072741076"-014
NUMBER ITERATIONS : 6
RANK : 3
SOLUTION VECTOR
+1.6410256410255"-002
+1.4807692307694"-002
-4.8397435897438"-002
+1.0000000000002"-002
-6.7948717948740"-003
+1.1602564102565"-002
+2.9999999999996"-002
-8.3974358974328"-003
1SECTION 3.1.1.3.1.2 (MAY 1974) PAGE 5
SOURCE TEXT(S):
0"CODE" 34282;
"PROCEDURE" SOLSVDUND(U, VAL, V, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V, X, EM;
"BEGIN" "INTEGER" I;
"REAL" MIN;
"ARRAY" X1[1:N];
MIN:= EM[6];
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
X1[I]:= "IF" VAL[I] <= MIN "THEN" 0 "ELSE" TAMVEC(1, N, I, V, X) /
VAL[I];
"FOR" I:= 1 "STEP" 1 "UNTIL" M "DO"
X[I]:= MATVEC(1, N, I, U, X1)
"END" SOLSVDUND;
"EOP"
0"CODE" 34283;
"INTEGER" "PROCEDURE" SOLUND(A, M, N, X, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, X, EM;
"BEGIN" "INTEGER" I;
"ARRAY" VAL[1:N], V[1:N,1:N];
SOLUND:= I:= QRISNGVALDEC(A, M, N, VAL, V, EM);
"IF" I = 0 "THEN" SOLSVDUND(A, VAL, V, M, N, X, EM)
"END" SOLUND;
"EOP"
1SECTION 3.1.1.3.1.3 (DECEMBER 1975) PAGE 1
AUTHOR : D.T.WINTER
INSTITUTE : MATHEMATICAL CENTRE
RECEIVED : 731217
BRIEF DESCRIPTION :
THIS SECTION CONTAINS 2 PROCEDURES FOR THE
CALCULATION OF THE HOMOGENEOUS EQUATIONS A * X = 0 AND X' * A = 0,
WHERE A DENOTES A MATRIX, AND X A VECTOR:
HOMSOLSVD ASSUMES THAT THE SINGULAR VALUES DECOMPOSITION OF A HAS
BEEN GIVEN.
HOMSOL FIRST CALCULATES THE SINGULAR VALUES
DECOMPOSITION BY MEANS OF THE PROCEDURE QRISNGVALDEC.
KEYWORDS :
HOMOGENEOUS SOLUTION
SINGULAR VALUES
SUBSECTION : HOMSOLSVD
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" HOMSOLSVD(U, VAL, V, M, N);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V;
"CODE" 34284;
THE MEANING OF THE FORMAL PARAMETERS IS :
U: <ARRAY IDENTIFIER>;
"ARRAY" U[1:M,1:N];
ENTRY:
THE MATRIX U IN THE SINGULAR VALUES DECOMPOSITION A=U*S*V'.
EXIT: THE COLUMNS OF U THAT CORRESPOND TO THE ELEMENTS
OF VAL WITH A VALUE SMALLER THAN SOME SMALL CONSTANT MAY BE
SEEN AS THE SOLUTIONS OF X' * A = 0;
VAL: <ARRAY IDENTIFIER>;
"ARRAY" VAL[1:N];
ENTRY:THE SINGULAR VALUES;
EXIT :THE ARRAY WILL BE ORDERED IN SUCH A WAY THAT
VAL[I] < VAL[J] IF J < I;
V: <ARRAY IDENTIFIER>;
"ARRAY" V[1:N,1:N];
ENTRY:THE MATRIX V IN THE SINGULAR VALUES DECOMPOSITION;
EXIT:THE COLUMNS OF V THAT CORRESPOND TO THE ELEMENTS OF VAL
THAT ARE SMALLER THAN SOME SMALL CONSTANT MAY BE SEEN AS THE
SOLUTIONS OF THE EQUATION A * X = 0;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF U;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF U;
1SECTION 3.1.1.3.1.3 (DECEMBER 1975) PAGE 2
PROCEDURES USED :
ICHCOL = CP34031
RUNNING TIME : PROPORTIONAL TO N ' 2
METHOD AND PERFORMANCE :
THE PROCEDURE DOES NOTHING MORE THAN A SIMPLE SORTING PROCESS ON
THE ELEMENTS OF THE ARRAY VAL, AT THE SAME TIME THE COLUMNS OF U
AND V ARE INTERCHANGED, ACCORDING TO THE INTERCHANGING OF THE
ELEMENTS VAL.
LANGUAGE : ALGOL 60
SUBSECTION : HOMSOL
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"INTEGER" "PROCEDURE" HOMSOL(A, M, N, V, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, V, EM;
"CODE" 34285;
HOMSOL:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. ZERO IF ALL
SINGULAR VALUES ARE CALCULATED.
THE MEANING OF THE FORMAL PAREMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:M,1:N];
ENTRY: THE MATRIX;
EXIT: THE COLUMNS OF A THAT CORRESPOND TO THE ELEMENTS OF
VAL THAT ARE SMALLER THAN SOME SMALL CONSTANT, MAY BE SEEN
AS THE SOLUTIONS OF THE EQUATION X' * A = 0.
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF A.
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF A.
1SECTION 3.1.1.3.1.3 (DECEMBER 1975) PAGE 3
V: <ARRAY IDENTIFIER>;
"ARRAY" V[1:N,1:N];
EXIT: THE COLUMNS OF V THAT CORRESPOND TO ELEMENTS OF VAL
SMALLER THAN SOME SMALL CONSTANT MAY BE SEEN AS THE
SOLUTIONS OF THE EQUATION A * X = 0.
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[0:7];
ENTRY: EM[0]: THE MACHINE PRECISION;
EM[2]: THE RELATIVE PRECISION FOR THE SINGULAR VALUES;
EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED IN
THE SINGULAR VALUES DECOMPOSITION.
EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE;
EXIT: EM[1]: THE INFINITY NORM OF THE MATRIX;
EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT;
EM[5]: THE NUMBER OF ITERATIONS PERFORMED IN THE SINGULAR
VALUES DECOMPOSITION;
EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF
SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6].
PROCEDURES USED :
QRISNGVALDEC = CP34273
HOMSOLSVD = CP34284
METHOD AND PERFORMANCE :
THE SOLUTION IS FOUND IN TWO STEPS :
1. THE SINGULAR VALUES DECOMPOSITION IS CALCULATED BY MEANS OF THE
PROCEDURE QRISNGVALDEC;
2. THE SINGULAR VALUES ARE ORDERED BY MEANS OF THE PROCEDURE
HOMSOLSVD (PROVIDED THAT ALL SINGULAR VALUES HAVE BEEN FOUND).
RUNNING TIME : ROUGHLY PROPORTIONAL TO (M + N) * N * N
LANGUAGE : ALGOL 60
REFERENCES :
WILKINSON, J.H. AND C.REINSCH
HANDBOOK OF AUTOMATIC COMPUTATION, VOL. 2
LINEAR ALGEBRA
HEIDELBERG (1971)
1SECTION 3.1.1.3.1.3 (NOVEMBER 1976) PAGE 4
EXAMPLE OF USE :
FIRST WE GIVE A PROGRAM, AND THAN THE RESULTS OF THIS PROGRAM :
"BEGIN" "ARRAY" A[1:8,1:5], V[1:5,1:5], EM[0:7];
"INTEGER" I, J;
A[1,1]:=22; A[1,2]:= A[2,3]:=10; A[1,3]:= A[7,1]:= A[8,5]:=2;
A[1,4]:= A[3,5]:=3; A[1,5]:= A[2,2]:=7; A[2,1]:=14; A[2,5]:=8;
A[2,4]:= A[8,3]:=0; A[3,1]:= A[3,3]:= A[6,5]:=-1; A[3,2]:=13;
A[3,4]:=-11; A[4,1]:=-3; A[4,2]:= A[4,4]:= A[5,4]:= A[8,4]:=-2;
A[4,3]:=13; A[4,5]:= A[5,5]:= A[8,1]:=4; A[5,1]:= A[6,1]:=9;
A[5,2]:=8; A[5,3]:= A[6,2]:= A[7,5]:=1; A[6,3]:=-7;
A[6,4]:= A[7,4]:= A[8,2]:=5; A[7,2]:=-6; A[7,3]:=6;
EM[0]:="-14; EM[2]:="-12; EM[4]:=80; EM[6]:="-10;
I:= HOMSOL(A, 8, 5, V, EM);
OUTPUT(61, "("4B, "("NUMBER SINGULAR VALUES NOT FOUND : ")",
3ZD,/, 4B, "("NORM : ")", N,/, 4B, "("MAX NEGL SUBD ELEM : ")",
N,/, 4B, "("NUMBER ITERATIONS : ")", 3ZD, /, 4B, "("RANK : ")",
3ZD, /")", I, EM[1], EM[3], EM[5], EM[7]);
"FOR" J:= EM[7] + 1 "STEP" 1 "UNTIL" 5 "DO"
OUTPUT(61, "("/, 4B, "("COLUMN NUMBER : ")", D, 5(/ ,4B, 2(N)),
3(/, 4B, N), /")", J, A[1,J], V[1,J], A[2,J], V[2,J], A[3,J],
V[3,J], A[4,J], V[4,J], A[5,J], V[5,J], A[6,J], A[7,J], A[8,J])
"END"
NUMBER SINGULAR VALUES NOT FOUND : 0
NORM : +4.4000000000000"+001
MAX NEGL SUBD ELEM : +4.3977072741076"-014
NUMBER ITERATIONS : 6
RANK : 3
COLUMN NUMBER : 4
+3.4708599800002"-001 -4.1909548511171"-001
-6.0723369623011"-001 +4.4050912303713"-001
+1.2207461910546"-001 -5.2004549247434"-002
+6.1882574433898"-001 +6.7605914021670"-001
-4.6344371870996"-003 +4.1297730284731"-001
+3.3409859838125"-001
-3.3528410857408"-002
-1.3547246422274"-002
COLUMN NUMBER : 5
-2.5533109413182"-001 +0.0000000000000"+000
-1.7359809248754"-001 -4.1854806384909"-001
-2.2081225414163"-001 -3.4879005320758"-001
+4.1165471593410"-002 -2.4415303724531"-001
+9.2044247057656"-001 +8.0221712237742"-001
-2.8895953996492"-002
+6.1327596621994"-002
-4.9058079025100"-002
1SECTION 3.1.1.3.1.3 (MAY 1974) PAGE 5
SOURCE TEXT(S):
"CODE" 34284;
"PROCEDURE" HOMSOLSVD(U, VAL, V, M, N);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V;
"BEGIN" "INTEGER" I, J;
"REAL" X;
"FOR" I:= N "STEP" - 1 "UNTIL" 2 "DO"
"FOR" J:= I - 1 "STEP" - 1 "UNTIL" 1 "DO"
"IF" VAL[I] > VAL[J] "THEN"
"BEGIN" X:= VAL[I]; VAL[I]:= VAL[J]; VAL[J]:= X;
ICHCOL(1, M, I, J, U); ICHCOL(1, N, I, J, V)
"END"
"END" HOMSOLSVD;
"EOP"
0"CODE" 34285;
"INTEGER" "PROCEDURE" HOMSOL(A, M, N, V, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, V, EM;
"BEGIN" "INTEGER" I;
"ARRAY" VAL[1:N];
HOMSOL:= I:= QRISNGVALDEC(A, M, N, VAL, V, EM);
"IF" I = 0 "THEN" HOMSOLSVD(A, VAL, V, M, N)
"END" HOMSOL;
"EOP"
1SECTION 3.1.1.3.1.4 (DECEMBER 1979) PAGE 1
AUTHOR : D.T.WINTER
INSTITUTE : MATHEMATICAL CENTRE
RECEIVED : 731217
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES FOR THE
CALCULATION OF THE PSEUDO-INVERSE OF A MATRIX:
PSDINVSVD ASSUMES THAT THE MATRIX IS GIVEN AS SINGULAR VALUES
DECOMPOSITION.
PSDINV FIRST CALCULATES THIS DECOMPOSITION.
KEYWORDS :
PSEUDO-INVERSE
SINGULAR VALUES
SUBSECTION : PSDINVSVD
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" PSDINVSVD(U, VAL, V, M, N, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V, EM; "CODE" 34286;
THE MEANING OF THE FORMAL PARAMETERS IS :
U: <ARRAY IDENTIFIER>;
"ARRAY" U[1:M,1:N];
ENTRY: THE MATRIX U IN THE SINGULAR VALUES DECOMPOSITION
U * S * V';
EXIT: THE TRANSPOSE OF THE PSEUDO-INVERSE.
VAL: <ARRAY IDENTIFIER>;
"ARRAY" VAL[1:N];
THE SINGULAR VALUES.
V: <ARRAY IDENTIFIER>;
"ARRAY" V[1:N,1:N];
THE MATRIX V IN THE SINGULAR VALUES DECOMPOSITION U * S * V'.
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF U.
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF V.
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[6:6];
ENTRY: EM[6]:THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE.
1SECTION 3.1.1.3.1.4 (DECEMBER 1979) PAGE 2
PROCEDURES USED :
MATVEC = CP34011
REQUIRED CENTRAL MEMORY : AN AUXILIARY ARRAY OF N REALS IS DECLARED
RUNNING TIME : ROUGHLY PROPORTIONAL TO M * N * N
METHOD AND PERFORMANCE :
THE PSEUDO-INVERSE IS CALCULATED IN TWO STEPS :
1. THE MATRIX X = VAL+ * U' IS CALCULATED, WHERE VAL+ DENOTES THE
DIAGONAL MATRIX OBTAINED FROM VAL BY PUTTING
VAL+[I,I] = 1/VAL[I] IF VAL[I] GREATER THAN OR EQUAL TO EM[6],
AND VAL+[I,I] = 0 OTHERWISE.
2. THE PSEUDO INVERSE (V * X) IS CALCULATED.
SUBSECTION : PSDINV
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"INTEGER" "PROCEDURE" PSDINV(A, M, N, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, EM; "CODE" 34287;
PSDINV:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. ZERO IF ALL
SINGULAR VALUES ARE CALCULATED.
THE MEANING OF THE FORMAL PARAMETERS IS :
A <ARRAY IDENTIFIER>;
"ARRAY" A[1:M,1:N];
ENTRY : THE GIVEN MATRIX;
EXIT : THE TRANSPOSE OF THE PSEUDO-INVERSE;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF A;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF A, N<= M;
EM: <ARRAY IDENTIFIER>;
"ARRAY" EM[0:7];
ENTRY: EM[0]: THE MACHINE PRECISION;
EM[2]: THE RELATIVE PRECISION FOR THE SINGULAR VALUES;
EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED;
EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE;
EXIT: EM[1]: THE INFINITY NORM OF THE MATRIX;
EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT;
EM[5]: THE NUMBER OF ITERATIONS PERFORMED IN THE SINGULAR
VALUES DECOMPOSITION;
EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF
SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6];
1SECTION 3.1.1.3.1.4 (DECEMBER 1979) PAGE 3
PROCEDURES USED :
QRISNGVALDEC = CP34273
PSDINVSVD = CP34286
REQUIRED CENTRAL MEMORY :
AUXILIARY ARRAYS ARE DECLARED TO A TOTAL OF (N + 1) * N REALS
RUNNING TIME : ROUGHLY PROPORTIONAL TO (2M + N) * N * N
METHOD AND PERFORMANCE :
FIRST THE SINGULAR VALUES DECOMPOSITION IS CALCULATED, AND THEN THE
PSEUDO-INVERSE IS CALCULATED BY PSDINVSVD.
REFERENCES :
WILKINSON, J.H. AND C.REINSCH
HANDBOOK OF AUTOMATIC COMPUTATION, VOL.2
LINEAR ALGEBRA
HEIDELBERG (1971)
1SECTION 3.1.1.3.1.4 (DECEMBER 1979) PAGE 4
EXAMPLE OF USE :
FIRST WE GIVE A PROGRAM, AND THEN THE RESULTS OF THIS PROGRAM :
"BEGIN" "ARRAY" A[1:8,1:5], EM[0:7];
"INTEGER" I, J;
A[1,1]:=22; A[1,2]:= A[2,3]:=10; A[1,3]:= A[7,1]:= A[8,5]:=2;
A[1,4]:= A[3,5]:=3; A[1,5]:= A[2,2]:=7; A[2,1]:=14; A[2,5]:=8;
A[2,4]:= A[8,3]:=0; A[3,1]:= A[3,3]:= A[6,5]:=-1; A[3,2]:=13;
A[3,4]:=-11; A[4,1]:=-3; A[4,2]:= A[4,4]:= A[5,4]:= A[8,4]:=-2;
A[4,3]:=13; A[4,5]:= A[5,5]:= A[8,1]:=4; A[5,1]:= A[6,1]:=9;
A[5,2]:=8; A[5,3]:= A[6,2]:= A[7,5]:=1; A[6,3]:=-7;
A[6,4]:= A[7,4]:= A[8,2]:=5; A[7,2]:=-6; A[7,3]:=6;
EM[0]:="-14; EM[2]:="-12; EM[4]:=80; EM[6]:="-10;
I:= PSDINV(A, 8, 5, EM);
OUTPUT(61, "("4B, "("NUMBER SINGULAR VALUES NOT FOUND : ")",
3ZD,/, 4B, "("NORM : ")", N,/, 4B, "("MAX NEGL SUBD ELEM : ")",
N,/, 4B, "("NUMBER ITERATIONS : ")", 3ZD, /, 4B, "("RANK : ")",
3ZD, /")", I, EM[1], EM[3], EM[5], EM[7]);
OUTPUT(61, "("/, 4B, "("TRANSPOSE OF PSEUDO-INVERSE")", /,
4B, "("FIRST THREE COLUMNS")", /, /, 8(4B, 3(N), /), /, /,
4B, "("LAST TWO COLUMNS")", /, /, 8(15B, 2(N), /)")",
A[1,1], A[1,2], A[1,3], A[2,1], A[2,2], A[2,3], A[3,1], A[3,2],
A[3,3], A[4,1], A[4,2], A[4,3], A[5,1], A[5,2], A[5,3], A[6,1],
A[6,2], A[6,3], A[7,1], A[7,2], A[7,3], A[8,1], A[8,2], A[8,3],
A[1,4], A[1,5], A[2,4], A[2,5], A[3,4], A[3,5], A[4,4], A[4,5],
A[5,4], A[5,5], A[6,4], A[6,5], A[7,4], A[7,5], A[8,4], A[8,5])
"END"
NUMBER SINGULAR VALUES NOT FOUND : 0
NORM : +4.4000000000000"+001
MAX NEGL SUBD ELEM : +4.3977072741076"-014
NUMBER ITERATIONS : 6
RANK : 3
TRANSPOSE OF PSEUDO-INVERSE
FIRST THREE COLUMNS
+2.1129807692308"-002 +4.6153846153850"-003 -2.1073717948727"-003
+9.3108974358974"-003 +2.2115384615376"-003 +2.0528846153848"-002
-1.1097756410256"-002 +2.7403846153848"-002 -3.8862179487199"-003
-7.9166666666667"-003 -5.0000000000007"-003 +3.3750000000001"-002
+5.5128205128205"-003 +9.8076923076935"-003 -8.9743589743826"-004
+1.4318910256410"-002 -2.5961538461548"-003 -2.0136217948716"-002
+4.8958333333335"-003 -1.4999999999998"-002 +1.5312499999996"-002
+1.5064102564102"-003 +7.4038461538447"-003 -1.6987179487147"-003
1SECTION 3.1.1.3.1.4 (DECEMBER 1979) PAGE 5
LAST TWO COLUMNS
+7.6041666666662"-003 +3.8060897435894"-003
-2.0833333333295"-004 +1.0016025641026"-002
-2.7604166666667"-002 +4.2067307692303"-003
-5.4166666666662"-003 +1.0416666666667"-002
-5.0000000000005"-003 +3.2051282051275"-003
+1.2812500000000"-002 -6.2099358974354"-003
+1.2395833333332"-002 +2.6041666666656"-003
-4.9999999999993"-003 +1.6025641025649"-003
"EOP"
SOURCE TEXT(S):
0"CODE" 34286;
"PROCEDURE" PSDINVSVD(U, VAL, V, M, N, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" U, VAL, V, EM;
"BEGIN" "INTEGER" I, J;
"REAL" MIN, VALI;
"ARRAY" X[1:N];
MIN:= EM[6];
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"IF" VAL[I] > MIN "THEN"
"BEGIN" VALI:= 1 / VAL[I];
"FOR" J:= 1 "STEP" 1 "UNTIL" M "DO" U[J,I]:= U[J,I] * VALI
"END"
"ELSE" "FOR" J:= 1 "STEP" 1 "UNTIL" M "DO" U[J,I]:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" X[J]:= U[I,J];
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
U[I,J]:= MATVEC(1, N, J, V, X)
"END"
"END" PSDINVSVD;
"EOP"
0"CODE" 34287;
"INTEGER" "PROCEDURE" PSDINV(A, M, N, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, EM;
"BEGIN" "INTEGER" I;
"ARRAY" VAL[1:N], V[1:N,1:N];
PSDINV:= I:= QRISNGVALDEC(A, M, N, VAL, V, EM);
"IF" I = 0 "THEN" PSDINVSVD(A, VAL, V, M, N, EM)
"END" PSDINV;
"EOP"
1SECTION 3.1.2.1.1.1.1.1 (DECEMBER 1975) PAGE 1
AUTHOR : T.J. DEKKER.
REVISOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 730903.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURE DECBND
FOR THE DECOMPOSITION OF A BAND MATRIX BY GAUSSIAN ELIMINATION
WITH STABILIZING ROW INTERCHANGES (PARTIAL PIVOTING).
KEY WORDS :
LINEAR EQUATIONS,
PARTIAL PIVOTING,
GAUSSIAN ELIMINATION,
BAND MATRIX.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" DECBND(A, N, LW, RW, AUX, M, P); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "INTEGER""ARRAY" P; "ARRAY" A, M, AUX;
"CODE" 34320;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1 : (LW + RW) * (N - 1) + N];
ENTRY : A CONTAINS ROWWISE THE BAND ELEMENTS OF THE
BAND MATRIX IN SUCH A WAY THAT THE (I,J)-TH ELEMENT OF
THE MATRIX IS GIVEN IN A[( LW + RW ) * (I - 1) + J],
I=1,...,N AND J=MAX(1,I-LW),...,MIN(N,I+RW).
THE VALUES OF THE REMAINING ELEMENTS OF A ARE
IRRELEVANT.
EXIT : THE BAND ELEMENTS OF THE GAUSSIAN ELIMINATED
MATRIX, WHICH IS AN UPPERTRIANGULAR BAND MATRIX WITH
(LW + RW) CODIAGONALS, ARE ROWWISE DELIVERED IN A AS
FOLLOWS: THE (I,J)-TH ELEMENT OF U IS
A [ ( LW + RW ) * (I - J) + J ],I=1,...,N AND
J=I,...,MIN(N,I + LW + RW).
N : <ARITHMETIC EXPRESSION>;
ORDER OF THE BAND MATRIX;
LW : <ARITHMETIC EXPRESSION>;
NUMBER OF LEFT CODIAGONALS OF A;
1SECTION 3.1.2.1.1.1.1.1 (JUNE 1974) PAGE 2
RW : <ARITHMETIC EXPRESSION>;
NUMBER OF RIGHT CODIAGONALS OF A;
AUX : <ARRAY IDENTIFIER>;
"ARRAY" AUX[1 : 5];
ENTRY :AUX[2] = EPS IS A RELATIVE TOLERANCE TO CONTROL
THE ELIMINATION; THE PROCESS IS DISCONTINUED IF
(EPS > PIVOT[I] / EUCLIDEAN NORM OF I-TH ROW)
IN THE I-TH ELIMINATION STEP;
NORMAL EXIT :
AUX[1] = SIGN OF THE DETERMINANT OF THE MATRIX
(+1 OR -1);
AUX[3] = N;
AUX[5] = MINIMUM ABSOLUTE VALUE OF
PIVOT[I] / EUCLIDEAN NORM OF THE I-TH ROW;
ABNORMAL EXIT : IF THE ELIMINATION CANNOT BE CARRIED
OUT, I.E. IF TEMP (THE QUANTITY
ABS(PIVOT[I] / EUCLIDEAN NORM OF THE I-TH ROW))
IS TOO SMALL IN THE I-TH ELIMINATION STEP :
AUX[3] = I - 1,
AUX[5] = TEMP;
M : <ARRAY IDENTIFIER>;
"ARRAY" M[1 : LW * (N - 2) + 1];
EXIT : THE GAUSSIAN MULTIPLIERS OF ALL ELIMINATIONS
IN SUCH A WAY THAT THE I-TH MULTIPLIER OF THE J-TH
STEP IS M [ LW * (J - 1) + I - J ].
P : <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" P[1 : N];
EXIT : THE PIVOTAL INDICES.
PROCEDURES USED :
VECVEC = CP34010,
ELMVEC = CP34020,
ICHVEC = CP34030.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : A REAL ARRAY OF N ELEMENTS IS DECLARED.
RUNNING TIME :
(C1 * LW + C2) * (LW + RW + 1) * N;
THE CONSTANTS C1 AND C2 DEPEND UPON THE
ARITHMETIC OF THE COMPUTER.
1SECTION 3.1.2.1.1.1.1.1 (JUNE 1974) PAGE 3
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
DECBND PERFORMS THE DECOMPOSITION OF A MATRIX WHOSE NON-ZERO
ELEMENTS ARE IN BAND FORM, AND WHOSE BAND ELEMENTS ARE STORED
ROWWISE IN A ONE-DIMENSIONAL ARRAY.
THE METHOD USED IS GAUSSIAN ELIMINATION WITH STABILIZING ROW
INTERCHANGES (PARTIAL PIVOTING).
THE GAUSSIAN ELIMINATION IS PERFORMED IN N STEPS. IN THE K-TH
STEP, K = 1, ... , N, A PIVOT IS SELECTED IN THE K-TH COLUMN OF
THE REMAINING SUBMATRIX OF ORDER N - K + 1 (THIS COLUMN
CONTAINS AT MOST LW + 1 NON-ZERO ELEMENTS); THEN THE PIVOTAL
ROW IS INTERCHANGED WITH THE K-TH ROW; SUBSEQUENTLY THE K-TH
UNKNOWN IS ELIMINATED IN THE LAST N - K ROWS (ONLY THE FIRST
LW OF THESE LAST ROWS ARE INVOLVED HERE).
THE PIVOT IS SELECTED IN SUCH A WAY THAT ITS ABSOLUTE
VALUE DIVIDED BY THE EUCLIDEAN NORM OF THE CORRESPONDING ROW OF
THE MATRIX IS MAXIMAL. THUS, THE MATRIX IS EQUILIBRATED IN THIS
PIVOTING STRATEGY SUCH, THAT THE ROWS EFFECTIVELY OBTAIN UNIT
EUCLIDEAN NORM.
THE PROCEDURE DELIVERS THE BAND ELEMENTS OF THE ELIMINATED MATRIX
(WHICH IS AN UPPER TRIANGULAR MATRIX WITH LW + RW
SUPERDIAGONALS) AND THE GAUSSIAN MULTIPLIERS FOR EACH ELIMINATION
.
THE ELIMINATION CANNOT BE CARRIED OUT IF THE ABSOLUTE VALUE OF THE
PIVOT IS LESS THAN A GIVEN RELATIVE TOLERANCE (AUX[2]) TIMES THE
EUCLIDEAN NORM OF THE CORRESPONDING ROW OF THE MATRIX. THEN THE
PREVIOUS STEP NUMBER OF THE ELIMINATION IS DELIVERED (IN AUX[3],
WHICH ELSE TAKES THE VALUE N). SEE ALSO REF [1], SECTION 212.
REFERENCE :
[1] DEKKER, T.J. :
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MC TRACT 22, 1968, MATHEMATISCH CENTRUM, AMSTERDAM.
EXAMPLE OF USE :
SEE EXAMPLE OF USE OF SOLBND.
1SECTION 3.1.2.1.1.1.1.1 (JUNE 1974) PAGE 4
SOURCE TEXT(S) :
0"CODE" 34320;
"PROCEDURE" DECBND(A, N, LW, RW, AUX, M, P); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "INTEGER" "ARRAY" P; "ARRAY" A, M, AUX;
"BEGIN" "INTEGER" I, J, K, KK, KK1, PK, MK, IK, LW1, F, Q, W, W1,
W2, NRW, IW, SDET;
"REAL" R, S, EPS, MIN;
"ARRAY" V[1:N];
F:= LW; W1:= LW + RW; W:= W1 + 1; W2:= W - 2; IW:= 0; SDET:= 1;
NRW:= N - RW; LW1:= LW + 1; Q:= LW - 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" LW "DO"
"BEGIN" Q:= Q - 1; IW:= IW + W1;
"FOR" J:= IW - Q "STEP" 1 "UNTIL" IW "DO" A[J]:= 0
"END";
IW:= - W2; Q:= - LW;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" IW:= IW + W; "IF" I <= LW1 "THEN" IW:= IW - 1;
Q:= Q + W; "IF" I > NRW "THEN" Q:= Q - 1;
V[I]:= SQRT(VECVEC(IW, Q, 0, A, A))
"END";
EPS:= AUX[2]; MIN:= 1; KK:= - W1; MK:= - LW;
"IF" F > NRW "THEN" W2:= W2 + NRW - F;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN""IF" F < N "THEN" F:= F + 1; IK:= KK:= KK + W;
MK:= MK + LW; S:= ABS(A[KK]) / V[K]; PK:= K; KK1:= KK + 1;
"FOR" I:= K + 1 "STEP" 1 "UNTIL" F "DO"
"BEGIN" IK:= IK + W1; M[MK + I - K]:= R:= A[IK]; A[IK]:= 0;
R:= ABS(R) / V[I]; "IF" R > S "THEN"
"BEGIN" S:= R; PK:= I "END"
"END";
"IF" S < MIN "THEN" MIN:= S; "IF" S < EPS "THEN"
"BEGIN" AUX[3]:= K - 1; AUX[5]:= S; "GO TO" END "END";
"IF" K + W2 >= N "THEN" W2:= W2 - 1;
P[K]:= PK; "IF" PK ^= K "THEN"
"BEGIN" V[PK]:= V[K];
PK:= PK - K; ICHVEC(KK1, KK1 + W2, PK * W1, A);
SDET:= - SDET; R:= M[MK + PK]; M[MK + PK]:= A[KK];
A[KK]:= R
"END""ELSE" R:= A[KK]; "IF" R < 0 "THEN" SDET:= - SDET;
IW:= KK1; LW1:= F - K + MK;
"FOR" I:= MK + 1 "STEP" 1 "UNTIL" LW1 "DO"
"BEGIN" M[I]:= S:= M[I] / R; IW:= IW + W1;
ELMVEC(IW, IW + W2, KK1 - IW, A, A, - S)
"END"
"END";
AUX[3]:= N; AUX[5]:= MIN;
END: AUX[1]:= SDET
"END" DECBND;
"EOP"
1SECTION 3.1.2.1.1.1.1.2 (JUNE 1974) PAGE 1
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 730903.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURE DETERMBND
FOR THE CALCULATION OF THE DETERMINANT OF A BAND MATRIX.
KEY WORDS :
DETERMINANT,
BAND MATRIX.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"REAL""PROCEDURE" DETERMBND(A, N, LW, RW, SGNDET); "VALUE" N, LW,
RW, SGNDET; "INTEGER" N, LW, RW, SGNDET; "ARRAY" A;
"CODE" 34321;
DETERMBND DELIVERS THE DETERMINANT OF THE MATRIX.
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, LW, RW : SEE 'CALLING SEQUENCE' OF DECBND
(SECTION 3.1.2.1.1.1.1.1.);
ENTRY : THE CONTENTS OF A ARE AS PRODUCED BY DECBND OR
DECSOLBND (SECTION 3.1.2.1.1.1.1.3.);
SGNDET : <ARITHMETIC EXPRESSION>;
ENTRY : THE SIGN OF THE DETERMINANT AS DELIVERED IN
AUX[1] BY DECBND, IF THE ELIMINATION BY DECBND WAS
SUCCESSFUL.
1SECTION 3.1.2.1.1.1.1.2 (DECEMBER 1975) PAGE 2
PROCEDURES USED : NONE.
RUNNING TIME : PROPORTIONAL TO N.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
DETERMBND CAN BE CALLED AFTER DECBND OR DECSOLBND ONLY IF THE
GAUSSIAN ELIMINATION WAS SUCCESSFUL, I.E. IF AUX[3] = N.
THE FUNCTION VALUE OF DETERMBND IS
THE DETERMINANT OF THE GAUSSIAN ELIMINATED UPPER TRIANGULAR
MATRIX PROVIDED WITH THE CORRECT SIGN THAT IS DELIVERED BY
DECBND OR DECSOLBND IN AUX[1].
DETERMBND SHOULD NOT BE CALLED WHEN OVERFLOW CAN BE EXPECTED.
EXAMPLE OF USE :
SEE EXAMPLES OF USE OF SOLBND AND DECSOLBND.
SOURCE TEXT(S) :
0"CODE" 34321;
"REAL""PROCEDURE" DETERMBND(A, N, LW, RW, SGNDET);
"VALUE" N, LW, RW, SGNDET; "INTEGER" N, LW, RW, SGNDET; "ARRAY" A;
"BEGIN""INTEGER" I, L; "REAL" P;
L:= 1; P:= 1; LW:= LW + RW + 1;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" P:= A[L] * P; L:= L + LW "END";
DETERMBND:= ABS(P) * SGNDET
"END" DETERMBND;
"EOP"
1SECTION 3.1.2.1.1.1.1.3 (JUNE 1974) PAGE 1
AUTHOR : T.J. DEKKER.
REVISOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 730903.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES.
A) SOLBND, FOR THE SOLUTION OF ONE OR MORE SYSTEMS
OF LINEAR EQUATIONS WITH THE SAME COEFFICIENT MATRIX, IF THIS
MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE PROCEDURE DECBND
(SECTION 3.1.2.1.1.1.1.1.).
B) DECSOLBND, FOR THE SOLUTION OF ONE SYSTEM OF
LINEAR EQUATIONS BY GAUSSIAN ELIMINATION WITH STABILIZING ROW
INTERCHANGES (PARTIAL PIVOTING) IF THE COEFFICIENT MATRIX IS IN
BAND FORM AND IS STORED ROWWISE IN A ONE-DIMENSIONAL ARRAY.
KEY WORDS :
LINEAR EQUATIONS,
PARTIAL PIVOTING,
GAUSSIAN ELIMINATION,
BAND MATRIX.
SUBSECTION : SOLBND.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" SOLBND(A, N, LW, RW, M, P, B); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "INTEGER""ARRAY" P; "ARRAY" A, M, B;
"CODE" 34071;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, LW, RW, M, P : SEE 'CALLING SEQUENCE' OF DECBND,
ENTRY : THE CONTENTS OF THE ARRAYS A, M, P ARE AS
PRODUCED BY DECBND;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1 : N];
ENTRY : THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT : THE SOLUTION OF THE SYSTEM.
1SECTION 3.1.2.1.1.1.1.3 (JUNE 1974) PAGE 2
PROCEDURES USED :
VECVEC = CP34010,
ELMVEC = CP34020.
RUNNING TIME :
(C3 * LW + C4 * RW + C5) * N;
THE CONSTANTS C3, C4 AND C5 DEPEND UPON THE
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
SOLBND CALCULATES THE SOLUTION OF A SYSTEM OF LINEAR EQUATIONS,
PROVIDED THAT THE MATRIX WAS DECOMPOSED BY A SUCCESSFUL CALL OF
DECBND (SECTION 3.1.2.1.1.1.1.1.).
THE SOLUTION OF THE LINEAR SYSTEM IS OBTAINED BY CARRYING OUT THE
ELIMINATIONS, FOR
WHICH THE GAUSSIAN MULTIPLIERS ARE SAVED, ON THE RIGHT HAND SIDE,
AND BY SOLVING THE NEW SYSTEM WITH THE UPPER TRIANGULAR BAND MATRIX
, AS PRODUCED BY DECBND, BY BACK SUBSTITUTION. THE SOLUTIONS OF
SEVERAL SYSTEMS WITH THE SAME COEFFICIENT MATRIX CAN BE OBTAINED BY
SUCCESSIVE CALLS OF SOLBND.
EXAMPLE OF USE :
THE FOLLOWING PROGRAM SOLVES THE SYSTEM OF SIMULTANEOUS EQUATIONS
2 * X1 - X2 = 1
- X1 + 2 * X2 - X3 = 0
- X2 + 2 * X3 - X4 = 0
- X3 + 2 * X4 - X5 = 0
- X4 + 2 * X5 = 1
"BEGIN""COMMENT" 730822, TEST DECBND, SOLBND AND DETERMBND;
"INTEGER" I; "INTEGER""ARRAY" ROWIND[1 : 5];
"ARRAY" BAND[1 : 13], MULT[1 : 4], RIGHT, AUX[1 : 5];
"FOR" I:= 1 "STEP" 1 "UNTIL" 13 "DO"
BAND[I]:= "IF" (I + 1) // 3 * 3 < I "THEN" 2 "ELSE" - 1;
RIGHT[1]:= RIGHT[5]:= 1;
"FOR" I:= 2, 3, 4 "DO" RIGHT[I]:= 0; AUX[2]:= "- 12;
DECBND(BAND, 5, 1, 1, AUX, MULT, ROWIND);
"IF" AUX[3] = 5 "THEN"
"BEGIN" SOLBND(BAND, 5, 1, 1, MULT, ROWIND, RIGHT);
OUTPUT(61, "("5(+2Z.4D2B), /"("DETERMINANT IS ")" +.8D"+DD
")", RIGHT, DETERMBND(BAND, 5, 1, 1, AUX[1]))
"END"
"END"
DELIVERS :
+1.0000 +1.0000 +1.0000 +1.0000 +1.0000
DETERMINANT IS +.60000000"+01
1SECTION 3.1.2.1.1.1.1.3 (DECEMBER 1975) PAGE 3
SUBSECTION : DECSOLBND.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" DECSOLBND(A, N, LW, RW, AUX, B); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "ARRAY" A, AUX, B;
"CODE" 34322;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, LW, RW, AUX : SEE DECBND (SECTION : 3.1.2.1.1.1.1.1);
B : SEE SOLBND (THIS SECTION).
PROCEDURES USED :
VECVEC = CP34010,
ELMVEC = CP34020,
ICHVEC = CP34030.
REQUIRED CENTRAL MEMORY :
EXECUTION FIELD LENGTH : A REAL ARRAY OF N ELEMENTS AND A REAL
ARRAY OF LW + 1 ELEMENTS ARE DECLARED.
RUNNING TIME :
(C1 * LW + C6) * (LW + RW + 1) * N;
THE CONSTANTS C1 AND C6 DEPEND UPON THE
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
1SECTION 3.1.2.1.1.1.1.3 (JUNE 1974) PAGE 4
METHOD AND PERFORMANCE :
DECSOLBND PERFORMS GAUSSIAN ELIMINATION IN THE SAME WAY AS DECBND
, MEANWHILE ALSO CARRYING OUT THE ELIMINATION WITH THE GIVEN
RIGHT HAND SIDE. THE SOLUTION OF THE ELIMINATED SYSTEM IS OBTAINED
BY BACK SUBSTITUTION.
EXAMPLE OF USE :
THE PROGRAM
"BEGIN""COMMENT" 730822, TEST DECSOLBND AND DETERMBND;
"INTEGER" I;
"ARRAY" BAND[1 : 13], RIGHT, AUX[1 : 5];
"FOR" I:= 1 "STEP" 1 "UNTIL" 13 "DO"
BAND[I]:= "IF" (I + 1) // 3 * 3 < I "THEN" 2 "ELSE" - 1;
RIGHT[1]:= RIGHT[5]:= 1;
"FOR" I:= 2, 3, 4 "DO" RIGHT[I]:= 0; AUX[2]:= "- 12;
DECSOLBND(BAND, 5, 1, 1, AUX, RIGHT);
"IF" AUX[3] = 5 "THEN"
"BEGIN"
OUTPUT(61, "("5(+2Z.4D2B), /"("DETERMINANT IS ")" +.8D"+DD
")", RIGHT, DETERMBND(BAND, 5, 1, 1, AUX[1]))
"END"
"END"
WHICH SOLVES THE SAME PROBLEM AS THE PROGRAM IN THE EXAMPLE OF USE
OF SOLBND, DELIVERS :
+1.0000 +1.0000 +1.0000 +1.0000 +1.0000
DETERMINANT IS +.60000000"+01
1SECTION 3.1.2.1.1.1.1.3 (JUNE 1974) PAGE 5
SOURCE TEXT(S) :
0"CODE" 34071;
"PROCEDURE" SOLBND(A, N, LW, RW, M, P, B); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "INTEGER" "ARRAY" P; "ARRAY" A, B, M;
"BEGIN" "INTEGER" F, I, K, KK, W, W1, W2, SHIFT;
"REAL" S;
F:= LW; SHIFT:= - LW; W1:= LW - 1;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN""IF" F < N "THEN" F:= F + 1; SHIFT:= SHIFT + W1;
I:=P[K]; S:= B[I]; "IF" I ^= K "THEN"
"BEGIN" B[I]:= B[K]; B[K]:= S "END";
ELMVEC(K + 1, F, SHIFT, B, M, - S)
"END";
W1:= LW + RW; W:= W1 + 1; KK:= (N + 1) * W - W1; W2:= - 1;
SHIFT:= N * W1;
"FOR" K:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" KK:= KK - W; SHIFT:= SHIFT - W1;
"IF" W2 < W1 "THEN" W2:= W2 + 1;
B[K]:= (B[K] - VECVEC(K + 1, K + W2, SHIFT, B, A)) / A[KK]
"END"
"END" SOLBND;
"EOP"
0"CODE" 34322;
"PROCEDURE" DECSOLBND(A, N, LW, RW, AUX, B); "VALUE" N, LW, RW;
"INTEGER" N, LW, RW; "ARRAY" A, B, AUX;
"BEGIN""INTEGER" I, J, K, KK, KK1, PK, IK, LW1, F, Q, W, W1, W2,IW,
NRW, SHIFT, SDET;
"REAL" R, S, EPS, MIN; "ARRAY" M[0:LW], V[1:N];
F:= LW; SDET:= 1; W1:= LW + RW; W:= W1 + 1; W2:= W - 2; IW:= 0;
NRW:= N - RW; LW1:= LW + 1; Q:= LW - 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" LW "DO"
"BEGIN" Q:= Q - 1; IW:= IW + W1;
"FOR" J:= IW - Q "STEP" 1 "UNTIL" IW "DO" A[J]:= 0
"END";
"COMMENT"
1SECTION 3.1.2.1.1.1.1.3 (JUNE 1974) PAGE 6
;
IW:= - W2; Q:= - LW;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" IW:= IW + W; "IF" I <= LW1 "THEN" IW:= IW - 1;
Q:= Q + W; "IF" I > NRW "THEN" Q:= Q - 1;
V[I]:= SQRT(VECVEC(IW, Q, 0, A, A))
"END";
EPS:= AUX[2]; MIN:= 1; KK:= - W1;
"IF" F > NRW "THEN" W2:= W2 + NRW - F;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN""IF" F < N "THEN" F:= F + 1; IK:= KK:= KK + W;
S:= ABS(A[KK]) / V[K]; PK:= K; KK1:= KK + 1;
"FOR" I:= K + 1 "STEP" 1 "UNTIL" F "DO"
"BEGIN" IK:= IK + W1; M[I - K]:= R:= A[IK]; A[IK]:= 0;
R:= ABS(R) / V[I]; "IF" R > S "THEN"
"BEGIN" S:= R; PK:= I "END"
"END";
"IF" S < MIN "THEN" MIN:= S; "IF" S < EPS "THEN"
"BEGIN" AUX[3]:= K - 1; AUX[5]:= S; "GO TO" END "END";
"IF" K + W2 >= N "THEN" W2:= W2 - 1; "IF" PK ^= K "THEN"
"BEGIN" V[PK]:= V[K];
PK:= PK - K; ICHVEC(KK1, KK1 + W2, PK * W1, A);
SDET:= - SDET; R:= B[K]; B[K]:= B[PK + K];
B[PK + K]:= R; R:= M[PK]; M[PK]:= A[KK]; A[KK]:= R
"END"
"ELSE" R:= A[KK]; IW:= KK1; LW1:= F - K;
"IF" R < 0 "THEN" SDET:= - SDET;
"FOR" I:= 1 "STEP" 1 "UNTIL" LW1 "DO"
"BEGIN" M[I]:= S:= M[I] / R; IW:= IW + W1;
ELMVEC(IW, IW + W2, KK1 - IW, A, A, - S);
B[K + I]:= B[K + I] - B[K] * S
"END"
"END";
AUX[3]:= N; AUX[5]:= MIN;
KK:= (N + 1) * W - W1; W2:= - 1; SHIFT:= N * W1;
"FOR" K:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" KK:= KK - W; SHIFT:= SHIFT - W1;
"IF" W2 < W1 "THEN" W2:= W2 + 1;
B[K]:= (B[K] - VECVEC(K + 1, K + W2, SHIFT, B, A)) / A[KK]
"END";
END: AUX[1]:= SDET
"END" DECSOLBND;
"EOP"
1SECTION 3.1.2.1.1.1.2.1 (JUNE 1974) PAGE 1
AUTHOR: W. HOFFMANN.
CONTRIBUTOR: J. C. P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731210.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PREPARATORY PROCEDURES FOR THE
SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS WITH A
TRIDIAGONAL MATRIX;
DECTRI PERFORMS A TRIANGULAR DECOMPOSITION OF A TRIDIAGONAL
MATRIX.
DECTRIPIV PERFORMS A TRIANGULAR DECOMPOSITION OF A TRIDIAGONAL
MATRIX, USING PARTIAL PIVOTING TO STABILIZE THE PROCESS.
KEYWORDS:
LU DECOMPOSITION,
TRIANGULAR DECOMPOSITION,
TRIDIAGONAL MATRIX.
SUBSECTION: DECTRI.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECTRI(SUB, DIAG, SUPER, N, AUX);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX;
"CODE" 34423;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1: N - 1];
ENTRY: THE SUBDIAGONAL OF THE GIVEN MATRIX T, SAY;
T[I + 1, I] SHOULD BE GIVEN IN SUB[I], I = 1,
..., N - 1;
EXIT: SUPPOSE L DENOTES THE LOWER-BIDIAGONAL MATRIX, SUCH
THAT LU = T, FOR SOME UPPER-BIDIAGONAL MATRIX U,
WITH UNIT DIAGONAL ELEMENTS, THEN L[I + 1, I] WILL
BE DELIVERED IN SUB[I], I = 1, ..., AUX[3] - 1;
1SECTION 3.1.2.1.1.1.2.1 (JUNE 1974) PAGE 2
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF T;
EXIT: L[I, I] WILL BE DELIVERED IN DIAG[I], I = 1, ...,
AUX[3];
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY: THE SUPERDIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN
IN SUPER[I], I = 1, ..., N - 1;
EXIT: U[I, I + 1] WILL BE DELIVERED IN SUPER[I], I = 1,
..., AUX[3] - 1;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY :
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX, ELSE AUX[5] IS SET EQUAL TO
THE VALUE OF THAT ELEMENT WHICH CAUSES THE
BREAKDOWN OF THE DECOMPOSITION.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE METHOD USED IN DECTRI YIELDS A LOWER-BIDIAGONAL MATRIX L AND A
UNIT UPPER-BIDIAGONAL MATRIX U, SUCH THAT THE PRODUCT LU EQUALS THE
GIVEN TRIDIAGONAL MATRIX; THE PROCESS IS TERMINATED IN THE K-TH
STEP, IF THE MODULUS OF THE K-TH DIAGONAL ELEMENT IS SMALLER THAN A
CERTAIN SMALL VALUE, WHICH IS GIVEN BY AUX[2] MULTIPLIED BY THE
1-NORM OF THE K-TH ROW; IN THIS CASE AUX[3] WILL BE GIVEN THE VALUE
K - 1 AND AUX[5] WILL BE GIVEN THE VALUE OF THE K-TH DIAGONAL
ELEMENT.
EXAMPLE OF USE: SEE DECSOLTRI (SECTION 3.1.2.1.1.1.2.3).
1SECTION 3.1.2.1.1.1.2.1 (DECEMBER 1975) PAGE 3
SUBSECTION: DECTRIPIV.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECTRIPIV(SUB, DIAG, SUPER, N, AID, AUX, PIV);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AID, AUX;
"BOOLEAN" "ARRAY" PIV;
"CODE" 34426;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1: N - 1];
ENTRY: THE SUBDIAGONAL OF THE GIVEN MATRIX T, SAY;
T[I+1,I] SHOULD BE GIVEN IN SUB[I],I > 1,...,N - 2
EXIT: LET T' DENOTE THE MATRIX T WITH PERMUTED ROWS;
SUPPOSE L DENOTES THE LOWER-BIDIAGONAL MATRIX, SUCH
THAT LU = T', FOR SOME UNIT UPPER-TRIANGULAR MATRIX
U, THEN L[I + 1, I] WILL BE DELIVERED IN SUB[I],
I = 1, ..., AUX[3] - 1; NOTE THAT U HAS TWO
CODIAGONALS, BECAUSE OF THE PARTIAL PIVOTING DURING
THE DECOMPOSITION;
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF T;
EXIT: L[I,I] WILL BE DELIVERED IN DIAG[I],I=1,...,AUX[3];
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY: THE SUPERDIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN
IN SUPER[I], I = 1, ..., N - 1;
EXIT: U[I, I + 1] WILL BE DELIVERED IN SUPER[I], I = 1,
..., AUX[3] - 1;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AID: <ARRAY IDENTIFIER>;
"ARRAY" AID[1: N - 2];
EXIT:U[I,I+2] WILL BE DELIVERED IN AID[I],I=1,...,AUX[3]-2;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX, ELSE AUX[5] IS SET EQUAL TO
THE VALUE OF THAT ELEMENT WHICH CAUSES THE
BREAKDOWN OF THE DECOMPOSITION.
PIV: <ARRAY IDENTIFIER>;
"BOOLEAN""ARRAY" PIV[1 : N - 1];
THE VALUE OF PIV[I] WILL BE TRUE IF THE I-TH AND (I + 1)-TH
ROW ARE INTERCHANGED, I = 1, ..., MIN(AUX[3], N - 1), ELSE
PIV[I] WILL BE FALSE.
1SECTION 3.1.2.1.1.1.2.1 (JUNE 1974) PAGE 4
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE METHOD USED IN DECTRIPIV YIELDS A LOWER-BIDIAGONAL MATRIX L AND
A UNIT UPPER-TRIANGULAR MATRIX U WITH TWO CODIAGONALS, SUCH THAT
THE PRODUCT LU EQUALS THE GIVEN TRIDIAGONAL MATRIX WITH PERMUTED
ROWS; PARTIAL PIVOTING IS USED DURING THE TRIANGULAR DECOMPOSITION,
I.E. THAT ELEMENT OF THE K-TH COLUMN OF L IS CHOSEN AS PIVOT IN THE
K-TH STEP, WHOSE MODULUS DIVIDED BY THE 1-NORM OF THE CORRESPONDING
ROW OF THE GIVEN MATRIX IS MAXIMAL; THE PROCESS IS TERMINATED IN
THE K-TH STEP, IF THE MODULUS OF THE K-TH PIVOT ELEMENT IS LESS
THAN A CERTAIN SMALL VALUE, WHICH IS GIVEN BY AUX[2] MULTIPLIED BY
THE 1-NORM OF THE CORRESPONDING ROW; IN THIS CASE AUX[3] WILL BE
GIVEN THE VALUE K - 1, AND AUX[5] WILL BE GIVEN THE VALUE OF THE
K-TH PIVOT ELEMENT.
EXAMPLE OF USE: SEE SOLTRIPIV (SECTION 3.1.2.1.1.1.2.3).
SOURCE TEXTS:
0"CODE" 34423;
"PROCEDURE" DECTRI(SUB, DIAG, SUPER, N, AUX);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX;
"BEGIN" "INTEGER" I, N1;
"REAL" D, R, S, U, NORM, NORM1, TOL;
TOL:= AUX[2]; D:= DIAG[1]; R:= SUPER[1];
NORM:= NORM1:= ABS(D) + ABS(R);
"IF" ABS(D) <= NORM1 * TOL "THEN"
"BEGIN" AUX[3]:= 0; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[1]:= R / D; S:= SUB[1]; N1:= N - 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" N1 "DO"
"BEGIN" D:= DIAG[I]; R:= SUPER[I];
NORM1:= ABS(S) + ABS(D) + ABS(R);
D:= DIAG[I]:= D - U * S;
"IF" ABS(D) <= NORM1 * TOL "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[I]:= R / D; S:= SUB[I];
"IF" NORM1 > NORM "THEN" NORM:= NORM1
"END";
D:= DIAG[N]; NORM1:= ABS(D) + ABS(S);
D:= DIAG[N]:= D - U * S;
"IF" ABS(D) <= NORM1 * TOL "THEN"
"BEGIN" AUX[3]:= N1; AUX[5]:= D; "GOTO" EXIT "END";
"IF" NORM1 > NORM "THEN" NORM:= NORM1;
AUX[3]:= N; AUX[5]:= NORM;
EXIT:
"END" DECTRI
1SECTION 3.1.2.1.1.1.2.1 (JUNE 1974) PAGE 5
;
"EOP"
0"CODE" 34426;
"PROCEDURE" DECTRIPIV(SUB, DIAG, SUPER, N, AID, AUX, PIV);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AID, AUX;
"BOOLEAN" "ARRAY" PIV;
"BEGIN" "INTEGER" I, I1, N1, N2;
"REAL" D, R, S, U, T, Q, V, W, NORM, NORM1, NORM2, TOL;
TOL:= AUX[2]; D:= DIAG[1]; R:= SUPER[1];
NORM:= NORM2:= ABS(D) + ABS(R); N2:= N - 2;
"FOR" I:= 1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" I1:= I + 1; S:= SUB[I]; T:= DIAG[I1]; Q:= SUPER[I1];
NORM1:= NORM2; NORM2:= ABS(S) + ABS(T) + ABS(Q);
"IF" NORM2 > NORM "THEN" NORM:= NORM2;
"IF" ABS(D) * NORM2 < ABS(S) * NORM1 "THEN"
"BEGIN" "IF" ABS(S) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= S; "GOTO" EXIT "END";
DIAG[I]:= S; U:= SUPER[I]:= T / S;
V:= AID[I]:= Q / S; SUB[I]:= D;
W:= SUPER[I1]:= -V * D; D:= DIAG[I1]:= R - U * D;
R:= W; NORM2:= NORM1; PIV[I]:= "TRUE"
"END" "ELSE"
"BEGIN" "IF" ABS(D) <= TOL * NORM1 "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[I]:= R / D; D:= DIAG[I1]:= T - U * S;
AID[I]:= 0; PIV[I]:= "FALSE"; R:= Q
"END"
"END";
N1:= N - 1; S:= SUB[N1]; T:= DIAG[N]; NORM1:= NORM2;
NORM2:= ABS(S) + ABS(T); "IF" NORM2 > NORM "THEN" NORM:= NORM2;
"IF" ABS(D) * NORM2 < ABS(S) * NORM1 "THEN"
"BEGIN" "IF" ABS(S) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= N2; AUX[5]:= S; "GOTO" EXIT "END";
DIAG[N1]:= S; U:= SUPER[N1]:= T / S; SUB[N1]:= D;
D:= DIAG[N]:= R - U * D; NORM2:= NORM1; PIV[N1]:= "TRUE"
"END" "ELSE"
"BEGIN" "IF" ABS(D) <= TOL * NORM1 "THEN"
"BEGIN" AUX[3]:= N2; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[N1]:= R / D; D:= DIAG[N]:= T - U * S;
PIV[N1]:= "FALSE"
"END";
"IF" ABS(D) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= N1; AUX[5]:= D; "GOTO" EXIT "END";
AUX[3]:= N; AUX[5]:= NORM;
EXIT:
"END" DECTRIPIV;
"EOP"
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 1
AUTHOR: W. HOFFMANN.
CONTRIBUTOR: J. C. P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731210.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FOUR PROCEDURES
FOR SOLVING A SYSTEM OF LINEAR EQUATIONS WITH A
TRIDIAGONAL MATRIX;
SOLTRI CALCULATES A SOLUTION BY FORWARD AND BACK SUBSTITUTION IF
THE TRIANGULAR DECOMPOSED FORM AS DELIVERED BY DECTRI IS GIVEN.
DECSOLTRI PERFORMS THE TRIANGULAR DECOMPOSITION OF THE GIVEN
MATRIX ( NOT USING ANY PIVOTING STRATEGY DURING THE PROCESS ) AND
CALCULATES THE SOLUTION BY FORWARD AND BACK SUBSTITUTION.
SOLTRIPIV CALCULATES A SOLUTION BY FORWARD AND BACK SUBTITUTION,IF
THE TRIANGULAR DECOMPOSED FORM AS DELIVERED BY DECTRIPIV IS GIVEN.
DECSOLTRIPIV PERFORMS THE TRIANGULAR DECOMPOSITION OF THE GIVEN
MATRIX ( USING PARTIAL PIVOTING ) AND CALCULATES THE SOLUTION BY
FORWARD AND BACK SUBSTITUTION.
KEYWORDS:
ALGEBRAIC EQUATIONS,
LINEAR SYSTEMS,
TRIDIAGONAL MATRIX,
FORWARD AND BACK SUBSTITUTION.
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 2
SUBSECTION: SOLTRI.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" SOLTRI(SUB, DIAG, SUPER, N, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, B;
"CODE" 34424;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1, N - 1];
ENTRY : THE SUBDIAGONAL OF THE
LOWER-BIDIAGONAL MATRIX, AS DELIVERED BY DECTRI (SECTION
3.1.2.1.1.1.2.1);
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1:N];
ENTRY : THE DIAGONAL OF THE LOWER-
BIDIAGONAL MATRIX, AS DELIVERED BY DECTRI;
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY : THE SUPERDIAGONAL OF THE
UPPER-BIDIAGONAL MATRIX AS DELIVERED BY DECTRI;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE CALCULATED SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SOLTRI CALCULATES THE SOLUTION OF A LINEAR SYSTEM WITH A
TRIDIAGONAL MATRIX, WITH FORWARD AND BACK SUBSTITUTION; THE
TRIANGULARLY DECOMPOSED FORM OF THE MATRIX, AS PRODUCED BY DECTRI
(SECTION 3.1.2.1.1.1.2.1), SHOULD BE GIVEN; ONE CALL OF DECTRI
FOLLOWED BY SEVERAL CALLS OF SOLTRI MAY BE USED TO SOLVE SEVERAL
LINEAR SYSTEMS HAVING THE SAME TRIDIAGONAL MATRIX, BUT DIFFERENT
RIGHT-HAND SIDES.
EXAMPLE OF USE: SEE DECSOLTRI (THIS SECTION).
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 3
SUBSECTION: DECSOLTRI.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECSOLTRI(SUB, DIAG, SUPER, N, AUX, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX, B;
"CODE" 34425;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1: N - 1];
ENTRY: THE SUBDIAGONAL OF THE GIVEN MATRIX T, SAY;
T[I + 1, I] SHOULD BE GIVEN IN SUB[I], I = 1,
..., N - 1;
EXIT: SUPPOSE L DENOTES THE LOWER-BIDIAGONAL MATRIX, SUCH
THAT LU = T, FOR SOME UPPER-BIDIAGONAL MATRIX U,
WITH UNIT DIAGONAL ELEMENTS, THEN L[I + 1, I] WILL
BE DELIVERED IN SUB[I], I = 1, ..., AUX[3] - 1;
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF T;
EXIT: L[I, I] WILL BE DELIVERED IN DIAG[I], I = 1, ...,
AUX[3];
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY: THE SUPERDIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN
IN SUPER[I], I = 1, ..., N - 1;
EXIT: U[I, I + 1] WILL BE DELIVERED IN SUPER[I], I = 1,
..., AUX[3] - 1;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY :
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT :
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX (SEE SECTION 3.1.2.1.1.1.2.1,
SUBSECTION DECTRI);
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE SOLUTION OF THE LINEAR
SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 4
PROCEDURES USED:
DECTRI = CP34423,
SOLTRI = CP34424.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
DECSOLTRI CALCULATES THE SOLUTION OF A LINEAR SYSTEM WITH A
TRIDIAGONAL MATRIX; THE TRIANGULAR DECOMPOSITION IS DONE BY CALLING
DECTRI (SECTION 3.1.2.1.1.1.2.1) AND THE FORWARD AND BACK
SUBSTITUTION BY CALLING SOLTRI (THIS SECTION); IF AUX[3] < N, THEN
THE EFFECT OF DECSOLTRI IS MERELY THAT OF DECTRI.
EXAMPLE OF USE:
LET T BE A TRIDIAGONAL MATRIX WITH SUBDIAGONAL AND SUPERDIAGONAL
ELEMENTS I * 2 AND I RESPECTIVELY (I = 1, ..., N - 1), AND DIAGONAL
ELEMENTS I + 10 (I = 1, ..., N); LET B BE THE SECOND COLUMN OF T;
THEN THE SOLUTION OF THE LINEAR SYSTEM TX = B IS GIVEN BY THE
SECOND UNIT VECTOR; BY THE FOLLOWING PROGRAM WE MAY SOLVE THIS
SYSTEM AND PRINT THE ERROR IN THE CALCULATED SOLUTION.
"BEGIN"
"INTEGER" I;
"ARRAY" D, SUB, SUPER, B[1:30], AUX[2:5];
"FOR" I:= 1 "STEP" 1 "UNTIL" 30 "DO"
"BEGIN" SUB[I]:= I * 2; SUPER[I]:= I; D[I]:= I + 10;
B[I]:= 0
"END"; B[1]:= 1; B[2]:= 12; B[3]:= 4;
AUX[2]:= "-14;
DECSOLTRI(SUB, D, SUPER, 30, AUX, B);
OUTPUT(71, "("/,"("AUX[3] AND AUX[5]:")",2(/,N)")",
AUX[3], AUX[5]);
B[2]:= B[2] - 1;
OUTPUT(71, "("//"("ERROR IN THE SOLUTION: ")", N")",
SQRT(VECVEC(1, 30, 0, B, B)))
"END"
RESULTS:
AUX[3] AND AUX[5]:
+3.0000000000000"+001
+1.2400000000000"+002
ERROR IN THE SOLUTION: +0.0000000000000"+000
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 5
SUBSECTION: SOLTRIPIV.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" SOLTRIPIV(SUB, DIAG, SUPER, N, AID, PIV, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AID, B;
"BOOLEAN" "ARRAY" PIV;
"CODE" 34427;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1, N - 1];
ENTRY : THE SUBDIAGONAL OF THE
LOWER-BIDIAGONAL MATRIX, AS DELIVERED BY DECTRIPIV (SECTION
3.1.2.1.1.1.2.1);
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1:N];
ENTRY: THE DIAGONAL OF THE LOWER-
BIDIAGONAL MATRIX, AS DELIVERED BY DECTRIPIV;
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY : THE FIRST CODIAGONAL OF
THE UPPER-TRIANGULAR MATRIX AS DELIVERED BY DECTRIPIV;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AID: <ARRAY IDENTIFIER>;
"ARRAY" AID[1: N - 2];
ENTRY: THE SECOND CODIAGONAL OF
THE UPPER-TRIANGULAR MATRIX AS DELIVERED BY DECTRIPIV;
PIV: <ARRAY IDENTIFIER>;
"BOOLEAN""ARRAY" PIV[1: N-1];
ENTRY: THE PIVOT-
INFORMATION AS DELIVERED BY DECTRIPIV;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE CALCULATED SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SOLTRIPIV CALCULATES THE SOLUTION OF A LINEAR SYSTEM WITH A
TRIDIAGONAL MATRIX, WITH FORWARD AND BACK SUBSTITUTION; THE
TRIANGULARLY DECOMPOSED FORM OF THE MATRIX,AS PRODUCED BY DECTRIPIV
(SECTION 3.1.2.1.1.1.2.1), SHOULD BE GIVEN; ONE CALL OF DECTRIPIV
FOLLOWED BY SEVERAL CALLS OF SOLTRIPIV MAY BE USED TO SOLVE SEVERAL
LINEAR SYSTEMS HAVING THE SAME TRIDIAGONAL MATRIX, BUT DIFFERENT
RIGHT-HAND SIDES.
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 6
EXAMPLE OF USE:
LET T BE THE MATRIX AS GIVEN IN THE EXAMPLE OF USE OF DECSOLTRI
(THIS SECTION) AND LET B1 AND B2 BE THE SECOND AND THIRD COLUMN OF
T, THEN THE SOLUTIONS OF THE LINEAR SYSTEMS TX = B1 AND TX = B2
ARE GIVEN BY THE SECOND AND THIRD UNIT VECTOR RESPECTIVELY; IN THE
FOLLOWING PROGRAM THESE SYSTEMS ARE SOLVED AND THE ERRORS IN THE
CALCULATED SOLUTIONS ARE PRINTED.
"BEGIN"
"INTEGER" I;
"ARRAY" D, SUB, SUPER, AID, B1, B2[1:30], AUX[2:5];
"BOOLEAN" "ARRAY" PIV[1:29];
"FOR" I:= 1 "STEP" 1 "UNTIL" 30 "DO"
"BEGIN" SUB[I]:= I * 2; SUPER[I]:= I; D[I]:= I + 10;
B1[I]:= B2[I]:= 0
"END"; B1[1]:= 1; B1[2]:= 12; B1[3]:= 4;
B2[2]:= 2; B2[3]:= 13; B2[4]:= 6;
AUX[2]:= "-14;
DECTRIPIV(SUB, D, SUPER, 30, AID, AUX, PIV);
SOLTRIPIV(SUB, D, SUPER, 30, AID, PIV, B1);
SOLTRIPIV(SUB, D, SUPER, 30, AID, PIV, B2);
OUTPUT(71, "("/,"("AUX[3] AND AUX[5]:")",2(/,N)")",
AUX[3], AUX[5]);
B1[2]:= B1[2] - 1; B2[3]:= B2[3] - 1;
OUTPUT(71, "("//"("ERROR IN B1: ")",N,/,"("ERROR IN B2: ")",N")",
SQRT(VECVEC(1, 30, 0, B1, B1)), SQRT(VECVEC(1, 30, 0, B2, B2)))
"END"
RESULTS:
AUX[3] AND AUX[5]:
+3.0000000000000"+001
+1.2400000000000"+002
ERROR IN B1: +0.0000000000000"+000
ERROR IN B2: +0.0000000000000"+000
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 7
SUBSECTION: DECSOLTRIPIV.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECSOLTRIPIV(SUB, DIAG, SUPER, N, AUX, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX, B;
"CODE" 34428;
THE MEANING OF THE FORMAL PARAMETERS IS:
SUB: <ARRAY IDENTIFIER>;
"ARRAY" SUB[1: N - 1];
ENTRY: THE SUBDIAGONAL OF THE GIVEN MATRIX T, SAY;
T[I + 1, I] SHOULD BE GIVEN IN SUB[I], I = 1,
..., N - 1;
EXIT: THE ELEMENTS OF SUB WILL BE OVERWRITTEN;
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF T;
EXIT: THE ELEMENTS OF DIAG WILL BE OVERWRITTEN;
SUPER: <ARRAY IDENTIFIER>;
"ARRAY" SUPER[1: N - 1];
ENTRY: THE SUPERDIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN
IN SUPER[I], I = 1, ..., N - 1;
EXIT: THE ELEMENTS OF SUPER WILL BE OVERWRITTEN;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX (SEE SECTION 3.1.2.1.1.1.2.1.,
SUBSECTION DECTRIPIV);
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1 : N];
ENTRY: THE RIGHT-HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N, THEN THE SOLUTION OF THE LINEAR
SYSTEM WILL BE OVERWRITTEN ON B, ELSE B WILL REMAIN
UNALTERED.
1SECTION 3.1.2.1.1.1.2.3 (DECEMBER 1975) PAGE 8
PROCEDURES USED: NONE.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: ONE AUXILIARY ARRAY OF TYPE BOOLEAN AND
ORDER N IS DECLARED IN DECSOLTRIPIV;
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
ONE CALL OF DECSOLTRIPIV IS EQUIVALENT WITH CALLING CONSECUTIVELY
DECTRIPIV (SECTION 3.1.2.1.1.1.2.1) AND SOLTRIPIV (THIS SECTION);
HOWEVER, DECSOLTRIPIV DOES NOT MAKE USE OF DECTRIPIV AND SOLTRIPIV,
TO SAVE MEMORY SPACE AND TIME; THIS IS ONLY TRUE IN THE CASE THAT
LINEAR SYSTEMS WITH DIFFERENT MATRICES HAVE TO BE SOLVED; IF
AUX[3] < N THEN
DECSOLTRIPIV IS TERMINATED PREMATURELY (SEE DECTRIPIV IN SECTION
3.1.2.1.1.1.2.1).
EXAMPLE OF USE:
THE SAME LINEAR SYSTEM AS GIVEN IN THE EXAMPLE OF USE OF DECSOLTRI
MAY BE SOLVED WITH DECSOLTRIPIV BY THE FOLLOWING PROGRAM:
"BEGIN"
"INTEGER" I;
"ARRAY" D, SUB, SUPER, B[1:30], AUX[2:5];
"FOR" I:= 1 "STEP" 1 "UNTIL" 30 "DO"
"BEGIN" SUB[I]:= I * 2; SUPER[I]:= I; D[I]:= I + 10;
B[I]:= 0
"END"; B[1]:= 1; B[2]:= 12; B[3]:= 4;
AUX[2]:= "-14;
DECSOLTRIPIV(SUB, D, SUPER, 30, AUX, B);
OUTPUT(71, "("/,"("AUX[3] AND AUX[5]:")",2(/,N)")",
AUX[3], AUX[5]);
B[2]:= B[2] - 1;
OUTPUT(71, "("//"("ERROR IN THE SOLUTION: ")", N")",
SQRT(VECVEC(1, 30, 0, B, B)))
"END"
RESULTS:
AUX[3] AND AUX[5]:
+3.0000000000000"+001
+1.2400000000000"+002
ERROR IN THE SOLUTION: +0.0000000000000"+000
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 9
SOURCE TEXTS:
0"CODE" 34424;
"PROCEDURE" SOLTRI(SUB, DIAG, SUPER, N, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, B;
"BEGIN" "INTEGER" I;
"REAL" R;
R:= B[1]:= B[1] / DIAG[1];
"FOR" I:= 2 "STEP" 1 "UNTIL" N "DO"
R:= B[I]:= (B[I] - SUB[I - 1] * R) / DIAG[I];
"FOR" I:= N - 1 "STEP" -1 "UNTIL" 1 "DO"
R:= B[I] := B[I] - SUPER[I] * R
"END" SOLTRI;
"EOP"
0"CODE" 34425;
"PROCEDURE" DECSOLTRI(SUB, DIAG, SUPER, N, AUX, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX, B;
"BEGIN"
DECTRI(SUB, DIAG, SUPER, N, AUX); "IF" AUX[3]= N "THEN"
SOLTRI(SUB, DIAG, SUPER, N, B)
"END" DECSOLTRI;
"EOP"
0"CODE" 34427;
"PROCEDURE" SOLTRIPIV(SUB, DIAG, SUPER, N, AID, PIV, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AID, B;
"BOOLEAN" "ARRAY" PIV;
"BEGIN" "INTEGER" I, N1;
"REAL" BI, BI1, R, S, T;
N1:= N - 1;
"FOR" I:= 1 "STEP" 1 "UNTIL" N1 "DO"
"BEGIN" "IF" PIV[I] "THEN"
"BEGIN" BI:= B[I+1]; BI1:= B[I] "END"
"ELSE"
"BEGIN" BI:= B[I]; BI1:= B[I+1] "END";
R:= B[I]:= BI / DIAG[I];
B[I+1]:= BI1 - SUB[I] * R
"END";
R:= B[N]:= B[N] / DIAG[N];
T:= B[N1]:= B[N1] - SUPER[N1] * R;
"FOR" I:= N - 2 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" S:= R; R:= T; T:= B[I]:= B[I] - SUPER[I] * R -
("IF" PIV[I] "THEN" AID[I] * S "ELSE" 0)
"END"
"END" SOLTRIPIV
1SECTION 3.1.2.1.1.1.2.3 (JUNE 1974) PAGE 10
;
"EOP"
0"CODE" 34428;
"PROCEDURE" DECSOLTRIPIV(SUB, DIAG, SUPER, N, AUX, B);
"VALUE" N; "INTEGER" N; "ARRAY" SUB, DIAG, SUPER, AUX, B;
"BEGIN" "INTEGER" I, I1, N1, N2;
"REAL" D, R, S, U, T, Q, V, W, NORM, NORM1, NORM2, TOL,
BI, BI1, BI2;
"BOOLEAN" "ARRAY" PIV[1:N];
TOL:= AUX[2]; D:= DIAG[1]; R:= SUPER[1]; BI:= B[1];
NORM:= NORM2:= ABS(D) + ABS(R); N2:= N - 2;
"FOR" I:= 1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" I1:= I + 1; S:= SUB[I]; T:= DIAG[I1]; Q:= SUPER[I1];
BI1:= B[I1];
NORM1:= NORM2; NORM2:= ABS(S) + ABS(T) + ABS(Q);
"IF" NORM2 > NORM "THEN" NORM:= NORM2;
"IF" ABS(D) * NORM2 < ABS(S) * NORM1 "THEN"
"BEGIN" "IF" ABS(S) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= S; "GOTO" EXIT "END";
U:= SUPER[I]:= T / S; BI1:= B[I]:= BI1 / S;
BI:= BI - BI1 * D; V:= SUB[I]:= Q / S;
W:= SUPER[I1]:= -V * D; D:= DIAG[I1]:= R - U * D;
R:= W; NORM2:= NORM1; PIV[I]:= "TRUE"
"END" "ELSE"
"BEGIN" "IF" ABS(D) <= TOL * NORM1 "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[I]:= R / D; BI:= B[I]:= BI / D;
BI:= BI1 - BI * S; D:= DIAG[I1]:= T - U * S;
PIV[I]:= "FALSE"; R:= Q
"END"
"END";
N1:= N - 1; S:= SUB[N1]; T:= DIAG[N]; NORM1:= NORM2; BI1:= B[N];
NORM2:= ABS(S) + ABS(T); "IF" NORM2 > NORM "THEN" NORM:= NORM2;
"IF" ABS(D) * NORM2 < ABS(S) * NORM1 "THEN"
"BEGIN" "IF" ABS(S) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= N2; AUX[5]:= S; "GOTO" EXIT "END";
U:= SUPER[N1]:= T / S; BI1:= B[N1]:= BI1 / S;
BI:= BI - BI1 * D; D:= R - U * D; NORM2:= NORM1
"END" "ELSE"
"BEGIN" "IF" ABS(D) <= TOL * NORM1 "THEN"
"BEGIN" AUX[3]:= N2; AUX[5]:= D; "GOTO" EXIT "END";
U:= SUPER[N1]:= R / D; BI:= B[N1]:= BI / D;
BI:= BI1 - BI * S; D:= T - U * S
"END";
"IF" ABS(D) <= TOL * NORM2 "THEN"
"BEGIN" AUX[3]:= N1; AUX[5]:= D; "GOTO" EXIT "END";
AUX[3]:= N; AUX[5]:= NORM;
BI1:= B[N]:= BI / D; BI:= B[N1]:= B[N1] - SUPER[N1] * BI1;
"FOR" I:= N - 2 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" BI2:= BI1; BI1:= BI;
BI:= B[I]:= B[I] - SUPER[I] * BI1 -
("IF" PIV[I] "THEN" SUB[I] * BI2 "ELSE" 0)
"END";
EXIT:
"END" DECSOLTRIPIV;
"EOP"
1SECTION 3.1.2.1.1.2.1.1 (DECEMBER 1975) PAGE 1
AUTHOR : T.J. DEKKER.
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 731001.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURE CHLDECBND
FOR THE CHOLESKY DECOMPOSITION OF A SYMMETRIC POSITIVE DEFINITE
BAND MATRIX.
KEYWORDS :
LINEAR EQUATIONS,
CHOLESKY DECOMPOSITION,
SYMMETRIC POSITIVE DEFINITE MATRIX,
BAND MATRIX.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" CHLDECBND(A, N, W, AUX); "VALUE" N, W; "INTEGER" N, W;
"ARRAY" A, AUX;
"CODE" 34330;
THE MEANING OF THE FORMAL PARAMETERS IS :
A : <ARRAY IDENTIFIER>;
"ARRAY" A[1 : W * (N - 1) + N];
ENTRY : A CONTAINS COLUMNWISE (I.E. THE (I,J)-TH
ELEMENT OF THE MATRIX IS A[(J-1)*W+I], J=1,..,N,
I=MAX(1,J-W),..,J) THE UPPER-TRIANGULAR BAND
ELEMENTS OF THE SYMMETRIC BAND MATRIX;
EXIT : THE BAND ELEMENTS OF THE CHOLESKY
MATRIX, WHICH IS AN UPPER-TRIANGULAR BAND MATRIX WITH
W SUPERDIAGONALS, ARE DELIVERED COLUMNWISE IN A;
N : <ARITHMETIC EXPRESSION>;
ORDER OF THE BAND MATRIX;
W : <ARITHMETIC EXPRESSION>;
NUMBER OF SUPERDIAGONALS OF THE MATRIX;
AUX : <ARRAY IDENTIFIER>;
"ARRAY" AUX[2 : 3];
ENTRY : AUX[2] IS A RELATIVE TOLERANCE TO CONTROL THE
CALCULATION OF THE DIAGONAL ELEMENTS OF THE
CHOLESKY MATRIX (SEE METHOD AND PERFORMANCE);
NORMAL EXIT :
AUX[3] = N;
ABNORMAL EXIT :
AUX[3] = K - 1, WHERE K IS THE INDEX OF THE DIAGONAL
ELEMENT OF THE CHOLESKY MATRIX THAT CANNOT BE
CALCULATED.
1SECTION 3.1.2.1.1.2.1.1 (JUNE 1974) PAGE 2
PROCEDURES USED :
VECVEC = CP34010.
RUNNING TIME :
(C1 * W + C2) * W * N;
THE CONSTANTS C1 AND C2 DEPEND UPON THE
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
CHLDECBND PERFORMS THE CHOLESKY DECOMPOSITION OF A SYMMETRIC
POSITIVE DEFINITE MATRIX, WHOSE NON-ZERO ELEMENTS ARE IN BAND FORM,
AND WHOSE UPPER-TRIANGULAR BAND ELEMENTS ARE STORED COLUMNWISE IN
IONAL ARRAY.
THE METHOD USED IS CHOLESKY'S SQUARE ROOT METHOD. IF THE GIVEN
MATRIX IS POSITIVE DEFINITE, THEN THIS METHOD YIELDS AN UPPER-
TRIANGULAR BAND MATRIX, THE CHOLESKY MATRIX. THE NUMBER OF
NON-ZERO SUPERDIAGONALS OF THE GIVEN MATRIX AND ITS CHOLESKY
MATRIX ARE EQUAL. THE PROCESS IS COMPLETED IN N STAGES, AT EACH
STAGE PRODUCING A ROW OF THE CHOLESKY MATRIX. HOWEVER, THE
PROCESS IS DISCONTINUED IF AT SOME STAGE, SAY K, THE K-TH DIAGONAL
ELEMENT OF THE GIVEN MATRIX MINUS THE SUM OF SQUARES OF THE
SUPERDIAGONAL ELEMENTS OF THE K-TH COLUMN OF THE CHOLESKY MATRIX
(THE SQUARE ROOT OF THIS QUANTITY BEING THE K-TH DIAGONAL ELEMENT
OF THE CHOLESKY MATRIX) IS EITHER NEGATIVE OR LESS THAN A GIVEN
RELATIVE TOLERANCE (AUX[2]) TIMES THE MAXIMAL DIAGONAL ELEMENT
THE GIVEN MATRIX. IN THIS CASE THE GIVEN MATRIX, POSSIBLY MODIFIED
BY ROUNDING ERRORS, IS NOT POSITIVE DEFINITE. THIS IS INDICATED IN
AUX[3], BY WHICH THE VALUE K - 1 IS DELIVERED. IF THE
DECOMPOSITION IS CARRIED OUT FULLY, AUX[3] BECOMES N.
THE PROCEDURE DELIVERS THE BAND ELEMENTS OF THE CHOLESKY MATRIX.
SEE ALSO REF [1], SECTION 222.
REFERENCE :
[1] DEKKER, T.J. :
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1,
MC TRACT 22, 1968, MATHEMATISCH CENTRUM, AMSTERDAM.
EXAMPLE OF USE :
SEE EXAMPLE OF USE OF CHLSOLBND.
1SECTION 3.1.2.1.1.2.1.1 (JUNE 1974) PAGE 3
SOURCE TEXT(S) :
0"CODE" 34330;
"PROCEDURE" CHLDECBND(A, N, W, AUX); "VALUE" N, W; "INTEGER" N, W;
"ARRAY" A, AUX;
"BEGIN" "INTEGER" J, K, JMAX, KK, KJ, W1, START;
"REAL" R, EPS, MAX;
MAX:= 0; KK:= - W; W1:= W + 1;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + W1; "IF" A[KK] > MAX "THEN" MAX:= A[KK]"END";
JMAX:= W; W1:= W + 1; KK:= - W; EPS:= AUX[2] * MAX;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN""IF" K + W > N "THEN" JMAX:= JMAX - 1; KK:= KK + W1;
START:= KK - K + 1;
R:= A[KK] - VECVEC("IF" K <= W1 "THEN" START "ELSE" KK - W,
KK - 1, 0, A, A); "IF" R <= EPS "THEN"
"BEGIN" AUX[3]:= K - 1; "GO TO" END "END";
A[KK]:= R:= SQRT(R); KJ:= KK;
"FOR" J:= 1 "STEP" 1 "UNTIL" JMAX "DO"
"BEGIN" KJ:= KJ + W;
A[KJ]:= (A[KJ] - VECVEC("IF" K + J <= W1 "THEN" START
"ELSE" KK - W + J, KK - 1, KJ - KK, A, A)) / R
"END"
"END";
AUX[3]:= N;
END:
"END" CHLDECBND;
"EOP"
1SECTION 3.1.2.1.1.2.1.2 (JUNE 1974) PAGE 1
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 731001.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS ONE PROCEDURE,
CHLDETERMBND, FOR THE CALCULATION OF THE DETERMINANT OF A SYMMETRIC
POSITIVE DEFINITE BAND MATRIX.
KEY WORDS :
DETERMINANT,
SYMMETRIC POSITIVE DEFINITE MATRIX,
BAND MATRIX.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"REAL""PROCEDURE" CHLDETERMBND(A, N, W); "VALUE" N, W;
"INTEGER" N, W; "ARRAY" A;
"CODE" 34331;
CHLDETERMBND DELIVERS THE DETERMINANT OF THE SYMMETRIC POSITIVE
DEFINITE BAND MATRIX WHOSE CHOLESKY MATRIX IS STORED IN A.
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, W : SEE 'CALLING SEQUENCE' OF CHLDECBND
(SECTION 3.1.2.1.1.2.1.1.);
THE CONTENTS OF A ARE AS PRODUCED BY CHLDECBND OR
CHLDECSOLBND (SECTION 3.1.2.1.1.2.1.3.).
PROCEDURES USED : NONE.
RUNNING TIME : PROPORTIONAL TO N.
LANGUAGE : ALGOL 60.
1SECTION 3.1.2.1.1.2.1.2 (JUNE 1974) PAGE 2
METHOD AND PERFORMANCE :
CHLDETERMBND CAN BE CALLED AFTER CHLDECBND OR CHLDECSOLBND ONLY IF
THE CHOLESKY DECOMPOSITION WAS SUCCESSFUL, I.E. IF
AUX[3] = N.
THE FUNCTION VALUE OF CHLDETERMBND IS THE SQUARE OF THE DETERMINANT
OF THE CHOLESKY MATRIX.
CHLDETERMBND SHOULD NOT BE CALLED WHEN OVERFLOW CAN BE EXPECTED.
EXAMPLE OF USE :
SEE EXAMPLES OF USE OF CHLSOLBND AND CHLDECSOLBND.
SOURCE TEXT(S) :
0"CODE" 34331;
"REAL""PROCEDURE" CHLDETERMBND(A, N, W); "VALUE" N, W; "INTEGER" N,W;
"ARRAY" A;
"BEGIN""INTEGER" J, KK, W1; "REAL" P;
W1:= W + 1; KK:= - W; P:= 1;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + W1; P:= A[KK] * P "END";
CHLDETERMBND:= P * P
"END" CHLDETERMBND;
"EOP"
1SECTION 3.1.2.1.1.2.1.3 (DECEMBER 1975) PAGE 1
AUTHOR : T.J. DEKKER.
CONTRIBUTOR : J. KOK.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 731001.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS TWO PROCEDURES.
A) CHLSOLBND, FOR THE SOLUTION OF ONE OR MORE SYSTEMS OF LINEAR
EQUATIONS WITH THE SAME COEFFICIENT MATRIX, WHICH IS SYMMETRIC,
POSITIVE DEFINITE AND IN BANDFORM, PROVIDED THAT THIS
MATRIX HAS BEEN DECOMPOSED BY A CALL OF THE PROCEDURE CHLDECBND
(SECTION 3.1.2.1.1.2.1.1.).
B) CHLDECSOLBND, FOR THE SOLUTION OF ONE SYSTEM OF LINEAR
EQUATIONS BY CHOLESKY'S SQUARE ROOT METHOD, PROVIDED THAT THE
SYMMETRIC POSITIVE DEFINITE COEFFICIENT MATRIX IS IN BAND FORM
AND IS STORED COLUMNWISE IN A ONE-DIMENSIONAL ARRAY.
KEYWORDS :
LINEAR EQUATIONS,
CHOLESKY DECOMPOSITION,
SYMMETRIC POSITIVE DEFINITE MATRIX,
BAND MATRIX.
SUBSECTION : CHLSOLBND.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" CHLSOLBND(A, N, W, B); "VALUE" N, W; "INTEGER" N, W;
"ARRAY" A, B;
"CODE" 34332;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, W : SEE 'CALLING SEQUENCE' OF CHLDECBND,
THE CONTENTS OF THE ARRAY A ARE AS PRODUCED BY
CHLDECBND;
B : <ARRAY IDENTIFIER>;
"ARRAY" B[1 : N];
ENTRY : THE RIGHT HAND SIDE OF THE SYSTEM OF LINEAR
EQUATIONS;
EXIT : THE SOLUTION OF THE SYSTEM.
PROCEDURES USED :
VECVEC = CP34010,
SCAPRD1= CP34017.
1SECTION 3.1.2.1.1.2.1.3 (JUNE 1974) PAGE 2
RUNNING TIME:
(C3 * W + C4) * N;
THE CONSTANTS C3 AND C4 DEPEND UPON THE
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
THE PROCEDURE CHLSOLBND CALCULATES THE SOLUTION OF A SYSTEM OF
LINEAR EQUATIONS, PROVIDED THAT THE COEFFICIENT MATRIX WAS
DECOMPOSED BY A SUCCESSFUL CALL OF CHLDECBND (SECTION
3.1.2.1.1.2.1.1.).
THE SOLUTION OF THE LINEAR SYSTEM IS OBTAINED BY CARRYING OUT THE
FORWARD AND BACK SUBSTITUTION WITH THE CHOLESKY MATRIX AND THE
RIGHT HAND SIDE. THE LATTER IS OVERWRITTEN BY THE SOLUTION.
THE SOLUTIONS OF SEVERAL SYSTEMS WITH THE SAME COEFFICIENT MATRIX
CAN BE OBTAINED BY SUCCESSIVE CALLS OF CHLSOLBND.
EXAMPLE OF USE :
THE FOLLOWING PROGRAM SOLVES THE SYSTEM OF SIMULTANEOUS EQUATIONS
2 * X1 - X2 = 1
- X1 + 2 * X2 - X3 = 0
- X2 + 2 * X3 - X4 = 0
- X3 + 2 * X4 - X5 = 0
- X4 + 2 * X5 = 1
"BEGIN""COMMENT" 730829, TEST CHLDECBND, CHLSOLBND AND
CHLDETERMBND;
"INTEGER" I;
"ARRAY" SYMBAND[1 : 9], RIGHT[1 : 5], AUX[2 : 3];
"FOR" I:= 1 "STEP" 1 "UNTIL" 9 "DO"
SYMBAND[I]:= "IF" I // 2 * 2 < I "THEN" 2 "ELSE" - 1;
RIGHT[1]:= RIGHT[5]:= 1;
"FOR" I:= 2, 3, 4 "DO" RIGHT[I]:= 0; AUX[2]:= "- 12;
CHLDECBND(SYMBAND, 5, 1, AUX);
"IF" AUX[3] = 5 "THEN"
"BEGIN" CHLSOLBND(SYMBAND, 5, 1, RIGHT);
OUTPUT(61, "("5(+2Z.4D2B), /"("DETERMINANT IS ")" +.8D"+DD
")", RIGHT, CHLDETERMBND(SYMBAND, 5, 1))
"END"
"END".
THIS PROGRAM DELIVERS:
+1.0000 +1.0000 +1.0000 +1.0000 +1.0000
DETERMINANT IS +.60000000"+01
1SECTION 3.1.2.1.1.2.1.3 (JUNE 1974) PAGE 3
SUBSECTION : CHLDECSOLBND.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" CHLDECSOLBND(A, N, W, AUX, B); "VALUE" N, W;
"INTEGER" N, W; "ARRAY" A, AUX, B;
"CODE" 34333;
THE MEANING OF THE FORMAL PARAMETERS IS :
A, N, W, AUX : SEE CHLDECBND;
B : SEE CHLSOLBND.
PROCEDURES USED :
CHLDECBND = CP34330,
CHLSOLBND = CP34332.
RUNNING TIME:
(C1 * W + C5) * W * N;
THE CONSTANTS C1 AND C5 DEPEND UPON THE ELEMENTARY
ARITHMETIC OF THE COMPUTER.
LANGUAGE : ALGOL 60.
METHOD AND PERFORMANCE :
CHLDECSOLBND SOLVES A SYSTEM OF LINEAR EQUATIONS BY CALLING
CHLDECBND AND, PROVIDED THAT THE CHOLESKY DECOMPOSITION WAS
SUCCESSFUL, CHLSOLBND. THE COEFFICIENT MATRIX OF THIS SYSTEM HAS TO
BE A SYMMETRIC POSITIVE DEFINITE BAND MATRIX WHOSE UPPER-TRIANGULAR
BAND ELEMENTS ARE STORED COLUMNWISE IN IONAL ARRAY.
EXAMPLE OF USE :
THE PROGRAM
"BEGIN""COMMENT" 730829, TEST CHLDECSOLBND AND CHLDETERMBND;
"INTEGER" I;
"ARRAY" SYMBAND[1 : 9], RIGHT[1 : 5], AUX[2 : 3];
1SECTION 3.1.2.1.1.2.1.3 (JUNE 1974) PAGE 3
"FOR" I:= 1 "STEP" 1 "UNTIL" 9 "DO"
SYMBAND[I]:= "IF" I // 2 * 2 < I "THEN" 2 "ELSE" - 1;
RIGHT[1]:= RIGHT[5]:= 1;
"FOR" I:= 2, 3, 4 "DO" RIGHT[I]:= 0; AUX[2]:= "- 12;
CHLDECSOLBND(SYMBAND, 5, 1, AUX, RIGHT);
"IF" AUX[3] = 5 "THEN"
"BEGIN"
OUTPUT(61, "("5(+2Z.4D2B), /"("DETERMINANT IS ")" +.8D"+DD
")", RIGHT, CHLDETERMBND(SYMBAND, 5, 1))
"END"
"END"
WHICH SOLVES THE SAME PROBLEM AS THE PROGRAM IN THE EXAMPLE OF USE
OF CHLSOLBND, DELIVERS:
+1.0000 +1.0000 +1.0000 +1.0000 +1.0000
DETERMINANT IS +.60000000"+01
"EOP"
SOURCE TEXT(S) :
0"CODE" 34332;
"PROCEDURE" CHLSOLBND(A, N, W, B); "VALUE" N, W; "INTEGER" N, W;
"ARRAY" A, B;
"BEGIN" "INTEGER" I, K, IMAX, KK, W1;
KK:= - W; W1:= W + 1;
"FOR" K:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" KK:= KK + W1;
B[K]:= (B[K] - VECVEC("IF" K <= W1 "THEN" 1 "ELSE" K - W,
K - 1, KK - K, B, A)) / A[KK]
"END";
IMAX:= - 1;
"FOR" K:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN""IF" IMAX < W "THEN" IMAX:= IMAX + 1;
B[K]:= (B[K] - SCAPRD1(KK + W, W, K + 1, 1, IMAX, A, B)) /
A[KK]; KK:= KK - W1
"END"
"END" CHLSOLBND;
"EOP"
0"CODE" 34333;
"PROCEDURE" CHLDECSOLBND(A, N, W, AUX, B); "VALUE" N, W;
"INTEGER" N, W; "ARRAY" A, AUX, B;
"BEGIN"
CHLDECBND(A, N, W, AUX);
"IF" AUX[3] = N "THEN" CHLSOLBND(A, N, W, B)
"END" CHLDECSOLBND;
"EOP"
1SECTION 3.1.2.1.1.2.2.1 (JUNE 1974) PAGE 1
AUTHOR: W. HOFFMANN.
CONTRIBUTOR: J. C. P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731215.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS A PROCEDURE, DECSYMTRI, TO PERFORM A
TRIANGULAR DECOMPOSITION OF A SYMMETRIC TRIDIAGONAL MATRIX.
KEYWORDS:
LU DECOMPOSITION,
TRIANGULAR DECOMPOSITION,
SYMMETRIC TRIDIAGONAL MATRIX.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECSYMTRI(DIAG, CO, N, AUX);
"VALUE" N; "INTEGER" N; "ARRAY" DIAG, CO, AUX;
"CODE" 34420;
THE MEANING OF THE FORMAL PARAMETERS IS:
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF THE GIVEN MATRIX T, SAY;
EXIT: SUPPOSE U DENOTES THE UNIT UPPER-BIDIAGONAL MATRIX,
SUCH THAT U'DU = T FOR SOME DIAGONAL MATRIX D,
WHERE U' DENOTES THE TRANSPOSED MATRIX; THEN D[I,I]
WILL BE DELIVERED IN DIAG[I], I = 1, ..., AUX[3];
CO: <ARRAY IDENTIFIER>;
"ARRAY" CO[1: N - 1];
ENTRY: THE CODIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN IN
CO[I], I = 1, ..., N - 1;
EXIT: U[I, I + 1] WILL BE DELIVERED IN CO[I], I = 1, ...,
AUX[3] - 1;
N: <ARITHMETICAL EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX, ELSE AUX[5] IS SET EQUAL TO
THE VALUE OF THAT ELEMENT WHICH CAUSES THE
BREAKDOWN OF THE DECOMPOSITION.
1SECTION 3.1.2.1.1.2.2.1 (JUNE 1974) PAGE 2
PROCEDURES USED: NONE.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE METHOD USED IN DECSYMTRI YIELDS A UNIT UPPER-BIDIAGONAL MATRIX
U AND A DIAGONAL MATRIX D, SUCH THAT THE PRODUCT U'DU EQUALS THE
GIVEN SYMMETRIC TRIDIAGONAL MATRIX; THE PROCESS IS TERMINATED IN
THE K-TH STEP IF THE MODULUS OF THE K-TH DIAGONAL ELEMENT IS
SMALLER THAN A CERTAIN SMALL VALUE, WHICH IS GIVEN BY AUX[2]
MULTIPLIED BY THE 1-NORM OF THE K-TH ROW; IN THIS CASE AUX[3] WILL
BE GIVEN THE VALUE K - 1 AND AUX[5] WILL BE GIVEN THE VALUE OF
THE K-TH DIAGONAL ELEMENT.
EXAMPLE OF USE: SEE DECSOLSYMTRI (SECTION 3.1.2.1.1.2.2.3).
SOURCE TEXT:
0"CODE" 34420;
"PROCEDURE" DECSYMTRI(DIAG, CO, N, AUX); "VALUE" N; "INTEGER" N;
"ARRAY" DIAG, CO, AUX;
"BEGIN" "INTEGER" I, N1;
"REAL" D, R, S, U, TOL, NORM, NORMR;
TOL:= AUX[2]; D:= DIAG[1]; R:= CO[1];
NORM:= NORMR:= ABS(D) + ABS(R);
"IF" ABS(D) <= NORMR * TOL "THEN"
"BEGIN" AUX[3]:= 0; AUX[5]:= D; "GOTO" EXIT "END";
U:= CO[1]:= R / D; N1:= N - 1;
"FOR" I:= 2 "STEP" 1 "UNTIL" N1 "DO"
"BEGIN" S:= R; R:= CO[I]; D:= DIAG[I];
NORMR:= ABS(S) + ABS(D) + ABS(R);
D:= DIAG[I]:= D - U * S;
"IF" ABS(D) <= NORMR * TOL "THEN"
"BEGIN" AUX[3]:= I - 1; AUX[5]:= D; "GOTO" EXIT "END";
U:= CO[I]:= R / D; "IF" NORMR > NORM "THEN" NORM:= NORMR
"END";
D:= DIAG[N]; NORMR:= ABS(D) + ABS(R);
D:= DIAG[N]:= D - U * R;
"IF" ABS(D) <= NORMR * TOL "THEN"
"BEGIN" AUX[3]:= N1; AUX[5]:= D; "GOTO" EXIT "END";
"IF" NORMR > NORM "THEN" NORM:= NORMR;
AUX[3]:= N; AUX[5]:= NORM;
EXIT:
"END" DECSYMTRI;
"EOP"
1SECTION 3.1.2.1.1.2.2.3 (FEBRUARY 1979) PAGE 1
AUTHOR: W. HOFFMANN.
CONTRIBUTOR: J. C. P. BUS.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731215.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES
FOR SOLVING A SYSTEM OF LINEAR EQUATIONS WITH A SYMMETRIC
TRIDIAGONAL MATRIX;
SOLSYMTRI CALCULATES A SOLUTION IF THE
TRIANGULARLY DECOMPOSED FORM, AS DELIVERED BY DECSYMTRI (SECTION
3.1.2.1.1.2.2.1), IS GIVEN;
DECSOLSYMTRI PERFORMS THE TRIANGULAR
DECOMPOSITION AS WELL AS THE FORWARD AND BACK SUBSTITUTION TO
CALCULATE THE SOLUTION OF THE GIVEN LINEAR SYSTEM.
KEYWORDS:
ALGEBRAIC EQUATIONS,
LINEAR SYSTEMS,
SYMMETRIC TRIDIAGONAL MATRIX,
FORWARD AND BACK SUBSTITUTION.
1SECTION 3.1.2.1.1.2.2.3 (FEBRUARY 1979) PAGE 2
SUBSECTION: SOLSYMTRI.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" SOLSYMTRI(DIAG, CO, N, B);
"VALUE" N; "INTEGER" N; "ARRAY" DIAG, CO, B; "CODE" 34421;
THE MEANING OF THE FORMAL PARAMETERS IS:
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1:N];
ENTRY: THE DIAGONAL MATRIX, AS
DELIVERED BY DECSYMTRI (SECTION 3.1.2.1.1.2.2.1);
CO: <ARRAY IDENTIFIER>;
"ARRAY" CO[1: N - 1];
ENTRY: THE CODIAGONAL OF THE UNIT
UPPER-BIDIAGONAL MATRIX AS DELIVERED BY DECSYMTRI;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE LINEAR SYSTEM;
EXIT: THE CALCULATED SOLUTION OF THE LINEAR SYSTEM.
PROCEDURES USED: NONE.
METHOD AND PERFORMANCE:
SOLSYMTRI CALCULATES THE SOLUTION OF A LINEAR SYSTEM WITH A
SYMMETRIC TRIDIAGONAL MATRIX, WITH FORWARD AND BACK SUBSTITUTION;
THE TRIANGULARLY DECOMPOSED FORM OF THE MATRIX, AS PRODUCED BY
DECSYMTRI (SECTION 3.1.2.1.1.2.2.1), SHOULD BE GIVEN; ONE CALL OF
DECSYMTRI FOLLOWED BY SEVERAL CALLS OF SOLSYMTRI MAY BE USED TO
SOLVE SEVERAL LINEAR SYSTEMS HAVING THE SAME SYMMETRIC TRIDIAGONAL
MATRIX, BUT DIFFERENT RIGHT HAND SIDES.
EXAMPLE OF USE: SEE DECSOLSYMTRI (THIS SECTION).
1SECTION 3.1.2.1.1.2.2.3 (FEBRUARY 1979) PAGE 3
SUBSECTION: DECSOLSYMTRI.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" DECSOLSYMTRI(DIAG, CO, N, AUX, B);
"VALUE" N; "INTEGER" N; "ARRAY" DIAG, CO, AUX, B; "CODE" 34422;
THE MEANING OF THE FORMAL PARAMETERS IS:
DIAG: <ARRAY IDENTIFIER>;
"ARRAY" DIAG[1: N];
ENTRY: THE DIAGONAL OF THE GIVEN MATRIX T, SAY;
EXIT: SUPPOSE U DENOTES THE UNIT UPPER-BIDIAGONAL MATRIX,
SUCH THAT U'DU = T FOR SOME DIAGONAL MATRIX D,
WHERE U' DENOTES THE TRANSPOSED MATRIX; THEN D[I,I]
WILL BE DELIVERED IN DIAG[I], I = 1, ..., AUX[3];
CO: <ARRAY IDENTIFIER>;
"ARRAY" CO[1: N - 1];
ENTRY: THE CODIAGONAL OF T; T[I, I + 1] SHOULD BE GIVEN IN
CO[I], I = 1, ..., N - 1;
EXIT: U[I, I + 1] WILL BE DELIVERED IN CO[I], I = 1, ...,
AUX[3] - 1;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
AUX: <ARRAY IDENTIFIER>;
"ARRAY" AUX[2:5];
ENTRY:
AUX[2]: A RELATIVE TOLERANCE; A REASONABLE CHOICE FOR THIS
VALUE IS AN ESTIMATE OF THE RELATIVE PRECISION OF
THE MATRIX ELEMENTS, HOWEVER, IT SHOULD NOT BE
CHOSEN SMALLER THAN THE MACHINE PRECISION;
EXIT:
AUX[3]: THE NUMBER OF ELIMINATION STEPS PERFORMED;
AUX[5]: IF AUX[3] = N, THEN AUX[5] WILL EQUAL THE INFINITY-
NORM OF THE MATRIX, ELSE AUX[5] IS SET EQUAL TO
THE VALUE OF THAT ELEMENT WHICH CAUSES THE
BREAKDOWN OF THE DECOMPOSITION;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
ENTRY: THE RIGHT HAND SIDE OF THE LINEAR SYSTEM;
EXIT: IF AUX[3] = N THEN THE SOLUTION OF THE LINEAR
SYSTEM IS OVERWRITTEN ON B, ELSE B REMAINS
UNALTERED.
1SECTION 3.1.2.1.1.2.2.3 (FEBRUARY 1979) PAGE 4
PROCEDURES USED:
DECSYMTRI = CP34420,
SOLSYMTRI = CP34421.
METHOD AND PERFORMANCE:
DECSOLSYMTRI CALCULATES THE SOLUTION OF A LINEAR SYSTEM WITH A
SYMMETRIC TRIDIAGONAL MATRIX; THE TRIANGULAR DECOMPOSITION IS DONE
BY CALLING DECSYMTRI (SECTION 3.1.2.1.1.2.2.1) AND THE FORWARD AND
BACK SUBSTITUTION BY CALLING SOLSYMTRI (THIS SECTION); IF AUX[3]<N,
THEN THE EFFECT OF DECSOLSYMTRI IS MERELY THAT OF DECSYMTRI.
EXAMPLE OF USE:
LET T BE A SYMMETRIC TRIDIAGONAL MATRIX OF ORDER 100 WITH DIAGONAL
ELEMENTS I (I = 1, ..., 100) AND CODIAGONAL ELEMENTS I * 2 (I = 1,
..., 99); LET THE RIGHT HAND SIDE B BE GIVEN BY THE SECOND COLUMN
OF T; THEN THE SOLUTION OF THE LINEAR SYSTEM TX = B IS GIVEN BY THE
SECOND UNIT VECTOR; BY THE FOLLOWING PROGRAM WE MAY SOLVE THIS
SYSTEM AND PRINT THE ERROR IN THE CALCULATED SOLUTION.
"BEGIN"
"INTEGER" I; "ARRAY" D, CO, B[1:100], AUX[2:5];
"FOR" I:= 1 "STEP" 1 "UNTIL" 100 "DO"
"BEGIN" D[I]:= I; CO[I]:= I * 2; B[I]:= 0 "END";
B[1]:= B[2]:= 2; B[3]:= 4;
AUX[2]:= "-14;
DECSOLSYMTRI(D, CO, 100, AUX, B);
B[2]:= B[2] - 1;
OUTPUT(71, "("/,"(" AUX[3] AND AUX[5]:")",2(/4B,N)")",
AUX[3], AUX[5]);
OUTPUT(71, "("//,"(" ERROR IN THE SOLUTION:")",N,/")",
SQRT(VECVEC(1, 100, 0, B, B)))
"END"
RESULTS:
AUX[3] AND AUX[5]:
+1.0000000000000"+002
+4.9300000000000"+002
ERROR IN THE SOLUTION:+0.0000000000000"+000
1SECTION 3.1.2.1.1.2.2.3 (JUNE 1974) PAGE 5
SOURCE TEXTS:
0"CODE" 34421;
"PROCEDURE" SOLSYMTRI(DIAG, CO, N, B); "VALUE" N; "INTEGER" N;
"ARRAY" DIAG, CO, B;
"BEGIN" "INTEGER" I;
"REAL" R, S;
R:= B[1]; B[1]:= R / DIAG[1];
"FOR" I:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" R:= B[I] - CO[I-1] * R; B[I]:= R / DIAG[I] "END";
S:= B[N];
"FOR" I:= N - 1 "STEP" -1 "UNTIL" 1 "DO"
S:= B[I]:= B[I] - CO[I] * S
"END" SOLSYMTRI;
"EOP"
0"CODE" 34422;
"PROCEDURE" DECSOLSYMTRI(DIAG, CO, N, AUX, B); "VALUE" N;
"INTEGER" N; "ARRAY" DIAG, CO, AUX, B;
"BEGIN"
DECSYMTRI(DIAG, CO, N, AUX); "IF" AUX[3] = N "THEN"
SOLSYMTRI(DIAG, CO, N, B)
"END" DECSOLSYMTRI;
"EOP"
1SECTION 3.1.2.2.1 (DECEMBER 1975) PAGE 1
AUTHOR: P.W.HEMKER.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730615.
BRIEF DESCRIPTION:
CONJ GRAD SOLVES A LINEAR SYSTEM OF EQUATIONS BY THE METHOD OF
CONJUGATE GRADIENTS. THE SYSTEM HAS TO BE SYMMETRIC AND POSITIVE
DEFINITE.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" CONJ GRAD (MATVEC, X, R, L, N, GO ON, ITERATE, NORM2);
"VALUE" L, N; "BOOLEAN" GO ON; "INTEGER" L, N, ITERATE;
"REAL" NORM2; "ARRAY" X, R; "PROCEDURE" MATVEC;
"CODE" 34220;
THE MEANING OF THE FORMAL PARAMETERS IS:
MATVEC: <PROCEDURE IDENTIFIER>;
THE HEADING OF THIS PROCEDURE READS:
"PROCEDURE" MATVEC( P, Q); "ARRAY" P, Q;
THIS PROCEDURE DEFINES THE MATRIX A (THE COEFFICIENT MATRIX
OF THE SYSTEM) AS FOLLOWS :
AT EACH CALL MATVEC DELIVERS IN Q THE MATRIX-VECTOR PRODUCT
AP; P AND Q ARE ONE - DIMENSIONAL ARRAYS:
"ARRAY" P,Q[L:N];
X: <ARRAY IDENTIFIER>;
"ARRAY" X[L:N];
ENTRY: AN INITIAL APPROXIMATION TO THE SOLUTION X;
EXIT: THE SOLUTION;
R: <ARRAY IDENTIFIER>;
"ARRAY" R[L:N];
ENTRY: THE RIGHT-HAND SIDE OF THE SYSTEM;
EXIT: THE RESIDUE B - AX, COMPUTED RECURSIVELY;
L,N: <ARITHMETIC EXPRESSION>;
L AND N ARE RESPECTIVELY THE LOWER AND UPPER BOUND OF THE
ARRAYS X,R,P,Q;
GO ON: <BOOLEAN EXPRESSION>;
GO ON INDICATES THE CONTINUATION OF THE PROCESS.
THIS EXPRESSION MAY DEPEND ON THE JENSEN PARAMETERS ITERATE
AND NORM2. WITH THIS BOOLEAN EXPRESSION THE USER CONTROLS
THE CONTINUATION OF THE PROCESS. IF GO ON:= "FALSE" THE
ITERATION PROCESS IS STOPPED.
ITERATE:<IDENTIFIER>;
DELIVERS THE NUMBER OF ITERATION STEPS ALREADY PERFORMED;
NORM2: <IDENTIFIER>;
DELIVERS THE SQUARED EUCLIDEAN NORM OF THE RESIDUE,COMPUTED
RECURSIVELY
1SECTION 3.1.2.2.1 (JUNE 1974) PAGE 2
PROCEDURES USED:
VECVEC = CP34010 ,
ELMVEC = CP34020 .
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH: 7 + 2 * ( N - L + 1 ).
RUNNING TIME:
THE RUNNING TIME IS PROPORTIONAL TO THE NUMBER OF ITERATION STEPS
PERFORMED. EACH ITERATION STEP REQUIRES ONE EVALUATION OF THE
PROCEDURE MATVEC, THE EVALUATION OF TWO SCALAR - VECTOR - PRODUCTS
AND ONE VECTOR - VECTOR - PRODUCT.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE REF[1].
REFERENCES:
[1].J.K.REID.
ON THE METHOD OF CONJUGATE GRADIENTS FOR THE SOLUTION OF LARGE
SPARSE SYSTEMS OF LINEAR EQUATIONS.
IN:LARGE SPARSE SETS OF LINEAR EQUATIONS (J.K.REID ED)1971.
1SECTION 3.1.2.2.1 (DECEMBER 1979) PAGE 3
EXAMPLE OF USE:
"BEGIN"
"ARRAY" X,B[0:12];
"INTEGER" IT,I;
"REAL" NO;
"PROCEDURE" A(X,B);
"ARRAY" X,B;
"BEGIN" B[0]:=2*X[0]-X[1];
"FOR" I:=1 "STEP" 1 "UNTIL" 11 "DO"
B[I]:=-X[I-1]+2*X[I]-X[I+1];
B[12]:=2*X[12]-X[11]
"END" A;
"FOR" I:=0 "STEP" 1 "UNTIL" 12 "DO" X[I]:=B[I]:=0;
B[0]:=1;B[12]:=4;
CONJ GRAD(A,X,B,0,12,IT<20 "AND" NO>"-10,IT,NO);
OUTPUT(61,"(""("IT= ")",B3D5B,"("NO= ")",N,//,10B,
"("X")",20B,"("R")",//")",IT,NO);
"FOR" I:=0 "STEP" 1 "UNTIL" 12 "DO"
OUTPUT(61,"("N,5B,N,/")",X[I],B[I])
"END"
DELIVERS:
IT= 013 NO= +3.3424581859911"-027
X R
+1.2142857142857"+000 -7.1054273576010"-015
+1.4285714285715"+000 +1.5151278924296"-014
+1.6428571428572"+000 -1.3184703260130"-014
+1.8571428571429"+000 +1.6718441615946"-014
+2.0714285714286"+000 -1.5514524667596"-014
+2.2857142857144"+000 +2.2130179956186"-014
+2.5000000000001"+000 -2.2524167805437"-014
+2.7142857142858"+000 +2.0834049529361"-014
+2.9285714285715"+000 -1.8674557504802"-014
+3.1428571428572"+000 +1.9163204503355"-014
+3.3571428571429"+000 -1.2366043539824"-014
+3.5714285714286"+000 +8.2548347242718"-015
+3.7857142857143"+000 +4.4408920985006"-016 .
1SECTION 3.1.2.2.1 (JUNE 1974) PAGE 4
SOURCE TEXT(S):
0"CODE" 34220;
"PROCEDURE" CONJ GRAD( MATVEC, X, R, L, N, GO ON, ITERATE, NORM2);
"VALUE" L, N; "PROCEDURE" MATVEC; "ARRAY" X, R; "BOOLEAN" GO ON;
"INTEGER" L, N, ITERATE; "REAL" NORM2;
"BEGIN" "ARRAY" P, AP[ L: N];
"INTEGER" I;
"REAL" A, B, PRR, RRP;
"FOR" ITERATE:= 0, ITERATE + 1 "WHILE" GO ON "DO"
"BEGIN" "IF" ITERATE = 0 "THEN"
"BEGIN" MATVEC( X, P);
"FOR" I:= L "STEP" 1 "UNTIL" N "DO"
P[ I]:= R[ I]:= R[ I] - P[ I];
PRR:= VECVEC( L, N, 0, R, R)
"END" "ELSE"
"BEGIN" B:= RRP / PRR; PRR:= RRP;
"FOR" I:= L "STEP" 1 "UNTIL" N "DO"
P[ I]:= R[ I] + B * P[ I]
"END";
MATVEC( P, AP);
A:= PRR / VECVEC( L, N, 0, P, AP);
ELMVEC( L, N, 0, X, P, A);
ELMVEC( L, N, 0, R, AP, -A);
NORM2:= RRP:= VECVEC( L, N, 0, R, R)
"END"
"END" CONJ GRAD;
"EOP"
1SECTION:3.2.1.1.1 (JUNE 1974) PAGE 1
AUTHORS : T.J.DEKKER AND W.HOFFMANN.
CONTRIBUTORS: W.HOFFMANN, J.G.VERWER.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731022.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS TWO PROCEDURES.
A) EQILBR EQUILIBRATES A MATRIX BY MEANS OF A DIAGONAL SIMILARITY
TRANSFORMATION,
B) BAKLBR PERFORMS THE CORRESPONDING BACK TRANSFORMATION ON THE
COLUMNS OF A MATRIX AND SHOULD BE CALLED AFTER EQILBR.
KEYWORDS:
SIMILARITY TRANSFORMATION,
EQUILIBRATION.
SUBSECTION: EQILBR.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" EQILBR(A, N, EM, D, INT); "VALUE" N;
"INTEGER" N; "ARRAY" A, EM, D; "INTEGER" "ARRAY" INT;
"CODE" 34173;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE MATRIX TO BE EQUILIBRATED;
EXIT: THE EQUILIBRATED MATRIX;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:0];
ENTRY: EM[0], THE MACHINE PRECISION;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT: THE MAIN DIAGONAL OF THE TRANSFORMING DIAGONAL
MATRIX;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
EXIT: INFORMATION DEFINING THE POSSIBLE INTERCHANGING OF
SOME ROWS AND THE CORRESPONDING COLUMNS;
1SECTION:3.2.1.1.1 (JUNE 1974) PAGE 2
PROCEDURES USED:
TAMMAT = CP34014,
MATTAM = CP34015,
ICHCOL = CP34031,
ICHROW = CP34032.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE MATRIX IS EQUILIBRATED BY MEANS OF OSBORNE'S DIAGONAL
SIMILARITY TRANSFORMATION POSSIBLY WITH INTERCHANGES [2].
THE TRANSFORMING DIAGONAL MATRIX AND THE EQUILIBRATED MATRIX ARE
CALCULATED ITERATIVELY:
IN EACH STEP A CERTAIN COLUMN OF THE MATRIX IS MULTIPLIED BY, AND
THE CORRESPONDING ROW DIVIDED BY, A FACTOR WHICH IS CHOSEN IN SUCH
A WAY THAT THE CONSIDERED COLUMN AND ROW OBTAIN ROUGHLY THE SAME
EUCLIDEAN NORM (IN FACT, THE FACTOR IS ROUNDED TO THE NEAREST
INTEGRAL POWER OF 2, IN ORDER TO PREVENT ROUNDING ERRORS); THE
COLUMNS AND ROWS ARE HANDLED IN CYCLIC ORDER. IF THE MATRIX DOES
NOT CONTAIN COLUMNS OR ROWS WHOSE OFF-DIAGONAL ELEMENTS ARE 0 OR
NEARLY 0, THEN THE PROCESS (WITH UNROUNDED FACTORS) CONVERGES, AND
IN PRACTICE A FEW STEPS ARE NEEDED TO OBTAIN A REASONABLY
EQUILIBRATED MATRIX [2].
IF ALL OFF-DIAGONAL ELEMENTS OF SOME CONSIDERED COLUMN (ROW) ARE 0
OR NEARLY 0, THEN THIS COLUMN (ROW) IS INTERCHANGED WITH THE FIRST
NONZERO COLUMN (LAST NONZERO ROW) OF THE MATRIX, AND, IN ORDER TO
HAVE A SIMILARITY TRANSFORMATION, THE CORRESPONDING ROWS (COLUMNS)
ARE ALSO INTERCHANGED; THEN FOR THE FURTHER EQUILIBRATION, THE
SUBMATRIX IS CONSIDERED WHICH DOES NOT CONTAIN SUCH ZERO COLUMNS
AND ROWS AND THE CORRESPONDING ROWS AND COLUMNS. THE EQUILIBRATION
PROCESS IS CONTINUED UNTIL, IN A WHOLE CYCLE NO FACTOR > 2 OR < 0.5
AND NO ZERO COLUMN OR ROW IS FOUND, OR UNTIL (N + 1) * N ** 2 ROWS
AND COLUMNS HAVE BEEN CONSIDERED.
1SECTION:3.2.1.1.1 (JUNE 1974) PAGE 3
SUBSECTION: BAKLBR.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" BAKLBR(N, N1, N2, D, INT, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" D, VEC; "INTEGER" "ARRAY" INT;
"CODE" 34174;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE LENGTH OF THE VECTORS TO BE TRANSFORMED;
N1, N2: <ARITHMETIC EXPRESSION>;
THE SERIAL NUMBERS OF THE FIRST AND LAST VECTOR TO BE
TRANSFORMED;
VEC: <ARRAY IDENTIFIER>;
"ARRAY"VEC[1:N,N1:N2];
ENTRY: THE N2 - N1 + 1 VECTORS OF LENGTH N TO BE
TRANSFORMED;
EXIT: THE N2 - N1 + 1 VECTORS OF LENGTH N RESULTING FROM
THE BACK TRANSFORMATION;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
ENTRY: THE MAIN DIAGONAL OF THE TRANSFORMING DIAGONAL
MATRIX OF ORDER N, AS PRODUCED BY EQILBR;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
ENTRY: INFORMATION DEFINING THE POSSIBLE INTERCHANGING OF
SOME ROWS AND COLUMNS, AS PRODUCED BY EQILBR.
PROCEDURES USED:
ICHROW = CP34032.
RUNNING TIME: ROUGHLY PROPORTIONAL TO (N2 - N1 + 1) * N.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE BACK TRANSFORMATION, WHICH CORRESPONDS WITH THE DIAGONAL
SIMILARITY TRANSFORMATION AS PERFORMED BY EQILBR, TRANSFORMS
A VECTOR X INTO A VECTOR DX AND PERFORMS THE CORRESPONDING
INTERCHANGES. THE MATRIX D IS THE DIAGONAL MATRIX OF THE DIAGONAL
SIMILARITY TRANSFORMATION.
REFERENCES:
[1] DEKKER, T. J. AND HOFFMANN, W.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2,
MATHEMATICAL CENTRE TRACTS 23,
MATHEMATISCH CENTRUM, AMSTERDAM, 1968;
[2] OSBORNE, E. E, ON PRECONDITIONING OF MATRICES,
J. ACM 7(1960) 338-354.
EXAMPLES OF USE:
EXAMPLES OF USE OF EQILBR AND BAKLBR CAN BE FOUND IN THE PROCEDURES
FOR CALCULATING EIGENVALUES AND EIGENVECTORS AS DESCRIBED IN
SECTION 3.3.1.2.2.
1SECTION:3.2.1.1.1 (JUNE 1974) PAGE 4
SOURCE TEXT(S) :
0"CODE" 34173;
"COMMENT" MCA 2405;
"PROCEDURE" EQILBR(A, N, EM, D, INT); "VALUE" N; "INTEGER" N;
"ARRAY" A, EM, D; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, IM, I1, P, Q, J, T, COUNT, EXPONENT, NI;
"REAL" C, R, EPS, OMEGA, FACTOR;
"PROCEDURE" MOVE(K); "VALUE" K; "INTEGER" K;
"BEGIN" "REAL" DI;
NI:= Q - P; T:= T + 1; "IF" K ^= I "THEN"
"BEGIN" ICHCOL(1, N, K, I, A); ICHROW(1, N, K, I, A);
DI:= D[I]; D[I]:= D[K]; D[K]:= DI
"END"
"END" MOVE;
FACTOR:= 1 / (2 * LN(2)); "COMMENT" MORE GENERALLY: LN(BASE);
EPS:= EM[0]; OMEGA:= 1 / EPS; T:= P:= 1; Q:= NI:= I:= N;
COUNT:= (N + 1) * N // 2;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" D[J]:= 1; INT[J]:= 0 "END";
"FOR" I:= "IF" I < Q "THEN" I + 1 "ELSE" P
"WHILE" COUNT > 0 "AND" NI > 0 "DO"
"BEGIN" COUNT:= COUNT - 1; IM:= I - 1; I1:= I + 1;
C:= SQRT(TAMMAT(P, IM, I, I, A, A) +
TAMMAT(I1, Q, I, I, A, A));
R:= SQRT(MATTAM(P, IM, I, I, A, A) +
MATTAM(I1, Q, I, I, A, A));
"IF" C * OMEGA <= R * EPS "THEN"
"BEGIN" INT[T]:= I; MOVE(P); P:= P + 1 "END"
"ELSE" "IF" R * OMEGA <= C * EPS "THEN"
"BEGIN" INT[T]:= -I; MOVE(Q); Q:= Q - 1 "END"
"ELSE"
"BEGIN" EXPONENT:= LN(R / C) * FACTOR;
"IF" ABS(EXPONENT) > 1 "THEN"
"BEGIN" NI:= Q - P; C:= 2 ** EXPONENT; R:= 1 / C;
D[I]:= D[I] * C;
"FOR" J:= 1 "STEP" 1 "UNTIL" IM,
I1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" A[J,I]:= A[J,I] * C;
A[I,J]:= A[I,J] * R
"END"
"END" "ELSE" NI:= NI - 1
"END"
"END"
"END" EQILBR
1SECTION:3.2.1.1.1 (JUNE 1974) PAGE 5
;
"EOP"
0"CODE" 34174;
"COMMENT" MCA 2406;
"PROCEDURE" BAKLBR(N, N1, N2, D, INT, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" D, VEC; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, J, K, P, Q; "REAL" DI;
P:= 1; Q:= N;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" DI:= D[I]; "IF" DI ^= 1 "THEN"
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO" VEC[I,J]:=
VEC[I,J] * DI; K:= INT[I];
"IF" K > 0 "THEN" P:= P + 1 "ELSE"
"IF" K < 0 "THEN" Q:= Q - 1
"END";
"FOR" I:= P - 1 + N - Q "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" K:= INT[I]; "IF" K > 0 "THEN"
"BEGIN" P:= P - 1; "IF" K ^= P "THEN"
ICHROW(N1, N2, K, P, VEC)
"END" "ELSE"
"BEGIN" Q:= Q + 1; "IF" -K ^= Q "THEN"
ICHROW(N1, N2, -K, Q, VEC)
"END"
"END"
"END" BAKLBR;
"EOP"
1SECTION:3.2.1.1.2 (JUNE 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED: 731008.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURES EQILBRCOM AND BAKLBRCOM.
EQILBRCOM EQUILIBRATES A GIVEN MATRIX.
BAKLBRCOM TRANSFORMS THE EIGENVECTORS OF THE EQUILIBRATED MATRIX
INTO THE EIGENVECTORS OF THE ORIGINAL MATRIX.
KEYWORDS :
COMPLEX MATRIX,
EIGENVECTORS,
EQUILIBRATION.
SUBSECTION: EQILBRCOM.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" EQILBRCOM(A1, A2, N, EM, D, INT); "VALUE" N;
"INTEGER" N; "ARRAY" A1, A2, EM, D; "INTEGER" "ARRAY" INT;
"CODE" 34361;
THE MEANING OF THE FORMAL PARAMETERS IS:
A1,A2:: <ARRAY IDENTIFIER>;
"ARRAY" A1,A2[1:N,1:N];
ENTRY:
THE REAL PART AND IMAGINARY PART OF THE MATRIX TO BE
EQUILIBRATED MUST BE GIVEN IN THE ARRAYS A1 AND A2,
RESPECTIVELY;
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE EQUILIBRATED
MATRIX ARE DELIVERED IN THE ARRAYS A1 AND A2,
RESPECTIVELY;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:7];
ENTRY:
EM[0]: THE MACHINE PRECISION;
EM[6]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS;
EXIT:
EM[7]: THE NUMBER OF ITERATIONS PERFORMED;
1SECTION:3.2.1.1.2 (JUNE 1974) PAGE 2
D: <ARRAY IDENTIFIER>:
"ARRAY" D[1:N];
EXIT:
THE SCALING FACTORS OF THE DIAGONAL SIMILARITY
TRANSFORMATION;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
EXIT:
INFORMATION CONCERNING THE POSSIBLE INTERCHANGING OF
SOME ROWS AND CORRESPONDING COLUMNS.
PROCEDURES USED:
ICHCOL = CP34031,
ICHROW = CP34332,
TAMMAT = CP34014,
MATTAM = CP34015.
RUNNING TIME: PROPORTIONAL TO N * NUMBER OF ITERATIONS.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE BAKLBRCOM.
SUBSECTION: BAKLBRCOM.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" BAKLBRCOM(N, N1, N2, D, INT, VR, VI);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, VR, VI;
"INTEGER" "ARRAY" INT;
"CODE" 34362;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX OF WHICH THE EIGENVECTORS ARE
CALCULATED;
N1,N2: <ARITHMETIC EXPRESSION>;
THE EIGENVECTORS CORRESPONDING TO THE EIGENVALUES WITH
INDICES N1,...,N2 ARE TO BE TRANSFORMED;
D: <ARRAY IDENTIFIER>:
"ARRAY" D[1:N];
ENTRY: THE SCALING FACTORS OF THE DIAGONAL SIMILARITY
TRANSFORMATION AS DELIVERED BY EQILBRCOM;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
ENTRY: INFORMATION DEFINING THE INTERCHANGING OF SOME
ROWS AND COLUMNS, AS DELIVERED BY EQILBRCOM;
1SECTION:3.2.1.1.2 (DECEMBER 1975) PAGE 3
VR,VI: <ARRAY IDENTIFIER>;
"ARRAY" VR,VI[1:N,N1:N2];
ENTRY:
THE BACK TRANSFORMATION IS PERFORMED ON THE EIGENVECTORS
WITH THE REAL PARTS GIVEN IN ARRAY VR AND THE IMAGINARY
PARTS GIVEN IN ARRAY VI;
EXIT:
THE REAL PARTS AND IMAGINARY PARTS OF THE RESULTING
EIGENVECTORS ARE DELIVERED IN THE COLUMNS OF THE ARRAYS
VR AND VI, RESPECTIVELY.
PROCEDURES USED: BAKLBR = CP34174.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N * (N2-N1).
LANGUAGE: ALGOL 60.
THE FOLLOWING HOLDS FOR BOTH PROCEDURES:
METHOD AND PERFORMANCE:
A MATRIX M IS SAID TO BE EQUILIBRATED, WHEN THE DIAGONAL ELEMENTS
OF M''M - MM'' ARE ZERO, WHERE '' STANDS FOR CONJUGATING
AND TRANSPOSING. IN EQILBRCOM THE MATRIX M IS EQUILIBRATED
BY MEANS OF OSBORNE'S DIAGONAL SIMILARITY TRANSFORMATION WITH
POSSIBLE INTERCHANGES (OSBORNE, 1960).
BAKLBRCOM PERFORMS THE CORRESPONDING BACK TRANSFORMATION.
LET THE EIGENVECTORS OF THE EQUILIBRATED MATRIX BE GIVEN IN THE
COLUMNS OF MATRIX V. THE EIGENVECTORS OF THE ORIGINAL MATRIX ARE
OBTAINED BY MULTIPLYING (OR POSSIBLE INTERCHANGING) THE ROWS OF THE
MATRIX V WITH THE SCALING FACTORS. AS THE SCALING FACTORS ARE REAL
QUANTITIES, THE TRANSFORMATION IS PERFORMED BY CALLING THE
PROCEDURE BAKLBR FOR BOTH VR AND VI (DEKKER AND HOFFMANN, 1968).
REFERENCES:
DEKKER, T.J. AND W.HOFFMANN (1968),
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2,
MATH. CENTRE TRACTS 23, MATHEMATISCH CENTRUM;
OSBORNE, E.E. (1960),
ON PRECONDITIONING OF MATRICES,
JACM., 7, P.338-354;
PARLETT, B.N. AND C.REINSCH (1969),
BALANCING A MATRIX FOR CALCULATION OF EIGENVALUES AND
EIGENVECTORS,
NUM. MATH., 13, P.293-304;
1SECTION:3.2.1.1.2 (JUNE 1974) PAGE 4
EXAMPLE OF USE:
BAKLBRCOM IS USED IN THE PROCEDURE EIGCOM (SEE SECTION 3.3.2.2.2.).
AS A FORMAL TEST OF THE PROCEDURE EQILBRCOM, THE FOLLOWING MATRIX
WAS USED:
1 0 1024*I
0 1 0
I/1024 0 2
"BEGIN" "INTEGER" I,J; "INTEGER""ARRAY" INT[1:3];
"REAL" "ARRAY" A1,A2[1:3,1:3],EM[0:7],D[1:3];
EM[0]:=5"-14;EM[6]:=10;
INIMAT(1,3,1,3,A1,0);INIMAT(1,3,1,3,A2,0);
A1[1,1]:=A1[2,2]:=1;A1[3,3]:=2;
A2[1,3]:=2**10;A2[3,1]:=1/2**10;
EQILBRCOM(A1,A2,3,EM,D,INT);
OUTPUT(61,"(""("EQUILIBRATED MATRIX:")",/")");
OUTPUT(61,"("3(D2B),/,2(D2B),"("I")",/,D2B,"("I")",2BD/")",
A1[1,1],A1[1,2],A1[1,3],A1[2,1],A1[2,2],A1[3,1],A1[3,3]);
OUTPUT(61,"("/,"("EM[7]:")",5BD/")",EM[7]);
OUTPUT(61,"(""("D[1:3]: ")",3(3ZD2B),/")",D[1],D[2],D[3]);
OUTPUT(61,"(""("INT[1:3]: ")",BD,3B,2BD,B4ZD")",
INT[1],INT[2],INT[3])
"END"
OUTPUT:
EQUILIBRATED MATRIX:
1 0 0
0 1 I
0 I 2
EM[7]: 4
D[1:3]: 1 1024 1
INT[1:3]: 2 0 0
SOURCE TEXT(S) :
0"CODE" 34361;
"PROCEDURE" EQILBRCOM(A1, A2, N, EM, D, INT); "VALUE" N;
"INTEGER" N; "ARRAY" A1, A2, EM, D; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, P, Q, J, T, COUNT, EXPONENT, NI, IM, I1;
"REAL" C, R, EPS;
"COMMENT"
1SECTION:3.2.1.1.2 (JUNE 1974) PAGE 5
;
"PROCEDURE" MOVE(K); "VALUE" K; "INTEGER" K;
"BEGIN" "REAL" DI;
NI:= Q - P; T:= T + 1; "IF" K ^= I "THEN"
"BEGIN" ICHCOL(1, N, K, I, A1); ICHROW(1, N, K, I, A1);
ICHCOL(1, N, K, I, A2); ICHROW(1, N, K, I, A2);
DI:= D[I]; D[I]:= D[K]; D[K]:= DI
"END"
"END" MOVE;
EPS:= EM[0] ** 4; T:= P:= 1; Q:= NI:= I:= N;
COUNT:= EM[6];
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" D[J]:= 1; INT[J]:= 0 "END";
"FOR" I:= "IF" I < Q "THEN" I + 1 "ELSE" P "WHILE" COUNT > 0
"AND" NI > 0 "DO"
"BEGIN" COUNT:= COUNT - 1; IM:= I - 1; I1:= I + 1;
C:= TAMMAT(P, IM, I, I, A1, A1) + TAMMAT(I1, Q, I,
I, A1, A1) + TAMMAT(P, IM, I, I, A2, A2) +
TAMMAT(I1, Q, I, I, A2, A2);
R:= MATTAM(P, IM, I, I, A1, A1) + MATTAM(I1, Q, I,
I, A1, A1) + MATTAM(P, IM, I, I, A2, A2) +
MATTAM(I1, Q, I, I, A2, A2); "IF" C / EPS <= R "THEN"
"BEGIN" INT[T]:= I; MOVE(P); P:= P + 1 "END"
"ELSE" "IF" R / EPS <= C "THEN"
"BEGIN" INT[T]:= - I; MOVE(Q); Q:= Q - 1 "END"
"ELSE"
"BEGIN" EXPONENT:= LN(R / C) * 0.36067;
"IF" ABS(EXPONENT) > 1 "THEN"
"BEGIN" NI:= Q - P; C:= 2 ** EXPONENT;
D[I]:= D[I] * C;
"FOR" J:= 1 "STEP" 1 "UNTIL" IM, I1 "STEP" 1
"UNTIL" N "DO"
"BEGIN" A1[J,I]:= A1[J,I] * C;
A1[I,J]:= A1[I,J] / C;
A2[J,I]:= A2[J,I] * C;
A2[I,J]:= A2[I,J] / C
"END"
"END"
"ELSE" NI:= NI - 1
"END"
"END";
EM[7]:= EM[6] - COUNT
"END" EQILBRCOM;
"EOP"
0"CODE" 34362;
"PROCEDURE" BAKLBRCOM(N, N1, N2, D, INT, VR, VI);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, VR, VI;
"INTEGER" "ARRAY" INT;
"BEGIN"
BAKLBR(N, N1, N2, D, INT, VR);
BAKLBR(N, N1, N2, D, INT, VI)
"END" BAKLBRCOM;
"EOP"
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 1
AUTHORS: T.J.DEKKER, W.HOFFMANN.
CONTRIBUTORS: W.HOFFMANN, J.G.VERWER.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 730705.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS FIVE PROCEDURES.
A) TFMSYMTRI2 AND TFMSYMTRI1 TRANSFORM A REAL SYMMETRIC MATRIX INTO
A SIMILAR TRIDIAGONAL ONE BY MEANS OF HOUSEHOLDER'S TRANSFORMATION,
B) BAKSYMTRI2 AND BAKSYMTRI1 PERFORM THE CORRESPONDING BACK
TRANSFORMATION AND FINALLY,
C) TFMPREVEC (WHICH IS TO BE USED IN COMBINATION WITH TFMSYMTRI2)
CALCULATES THE TRANSFORMING MATRIX.
TFMSYMTRI2 AND BAKSYMTRI2 USE THE UPPER TRIANGLE OF A TWO-
DIMENSIONAL ARRAY FOR THE UPPER TRIANGLE OF THE GIVEN SYMMETRIC
MATRIX (TFMSYMTRI2) OR FOR THE DATA FOR HOUSEHOLDER'S BACK
TRANSFORMATION (BAKSYMTRI2). THE OTHER ELEMENTS ARE NEITHER USED
NOR CHANGED. TFMSYMTRI1 AND
BAKSYMTRI1 USE AN ARRAY A[1:(N+1)*N//2] FOR THE GIVEN
SYMMETRIC MATRIX (TFMSYMTRI1) OR FOR THE DATA FOR HOUSEHOLDER'S
TRANSFORMATION (BAKSYMTRI1).
KEYWORDS:
HOUSEHOLDER'S TRANSFORMATION,
TRIANGULARIZATION.
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 2
SUBSECTION: TFMSYMTRI2.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" TFMSYMTRI2(A, N, D, B, BB, EM); "VALUE" N;
"INTEGER" N; "ARRAY" A, D, B, BB, EM; "CODE" 34140;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE
GIVEN IN THE UPPER TRIANGULAR PART OF A (THE
ELEMENTS A[I,J],I<=J);
EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION IS
DELIVERED IN THE UPPER TRIANGULAR PART OF A. THE
ELEMENTS A[I,J], I>J ARE NEITHER USED NOR CHANGED;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL
MATRIX T (SAY), PRODUCED BY HOUSEHOLDER'S
TRANSFORMATION;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
EXIT: THE CODIAGONAL ELEMENTS OF T ARE DELIVERED IN B[1]
THROUGH B[N-1]; B[N] IS SET EQUAL TO ZERO;
BB: <ARRAY IDENTIFIER>;
"ARRAY"BB[1:N];
EXIT: THE SQUARES OF THE CODIAGONAL ELEMENTS OF T ARE
DELIVERED IN BB[1] THROUGH BB[N-1]; BB[N] IS SET
EQUAL TO ZERO;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0], THE MACHINE PRECISION;
EXIT: EM[1], THE INFINITY NORM OF THE ORIGINAL MATRIX.
PROCEDURES USED:
TAMVEC = CP34012,
MATMAT = CP34013,
TAMMAT = CP34014,
ELMVECCOL= CP34021,
ELMCOLVEC= CP34022,
ELMCOL = CP34023.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 3
METHOD AND PERFORMANCE:
A GIVEN SYMMETRIC MATRIX M IS TRANSFORMED INTO A TRIDIAGONAL ONE
BY MEANS OF N-1 ORTHOGONAL SIMILARITY TRANSFORMATIONS; THE P-TH
TRANSFORMATION IS CHOSEN IN SUCH A WAY THAT IN THE (N-P+1)-TH
COLUMN AND ROW OF M THE DESIRED ZEROES ARE INTRODUCED. HOWEVER,
IF, IN THIS COLUMN AND ROW, ALL ELEMENTS OUTSIDE THE MAIN DIAGONAL
AND THE ADJACENT CODIAGONALS ARE SMALLER IN ABSOLUTE VALUE THAN
THE INFINITY NORM OF M TIMES THE MACHINE PRECISION, THEN THE P-TH
TRANSFORMATION IS SKIPPED.
FOR FURTHER DETAILS SEE REF[1] AND REF[2].
SUBSECTION: BAKSYMTRI2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" BAKSYMTRI2(A, N, N1, N2, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC; "CODE" 34141;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE DATA FOR THE BACK TRANSFORMATION, AS PRODUCED
BY TFMSYMTRI2, MUST BE GIVEN IN THE UPPER
TRIANGULAR PART OF A;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
N1,N2: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND, RESPECTIVELY, OF THE COLUMN NUMBERS
OF VEC;
VEC: <ARRAY IDENTIFIER>;
"ARRAY"VEC[1:N,N1:N2];
ENTRY: THE VECTORS ON WHICH THE BACK TRANSFORMATION HAS TO
BE PERFORMED;
EXIT: THE TRANSFORMED VECTORS.
PROCEDURES USED:
TAMMAT = CP34014,
ELMCOL = CP34023.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED TIMES (N2-N1+1).
METHOD AND PERFORMANCE: SEE REF[1].
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 4
SUBSECTION: TFMPREVEC.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" TFMPREVEC(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"CODE" 34142;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE DATA FOR THE BACK TRANSFORMATION, AS PRODUCED
BY TFMSYMTRI2, MUST BE GIVEN IN THE UPPER
TRIANGULAR PART OF A;
EXIT: THE MATRIX WHICH TRANSFORMS THE ORIGINAL MATRIX
INTO A SIMILAR TRIDIAGONAL ONE.
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
PROCEDURES USED:
TAMMAT = CP34014,
ELMCOL = CP34023.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
METHOD AND PERFORMANCE: SEE REF[1].
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 5
SUBSECTION: TFMSYMTRI1.
CALLING SEQUENCE:
THE HEADING OF THIS PROCEDURE IS:
"PROCEDURE" TFMSYMTRI1(A, N, D, B, BB, EM); "VALUE" N;
"INTEGER" N; "ARRAY" A, D, B, BB, EM; "CODE" 34143;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:(N+1)*N//2];
ENTRY: THE UPPER TRIANGLE OF THE GIVEN MATRIX MUST BE
GIVEN IN SUCH A WAY THAT THE (I,J)-TH ELEMENT OF
THE MATRIX IS A[(J-1)*J//2+I], 1<=I<=J<=N;
EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION AS
USED BY BAKSYMTRI1;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL
MATRIX T (SAY), PRODUCED BY HOUSEHOLDER'S
TRANSFORMATION;
B: <ARRAY IDENTIFIER>;
"ARRAY" B[1:N];
EXIT: THE CODIAGONAL ELEMENTS OF T ARE DELIVERED IN B[1]
THROUGH B[N-1]; B[N] IS SET EQUAL TO ZERO;
BB: <ARRAY IDENTIFIER>;
"ARRAY"BB[1:N];
EXIT: THE SQUARES OF THE CODIAGONAL ELEMENTS OF T ARE
DELIVERED IN BB[1] THROUGH BB[N-1]; BB[N] IS SET
EQUAL TO ZERO;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0], THE MACHINE PRECISION;
EXIT: EM[1], THE INFINITY NORM OF THE ORIGINAL MATRIX.
PROCEDURES USED:
VECVEC = CP34010,
SEQVEC = CP34016,
ELMVEC = CP34020.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
METHOD AND PERFORMANCE: SEE TFMSYMTRI2 (THIS SECTION).
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 6
SUBSECTION: BAKSYMTRI1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" BAKSYMTRI1(A, N, N1, N2, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC; "CODE" 34144;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:(N+1)*N//2];
ENTRY: THE DATA FOR THE BACK TRANSFORMATION, AS PRODUCED
BY TFMSYMTRI1;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
N1,N2: <ARITHMETIC EXPRESSION>;
LOWER AND UPPER BOUND, RESPECTIVELY, OF THE COLUMN NUMBERS
OF VEC;
VEC: <ARRAY IDENTIFIER>;
"ARRAY"VEC[1:N,N1:N2];
ENTRY: THE VECTORS ON WHICH THE BACK TRANSFORMATION HAS TO
BE PERFORMED;
EXIT: THE TRANSFORMED VECTORS.
PROCEDURES USED:
VECVEC = CP34010,
ELMVEC = CP34020.
REQUIRED CENTRAL MEMORY:
AN AUXILIARY ONE-DIMENSIONAL REAL ARRAY OF LENGTH N IS USED.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED TIMES (N2-N1+1).
METHOD AND PERFORMANCE: SEE REF[1].
REFERENCES:
[1] DEKKER, T.J. AND HOFFMANN, W.
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2,
MATHEMATICAL CENTRE TRACTS 23,
MATHEMATISCH CENTRUM, AMSTERDAM, 1968;
[2] WILKINSON, J.H.
THE ALGEBRAIC EIGENVALUE PROBLEM,
CLARENDON PRESS, OXFORD 1965.
EXAMPLE OF USE:
THE FIVE PROCEDURES OF THIS SECTION ARE USED IN
SECTION 3.3.1.1.2:
EIGSYM2 USES TFMSYMTRI2 AND BAKSYMTRI2;
EIGSYM1 USES TFMSYMTRI1 AND BAKSYMTRI1;
QRISYM USES TFMSYMTRI2 AND TFMPREVEC.
1SECTION:3.2.1.2.1.1 (JUNE 1974) PAGE 7
SOURCE TEXT(S):
0"CODE" 34140;
"COMMENT" MCA 2300;
"PROCEDURE" TFMSYMTRI2(A, N, D, B, BB, EM); "VALUE" N;"INTEGER" N;
"ARRAY" A, B, BB, D, EM;
"BEGIN" "INTEGER" I, J, R, R1;
"REAL" W, X, A1, B0, BB0, D0, MACHTOL, NORM;
NORM:= 0;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" J "DO" W:= ABS(A[I,J]) + W;
"FOR" I:= J + 1 "STEP" 1 "UNTIL" N "DO" W:= ABS(A[J,I]) +
W; "IF" W > NORM "THEN" NORM:= W
"END";
MACHTOL:= EM[0] * NORM; EM[1]:= NORM; R:= N;
"FOR" R1:= N - 1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" D[R]:= A[R,R]; X:= TAMMAT(1, R - 2, R, R, A, A);
A1:= A[R1,R]; "IF" SQRT(X) <= MACHTOL "THEN"
"BEGIN" B0:= B[R1]:= A1; BB[R1]:= B0 * B0;A[R,R]:= 1 "END"
"ELSE"
"BEGIN" BB0:= BB[R1]:= A1 * A1 + X;
B0:= "IF" A1 > 0 "THEN" -SQRT(BB0) "ELSE" SQRT(BB0);
A1:= A[R1,R]:= A1 - B0; W:= A[R,R]:= 1 / (A1 * B0);
"FOR" J:= 1 "STEP" 1 "UNTIL" R1 "DO" B[J]:= (TAMMAT(1,
J, J, R, A, A) + MATMAT(J + 1, R1, J, R, A, A)) * W;
ELMVECCOL(1, R1, R, B, A, TAMVEC(1, R1, R, A, B) *
W * .5); "FOR" J:= 1 "STEP" 1 "UNTIL" R1 "DO"
"BEGIN" ELMCOL(1, J, J, R, A, A, B[J]);
ELMCOLVEC(1, J, J, A, B, A[J,R])
"END"; B[R1]:= B0
"END"; R:= R1
"END";
D[1]:= A[1,1]; A[1,1]:= 1; B[N]:= BB[N]:= 0
"END" TFMSYMTRI2
1SECTION:3.2.1.2.1.1 (JUNE 1974) PAGE 8
;
"EOP"
0"CODE" 34141;
"COMMENT" MCA 2301;
"PROCEDURE" BAKSYMTRI2(A, N, N1, N2, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC;
"BEGIN" "INTEGER" I, J, K; "REAL" W;
"FOR" J:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= A[J,J]; "IF" W < 0 "THEN"
"FOR" K:= N1 "STEP" 1 "UNTIL" N2 "DO"
ELMCOL(1, J - 1, K, J, VEC, A,
TAMMAT(1, J - 1, J, K, A, VEC) * W)
"END"
"END" BAKSYMTRI2;
"EOP"
0"CODE" 34142;
"COMMENT" MCA 2302;
"PROCEDURE" TFMPREVEC(A, N); "VALUE" N; "INTEGER" N; "ARRAY" A;
"BEGIN" "INTEGER" I, J, J1, K; "REAL" AB;
J1:= 1;
"FOR" J:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" J1 - 1 ,
J "STEP" 1 "UNTIL" N "DO" A[I,J1]:= 0;
A[J1,J1]:= 1; AB:= A[J,J];
"IF" AB < 0 "THEN"
"FOR" K:= 1 "STEP" 1 "UNTIL" J1 "DO"
ELMCOL(1, J1, K, J, A, A,
TAMMAT(1, J1, J, K, A, A) * AB); J1:= J
"END";
"FOR" I:= N - 1 "STEP" -1 "UNTIL" 1 "DO"
A[I,N]:= 0; A[N,N]:= 1
"END" TFMPREVEC
1SECTION:3.2.1.2.1.1 (DECEMBER 1979) PAGE 9
;
"EOP"
0"CODE" 34143;
"COMMENT" MCA 2305;
"PROCEDURE" TFMSYMTRI1(A, N, D, B, BB, EM); "VALUE" N;"INTEGER" N;
"ARRAY" A, B, BB, D, EM;
"BEGIN" "INTEGER" I, J, R, R1, P, Q, TI, TJ;
"REAL" S, W, X, A1, B0, BB0, D0, NORM, MACHTOL;
NORM:= 0; TJ:= 0;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" J "DO" W:= ABS(A[I + TJ]) +W;
TJ:= TJ + J; TI:= TJ + J;
"FOR" I:= J + 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= ABS(A[TI]) + W; TI:= TI + I "END";
"IF" W > NORM "THEN" NORM:= W
"END";
MACHTOL:= EM[0] * NORM; EM[1]:= NORM; Q:= (N + 1) * N // 2;
R:= N; "FOR" R1:= N - 1 "STEP" -1 "UNTIL" 1 "DO"
"BEGIN" P:= Q - R; D[R]:= A[Q];
X:= VECVEC(P + 1, Q - 2, 0, A, A);
A1:= A[Q - 1]; "IF" SQRT(X) <= MACHTOL "THEN"
"BEGIN" B0:= B[R1]:= A1; BB[R1]:= B0 * B0; A[Q]:= 1 "END"
"ELSE"
"BEGIN" BB0:= BB[R1]:= A1 * A1 + X;
B0:= "IF" A1 > 0 "THEN" -SQRT(BB0) "ELSE" SQRT(BB0);
A1:= A[Q - 1]:= A1 - B0; W:= A[Q]:= 1 / (A1 * B0);
TJ:= 0; "FOR" J:= 1 "STEP" 1 "UNTIL" R1 "DO"
"BEGIN" TI:= TJ + J; S:= VECVEC(TJ + 1, TI, P - TJ,
A, A); TJ:= TI + J;
B[J]:= (SEQVEC(J + 1, R1, TJ, P, A, A) + S) * W;
TJ:= TI
"END";
ELMVEC(1, R1, P, B, A, VECVEC(1,R1,P,B,A)* W *.5);
TJ:= 0; "FOR" J:= 1 "STEP" 1 "UNTIL" R1 "DO"
"BEGIN" TI:= TJ + J; ELMVEC(TJ + 1, TI, P - TJ, A, A,
B[J]);ELMVEC(TJ + 1, TI, -TJ, A, B, A[J + P]);
TJ:= TI
"END"; B[R1]:= B0
"END";
Q:= P; R:= R1
"END";
D[1]:= A[1]; A[1]:= 1; B[N]:= BB[N]:= 0
"END" TFMSYMTRI1
1SECTION:3.2.1.2.1.1 (JUNE 1974) PAGE 10
;
"EOP"
0"CODE" 34144;
"COMMENT" MCA 2306;
"PROCEDURE" BAKSYMTRI1(A, N, N1, N2, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC;
"BEGIN" "INTEGER" J, J1, K, TI, TJ;
"REAL" W; "ARRAY" AUXVEC[1:N];
"FOR" K:= N1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO"
AUXVEC[J]:= VEC[J,K]; TJ:= J1:= 1;
"FOR" J:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" TI:= TJ + J; W:= A[TI];
"IF" W < 0 "THEN" ELMVEC(1, J1, TJ, AUXVEC,A,VECVEC(1,
J1, TJ, AUXVEC, A) * W); J1:= J; TJ:= TI
"END";
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" VEC[J,K]:= AUXVEC[J]
"END"
"END" BAKSYMTRI1;
"EOP"
1SECTION:3.2.1.2.1.2 (JUNE 1974) PAGE 1
AUTHORS: T.J.DEKKER AND W.HOFFMANN.
CONTRIBUTORS: W.HOFFMANN, J.G.VERWER.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 731112.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THREE PROCEDURES.
A) TFMREAHES TRANSFORMS A MATRIX INTO A SIMILAR UPPER-HESSENBERG
MATRIX BY MEANS OF WILKINSON'S TRANSFORMATION,
B) BAKREAHES1 PERFORMS THE CORRESPONDING BACK TRANSFORMATION ON
A VECTOR AND SHOULD BE CALLED AFTER TFMREAHES,
C) BAKREAHES2 PERFORMS THE CORRESPONDING BACK TRANSFORMATION ON
THE COLUMNS OF A MATRIX AND SHOULD BE CALLED AFTER TFMREAHES.
KEYWORDS:
SIMILARITY TRANSFORMATION,
UPPER-HESSENBERG MATRIX.
1SECTION:3.2.1.2.1.2 (DECEMBER 1975) PAGE 2
SUBSECTION: TFMREAHES.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" TFMREAHES(A, N, EM, INT); "VALUE" N;
"INTEGER" N; "ARRAY" A, EM; "INTEGER" "ARRAY" INT;
"CODE" 34170;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE MATRIX TO BE TRANSFORMED;
EXIT: THE UPPER-HESSENBERG MATRIX IS DELIVERED IN THE
UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF A, THE
(NONTRIVIAL ELEMENTS OF THE) TRANSFORMING MATRIX,
L, IN THE REMAINING PART OF A, I.E. A[I,J] =
L[I,J + 1], FOR I = 3,...,N AND J = 1,...,I - 2;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0], THE MACHINE PRECISION;
EXIT: EM[1], THE INFINITY NORM OF THE ORIGINAL MATRIX;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
EXIT: THE PIVOTAL INDICES DEFINING THE STABILIZING ROW
AND COLUMN INTERCHANGES;
PROCEDURES USED:
MATVEC = CP34011,
MATMAT = CP34013,
ICHCOL = CP34031,
ICHROW = CP34032.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH:
A ONE-DIMENSIONAL REAL ARRAY OF LENGTH N IS DECLARED.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
WILKINSON'S TRANSFORMATION IS A TRIANGULAR SIMILARITY
TRANSFORMATION WITH STABILIZING ROW AND COLUMN INTERCHANGES
TRANSFORMING A MATRIX, M, INTO AN UPPER-HESSENBERG MATRIX, H.
THE TRANSFORMING MATRIX IS THE PRODUCT OF A PERMUTATION MATRIX, P,
AND A UNIT LOWER-TRIANGULAR MATRIX, L. THE NONDIAGONAL ELEMENTS IN
THE FIRST COLUMN OF L ARE 0, AND THE ROW AND COLUMN INTERCHANGES
ARE CHOSEN IN SUCH A WAY THAT THE ABSOLUTE VALUE OF EACH ELEMENT
OF L IS AT MOST 1.
BECAUSE OF THE SPECIAL FORM OF L, THE MATRICES H AND L CAN BE
STORED TOGETHER IN THE ARRAY USED FOR THE MATRIX M (SEE CALLING
SEQUENCE). FOR FURTHER DETAILS SEE REFERENCE [1] AND [2].
1SECTION:3.2.1.2.1.2 (JUNE 1974) PAGE 3
SUBSECTION: BAKREAHES1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" BAKREAHES1(A, N, INT, V); "VALUE" N;
"INTEGER" N; "ARRAY" A, V; "INTEGER" "ARRAY" INT;
"CODE" 34171;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE LENGTH OF THE VECTOR TO BE TRANSFORMED;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE (NONTRIVIAL ELEMENTS OF THE) TRANSFORMING
MATRIX, L, AS PRODUCED BY TFMREAHES MUST BE GIVEN
IN THE PART BELOW THE FIRST SUBDIAGONAL OF A,
I.E. A[I,J] = L[I,J + 1], FOR I = 3,...,N AND
J = 1,...,I - 2;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
ENTRY: PIVOTAL INDICES DEFINING THE STABILIZING ROW AND
COLUMN INTERCHANGES AS PRODUCED BY TFMREAHES;
V: <ARRAY IDENTIFIER>;
"ARRAY"V[1:N];
ENTRY: THE VECTOR TO BE TRANSFORMED;
EXIT: THE TRANSFORMED VECTOR.
PROCEDURES USED:
MATVEC = CP34011.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
THE BACK TRANSFORMATION WHICH CORRESPONDS TO WILKINSON'S
TRANSFORMATION AS PERFORMED BY TFMREAHES TRANSFORMS A VECTOR, X,
INTO THE VECTOR PLX, WHERE PL IS THE TRANSFORMING MATRIX.
1SECTION:3.2.1.2.1.2 (JUNE 1974) PAGE 4
SUBSECTION: BAKREAHES2.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE IS:
"PROCEDURE" BAKREAHES2(A, N, N1, N2, INT, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC; "INTEGER" "ARRAY" INT;
"CODE" 34172;
THE MEANING OF THE FORMAL PARAMETERS IS:
N: <ARITHMETIC EXPRESSION>;
THE LENGTH OF THE VECTORS TO BE TRANSFORMED;
N1, N2: <ARITHMETIC EXPRESSION>;
THE COLUMN NUMBERS OF THE FIRST AND LAST VECTOR TO BE
TRANSFORMED;
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE (NONTRIVIAL ELEMENTS OF THE) TRANSFORMING
MATRIX, L, AS PRODUCED BY TFMREAHES MUST BE GIVEN
IN THE PART BELOW THE FIRST SUBDIAGONAL OF A,
I.E. A[I,J] = L[I,J + 1], FOR I = 3,...,N AND
J = 1,...,I - 2;
INT: <ARRAY IDENTIFIER>;
"INTEGER""ARRAY" INT[1:N];
ENTRY: PIVOTAL INDICES DEFINING THE STABILIZING ROW AND
COLUMN INTERCHANGES AS PRODUCED BY TFMREAHES;
VEC: <ARRAY IDENTIFIER>;
"ARRAY"VEC[1:N,N1:N2];
ENTRY: THE N2 - N1 + 1 VECTORS OF LENGTH N TO BE
TRANSFORMED;
EXIT: THE N2 - N1 + 1 VECTORS OF LENGTH N RESULTING FROM
THE BACK TRANSFORMATION;
PROCEDURES USED:
TAMVEC = CP34012,
ICHROW = CP34032.
REQUIRED CENTRAL MEMORY:
EXECUTION FIELD LENGTH:
IONAL REAL ARRAY OF LENGTH N IS DECLARED.
RUNNING TIME: ROUGHLY PROPORTIONAL TO (N2 - N1 + 1) * N * N.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE:
SEE SUBSECTION BAKREAHES1.
REFERENCES:
[1] DEKKER, T. J. AND HOFFMANN, W,
ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2,
MATHEMATICAL CENTRE TRACTS 23,
MATHEMATISCH CENTRUM, AMSTERDAM, 1968;
[2] J. H. WILKINSON, THE ALGEBRAIC EIGENVALUE PROBLEM,
CLARENDON PRESS, OXFORD, 1965.
1SECTION:3.2.1.2.1.2 (JUNE 1974) PAGE 5
EXAMPLES OF USE:
EXAMPLES OF USE OF TFMREAHES, BAKREAHES1 AND BAKREAHES2 CAN BE
FOUND IN THE PROCEDURES FOR CALCULATING EIGENVALUES AND
EIGENVECTORS AS DESCRIBED IN SECTION 3.3.1.2.2.
SOURCE TEXT(S) :
0"CODE" 34170;
"COMMENT" MCA 2400;
"PROCEDURE" TFMREAHES(A, N, EM, INT); "VALUE" N; "INTEGER" N;
"ARRAY" A, EM; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, J, J1, K, L;
"REAL" S, T, MACHTOL, MACHEPS, NORM;
"ARRAY" B[0:N - 1];
MACHEPS:= EM[0]; NORM:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" S:= 0;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" S:= S + ABS(A[I,J]);
"IF" S > NORM "THEN" NORM:= S
"END";
EM[1]:= NORM; MACHTOL:= NORM * MACHEPS; INT[1]:= 0;
"FOR" J:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" J1:= J - 1; L:= 0; S:= MACHTOL;
"FOR" K:= J + 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" T:= ABS(A[K,J1]); "IF" T > S "THEN"
"BEGIN" L:= K; S:= T "END"
"END";
"IF" L ^= 0 "THEN"
"BEGIN" "IF" ABS(A[J,J1]) < S "THEN"
"BEGIN" ICHROW(1, N, J, L, A);
ICHCOL(1, N, J, L, A)
"END"
"ELSE" L:= J; T:= A[J,J1];
"FOR" K:= J + 1 "STEP" 1 "UNTIL" N "DO"
A[K,J1]:= A[K,J1] / T
"END"
"ELSE"
"FOR" K:= J + 1 "STEP" 1 "UNTIL" N "DO" A[K,J1]:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
B[I - 1]:= A[I,J]:= A[I,J] +
("IF" L = 0 "THEN" 0 "ELSE" MATMAT(J + 1, N, I, J1, A, A))-
MATVEC(1, "IF" J1 < I - 2 "THEN" J1 "ELSE" I - 2, I, A, B);
INT[J]:= L
"END"
"END" TFMREAHES
1SECTION:3.2.1.2.1.2 (JUNE 1974) PAGE 6
;
"EOP"
0"CODE" 34171;
"COMMENT" MCA 2401;
"PROCEDURE" BAKREAHES1(A, N, INT, V); "VALUE" N; "INTEGER" N;
"ARRAY" A, V; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, L;
"REAL" W; "ARRAY" X[1:N];
"FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" X[I - 1]:= V[I];
"FOR" I:= N "STEP" -1 "UNTIL" 2 "DO"
"BEGIN" V[I]:= V[I] + MATVEC(1, I - 2, I, A, X);
L:= INT[I]; "IF" L > I "THEN"
"BEGIN" W:= V[I]; V[I]:= V[L]; V[L]:= W "END"
"END"
"END" BAKREAHES1;
"EOP"
0"CODE" 34172;
"COMMENT" MCA 2402;
"PROCEDURE" BAKREAHES2(A, N, N1, N2, INT, VEC); "VALUE" N, N1, N2;
"INTEGER" N, N1, N2; "ARRAY" A, VEC; "INTEGER" "ARRAY" INT;
"BEGIN" "INTEGER" I, L, K; "ARRAY" U[1:N];
"FOR" I:= N "STEP" -1 "UNTIL" 2 "DO"
"BEGIN" "FOR" K:= I - 2 "STEP" -1 "UNTIL" 1 "DO"
U[K + 1]:= A[I,K];
"FOR" K:= N1 "STEP" 1 "UNTIL" N2 "DO"
VEC[I,K]:= VEC[I,K] + TAMVEC(2 , I - 1, K, VEC, U);
L:= INT[I]; "IF" L > I "THEN" ICHROW(N1, N2, I, L, VEC)
"END"
"END" BAKREAHES2;
"EOP"
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED : 730903.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THREE PROCEDURES;
A) HSHHRMTRI TRANSFORMS THE HERMITIAN MATRIX M INTO A SIMILAR
REAL SYMMETRIC TRIDIAGONAL MATRIX S;
B) BAKHRMTRI PERFORMS A BACK TRANSFORMATION CORRESPONDING TO
HSHHRMTRI;
C) HSHHRMTRIVAL DELIVERS THE MAIN DIAGONAL ELEMENTS AND THE
SQUARES OF THE CODIAGONAL ELEMENTS OF A HERMITIAN TRIDIAGONAL
MATRIX WHICH IS UNITARY SIMILAR WITH A GIVEN HERMITIAN MATRIX.
KEYWORDS :
HERMITIAN MATRIX ,
TRIDIAGONALIZATION ,
COMPLEX HOUSEHOLDER,S TRANSFORMATION .
SUBSECTION : HSHHRMTRI.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" HSHHRMTRI(A, N, D, B, BB, EM, TR, TI); "VALUE" N;
"INTEGER" N; "ARRAY" A, D, B, BB, EM, TR, TI;
"CODE" 34363;
THE MEANING OF THE FORMAL PARAMETERS IS :
A,TR,TI: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
"ARRAY" TR,TI[1:N-1];
ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE
HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER
TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J);
THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE
OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE
STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J);
EXIT: DATA FOR THE BACKTRANSFORMATION;
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 2
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT : THE MAIN DIAGONAL OF THE RESULTING SYMMETRIC
TRIDIAGONAL MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY"B[1:N-1];
EXIT: THE CODIAGONAL ELEMENTS OF THE RESULTING SYMMETRIC
TRIDIAGONAL MATRIX;
BB: <ARRAY IDENTIFIER>;
"ARRAY"BB[1:N-1];
EXIT : THE SQUARES OF THE MODULI OF THE CODIAGONAL
ELEMENTS OF THE RESULTING SYMMETRIC TRIDIAGONAL
MATRIX;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0], THE MACHINE PRECISION;
EXIT: EM[1], AN ESTIMATE FOR A NORM OF THE ORIGINAL
MATRIX.
PROCEDURES USED :
MATVEC = CP34011 ,
TAMVEC = CP34012 ,
MATMAT = CP34013 ,
TAMMAT = CP34014 ,
MATTAM = CP34015 ,
ELMVECCOL = CP34021 ,
ELMCOLVEC = CP34022 ,
ELMCOL = CP34023 ,
ELMROW = CP34024 ,
ELMVECROW = CP34026 ,
ELMROWVEC = CP34027 ,
ELMROWCOL = CP34028 ,
ELMCOLROW = CP34029 ,
CARPOL = CP34344 .
RUNNING TIME : PROPORTIONAL TO N CUBED .
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE HSHHRMTRIVAL (THIS SECTION).
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 3
SUBSECTION : BAKHRMTRI.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" BAKHRMTRI(A, N, N1, N2, VECR, VECI, TR, TI);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2;
"ARRAY" A, VECR, VECI, TR, TI;
"CODE" 34365;
THE MEANING OF THE FORMAL PARAMETERS IS :
A,TR,TI: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
"ARRAY" TR,TI[1:N-1];
ENTRY: THE DATA FOR THE BACKTRANSFORMATION AS PRODUCED
BY HSHHRMTRI;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX OF WHICH THE EIGENVECTORS ARE
CALCULATED;
N1,N2: <ARITHMETIC EXPRESSION>;
THE EIGENVECTORS CORRESPONDING TO THE EIGENVALUES WITH
INDICES N1,...,N2 ARE TO BE TRANSFORMED;
VECR,VECI: <ARRAY IDENTIFIER>;
"ARRAY" VECR,VECI[1:N,N1:N2];
ENTRY:
THE BACK TRANSFORMATION IS PERFORMED ON THE REAL
EIGENVECTORS GIVEN IN THE COLUMNS OF ARRAY VECR;
EXIT:
VECR : REAL PART OF THE TRANSFORMED EIGENVECTORS;
VECI : IMAGINARY PART OF THE TRANSFORMED EIGENVECTORS.
PROCEDURES USED :
MATMAT = CP34013 ,
TAMMAT = CP34014 ,
ELMCOL = CP34023 ,
ELMCOLROW = CP34029 ,
COMMUL = CP34341 ,
COMROWCST = CP34353 .
RUNNING TIME: PROPORTIONAL TO (N2-N1+1)*N**2 .
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE HSHHRMTRIVAL (THIS SECTION).
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 4
SUBSECTION : HSHHRMTRIVAL.
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE READS :
"PROCEDURE" HSHHRMTRIVAL(A, N, D, BB, EM); "VALUE" N; "INTEGER" N;
"ARRAY" A, D, BB, EM;
"CODE" 34364;
THE MEANING OF THE FORMAL PARAMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE
HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER
TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J);
THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE
OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE
STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J);
THE ELEMENTS OF A ARE ALTERED;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT: THE MAIN DIAGONAL OF THE RESULTING HERMITIAN
TRIDIAGONAL MATRIX;
BB: <ARRAY IDENTIFIER>;
"ARRAY"BB[1:N-1];
EXIT: THE SQUARES OF THE MODULI OF THE CODIAGONAL
ELEMENTS OF THE RESULTING HERMITIAN TRIDIAGONAL
MATRIX;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0],THE MACHINE PRECISION;
EXIT: EM[1], AN ESTIMATE FOR A NORM OF THE ORIGINAL
MATRIX;
PROCEDURES USED :
MATVEC = CP34011 ,
TAMVEC = CP34012 ,
MATMAT = CP34013 ,
TAMMAT = CP34014 ,
MATTAM = CP34015 ,
ELMVECCOL = CP34021 ,
ELMCOLVEC = CP34022 ,
ELMCOL = CP34023 ,
ELMROW = CP34024 ,
ELMVECROW = CP34026 ,
ELMROWVEC = CP34027 ,
ELMROWCOL = CP34028 ,
ELMCOLROW = CP34029 .
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 5
RUNNING TIME : PROPORTIONAL TO N CUBED .
LANGUAGE : ALGOL 60.
THE FOLLOWING HOLDS FOR THE THREE PROCEDURES :
METHOD AND PERFORMANCE :
HSHHRMTRIVAL TRANSFORMS A HERMITIAN MATRIX INTO A SIMILAR
HERMITIAN TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S
TRANSFORMATION.
HSHHRMTRI TRANSFORMS A HERMITIAN MATRIX INTO A SIMILAR
REAL TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S
TRANSFORMATION FOLLOWED BY A COMPLEX DIAGONAL UNITARY
SIMILARITY TRANSFORMATION IN ORDER TO MAKE THE RESULTING
TRIDIAGONAL MATRIX REAL SYMMETRIC;
HOUSEHOLDER'S TRANSFORMATION FOR COMPLEX HERMITIAN MATRICES
IS A UNITARY SIMILARITY TRANSFORMATION,TRANSFORMING A HERMITIAN
MATRIX INTO A SIMILAR COMPLEX TRIDIAGONAL ONE (SEE WILKINSON,
1965, P. 342-343). LET M BE A GIVEN HERMITIAN MATRIX OF ORDER
N, WITH REAL PART MR AND IMAGINARY PART MI, P THE
TRANSFORMING MATRIX AND T THE RESULTING HERMITIAN TRIDIAGONAL
MATRIX. SINCE P IS UNITARY, WE HAVE T = P"MP, WHERE
" STANDS FOR CONJUGATING AND TRANSPOSING. THE MATRIX P IS THE
PRODUCT OF N-2 HOUSEHOLDER MATRICES, THESE BEING UNITARY
HERMITIAN MATRICES OF THE FORM I-U"U/T, WHERE T IS A SCALAR
(>0), AND U A COMPLEX VECTOR. THE K-TH HOUSEHOLDER MATRIX,
K=1,...,N-2, IS CHOSEN IN SUCH A WAY THAT THE LAST K ELEMENTS OF U
VANISH, AND THE DESIRED ZEROS ARE INTRODUCED IN THE (N-K+1)-TH
COLUMN AND ROW OF THE MATRIX M. HOWEVER, IF THE EUCLIDIAN NORM
OF THE FIRST N-K-1 ELEMENTS OF COLUMN N-K+1 OF THE MATRIX M IS
SMALLER THAN THE MACHINE PRECISION TIMES THE INFINITY NORM OF THE
MATRIX ( NORM(MR) + NORM(MI) ), THEN THE K-TH TRANSFORMATION
IS SKIPPED (I.E. THE K-TH HOUSEHOLDER MATRIX IS REPLACED BY I).
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 6
THE COMPLEX DIAGONAL SIMILARITY TRANSFORMATION D TRANSFORMS THE
HERMITIAN TRIDIAGONAL MATRIX T INTO A REAL SYMMETRIC TRIDIAGONAL
MATRIX S (MUELLER, 1966). THE DIAGONAL OF D IS CHOSEN IN SUCH
A WAY THAT THE CODIAGONAL ELEMENTS OF T ARE TRANSFORMED INTO
THEIR ABSOLUTE VALUES.
BAKHRMTRI PERFORMS THE BACK TRANSFORMATION TO REPLACE THE
EIGENVECTORS OF THE TRIDIAGONAL SYMMETRIC MATRIX S BY THE
EIGENVECTORS OF THE ORIGINAL HERMITIAN MATRIX M. IF X IS AN
EIGENVECTOR OF S THEN PDX IS THE CORRESPONDING EIGENVECTOR OF
M. STARTING FROM THE VECTOR V=DX, THE VECTOR PDX IS
OBTAINED BY SUCCESSIVELY REPLACING V BY THE K-TH HOUSEHOLDER
MATRIX TIMES V, FOR K=N-2,...,1. THE RESULTING VECTOR V THEN EQUALS
PDX.
REFERENCES :
MUELLER, D.J. (1966),
HOUSEHOLDER,S METHOD FOR COMPLEX MATRICES AND EIGENSYSTEMS OF
HERMITIAN MATRICES,
NUMER.MATH., 8, P.72-92;
WILKINSON, J.H. (1965),
THE ALGEBRAIC EIGENVALUE PROBLEM,
CLARENDON PRESS, OXFORD.
EXAMPLE OF USE :
THE PROCEDURES HSHHRMTRIVAL AND BAKHRMTRI ARE USED IN SECTION
3.3.2.1. :
EIGVALHRM AND QRIVALHRM USE HSHHRMTRIVAL,
EIGHRM AND QRIHRM USE BAKHRMTRI .
AS A FORMAL TEST OF THE PROCEDURE HSHHRMTRI, THE FOLLOWING MATRIX
WAS USED (SEE GREGORY AND KARNEY, CHAPTER 6, EXAMPLE 6.6) :
3 1 0 +2I
1 3 -2I 0
0 +2I 1 1
-2I 0 1 1
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 7
"BEGIN"
"COMMENT" GREGORY AND KARNEY,CHAPTER 6, EXAMPLE 6.6;
"REAL" "ARRAY" A[1:4,1:4],D,B,BB[1:4],TR,TI[1:3],EM[0:1];
"INTEGER" I,J;
"PROCEDURE" OUT(S,A,N);
"VALUE" N;"INTEGER" N;"ARRAY" A;"STRING" S;
"BEGIN" "INTEGER" I,J;
OUTPUT(61,"("10S")",S);
"FOR" I:=1 "STEP" 1 "UNTIL" N "DO"
OUTPUT(61,"("+D.3DBB")",A[I]);
OUTPUT(61,"("/")")
"END" OUT;
INIMAT(1,4,1,4,A,0);
A[1,1]:=A[2,2]:=3;
A[1,2]:=A[3,3]:=A[3,4]:=A[4,4]:=1;
A[3,2]:=2;A[4,1]:=-2;
EM[0]:="-14;
OUTPUT(61,"(""("INITIAL MATRIX GIVEN IN ARRAY A[1:4,1:4]:")",/")");
"FOR" I:=1 "STEP" 1 "UNTIL" 4 "DO"
"BEGIN" "FOR" J:=1 "STEP" 1 "UNTIL" 4 "DO"
OUTPUT(61,"("-DBBB")",A[I,J]);
OUTPUT(61,"("/")")
"END";
OUTPUT(61,"("/,"("HSHHRMTRI DELIVERS:")",//")");
HSHHRMTRI(A,4,D,B,BB,EM,TR,TI);
OUT("("D[1:4]: ")",D,4);
OUT("("B[1:3]: ")",B,3);
OUT("("BB[1:3]: ")",BB,3);
OUT("("EM[1]: ")",EM,1);
"END"
OUTPUT :
INITIAL MATRIX GIVEN IN ARRAY A[1:4,1:4]:
3 1 0 0
0 3 0 0
0 2 1 1
-2 0 0 1
HSHHRMTRI DELIVERS:
D[1:4]: +3.000 +1.400 +2.600 +1.000
B[1:3]: +2.236 +0.800 +2.236
BB[1:3]: +5.000 +0.640 +5.000
EM[1]: +6.000
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 8
SOURCE TEXT(S) :
0"CODE" 34363;
"PROCEDURE" HSHHRMTRI(A, N, D, B, BB, EM, TR, TI); "VALUE" N;
"INTEGER" N; "ARRAY" A, D, B, BB, EM, TR, TI;
"BEGIN" "INTEGER" I, J, J1, JM1, R, RM1;
"REAL" NRM, W, TOL2, X, AR, AI, MOD, C, S, H, K, T, Q,
AJR, ARJ, BJ, BBJ;
NRM:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= ABS(A[I,I]);
"FOR" J:= I - 1 "STEP" - 1 "UNTIL" 1, I + 1 "STEP" 1
"UNTIL" N "DO" W:= W + ABS(A[I,J]) + ABS(A[J,I]);
"IF" W > NRM "THEN" NRM:= W
"END" I;
TOL2:= (EM[0] * NRM) ** 2; EM[1]:= NRM; R:= N;
"FOR" RM1:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" X:= TAMMAT(1, R - 2, R, R, A, A) + MATTAM(1, R -
2, R, R, A, A); AR:= A[RM1,R]; AI:= - A[R,RM1];
D[R]:= A[R,R]; CARPOL(AR, AI, MOD, C, S);
"IF" X < TOL2 "THEN"
"BEGIN" A[R,R]:= - 1; B[RM1]:= MOD;
BB[RM1]:= MOD * MOD
"END"
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 9
"ELSE"
"BEGIN" H:= MOD * MOD + X; K:= SQRT(H);
T:= A[R,R]:= H + MOD * K;
"IF" AR = 0 "AND" AI = 0 "THEN" A[RM1,R]:= K "ELSE"
"BEGIN" A[RM1,R]:= AR + C * K;
A[R,RM1]:= - AI - S * K; S:= - S
"END";
C:= - C; J:= 1; JM1:= 0;
"FOR" J1:= 2 "STEP" 1 "UNTIL" R "DO"
"BEGIN" B[J]:= (TAMMAT(1, J, J, R, A, A) +
MATMAT(J1, RM1, J, R, A, A) + MATTAM(1,
JM1, J, R, A, A) - MATMAT(J1, RM1, R, J,
A, A)) / T;
BB[J]:= (MATMAT(1, JM1, J, R, A, A) -
TAMMAT(J1, RM1, J, R, A, A) - MATMAT(1, J,
R, J, A, A) - MATTAM(J1, RM1, J, R, A, A))
/ T; JM1:= J; J:= J1
"END" J1;
Q:= (TAMVEC(1, RM1, R, A, B) - MATVEC(1, RM1,
R, A, BB)) / T / 2;
ELMVECCOL(1, RM1, R, B, A, - Q);
ELMVECROW(1, RM1, R, BB, A, Q); J:= 1;
"FOR" J1:= 2 "STEP" 1 "UNTIL" R "DO"
"BEGIN" AJR:= A[J,R]; ARJ:= A[R,J]; BJ:= B[J];
BBJ:= BB[J];
ELMROWVEC(J, RM1, J, A, B, - AJR);
ELMROWVEC(J, RM1, J, A, BB, ARJ);
ELMROWCOL(J, RM1, J, R, A, A, - BJ);
ELMROW(J, RM1, J, R, A, A, BBJ);
ELMCOLVEC(J1, RM1, J, A, B, - ARJ);
ELMCOLVEC(J1, RM1, J, A, BB, - AJR);
ELMCOL(J1, RM1, J, R, A, A, BBJ);
ELMCOLROW(J1, RM1, J, R, A, A, BJ); J:= J1;
"END" J1;
BB[RM1]:= H; B[RM1]:= K;
"END";
TR[RM1]:= C; TI[RM1]:= S; R:= RM1;
"END" RM1;
D[1]:= A[1,1];
"END" HSHHRMTRI
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 10
;
"EOP"
0"CODE" 34365;
"PROCEDURE" BAKHRMTRI(A, N, N1, N2, VECR, VECI, TR, TI);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2;
"ARRAY" A, VECR, VECI, TR, TI;
"BEGIN" "INTEGER" I, J, R, RM1;
"REAL" C, S, T, QR, QI;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO" VECI[I,J]:= 0; C:= 1;
S:= 0;
"FOR" J:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" COMMUL(C, S, TR[J], TI[J], C, S);
COMROWCST(N1, N2, J, VECR, VECI, C, S)
"END" J;
RM1:= 2;
"FOR" R:= 3 "STEP" 1 "UNTIL" N "DO"
"BEGIN" T:= A[R,R]; "IF" T > 0 "THEN"
"FOR" J:= N1 "STEP" 1 "UNTIL" N2 "DO"
"BEGIN" QR:= (TAMMAT(1, RM1, R, J, A, VECR) -
MATMAT(1, RM1, R, J, A, VECI)) / T;
QI:= (TAMMAT(1, RM1, R, J, A, VECI) +
MATMAT(1, RM1, R, J, A, VECR)) / T;
ELMCOL(1, RM1, J, R, VECR, A, - QR);
ELMCOLROW(1, RM1, J, R, VECR, A, - QI);
ELMCOLROW(1, RM1, J, R, VECI, A, QR);
ELMCOL(1, RM1, J, R, VECI, A, - QI)
"END";
RM1:= R;
"END" R
"END" BAKHRMTRI
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 11
;
"EOP"
0"CODE" 34364;
"PROCEDURE" HSHHRMTRIVAL(A, N, D, BB, EM); "VALUE" N; "INTEGER" N;
"ARRAY" A, D, BB, EM;
"BEGIN" "INTEGER" I, J, J1, JM1, R, RM1;
"REAL" NRM, W, TOL2, X, AR, AI, H, T, Q, AJR, ARJ, DJ,
BBJ, MOD2;
NRM:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" W:= ABS(A[I,I]);
"FOR" J:= I - 1 "STEP" - 1 "UNTIL" 1, I + 1 "STEP" 1
"UNTIL" N "DO" W:= W + ABS(A[I,J]) + ABS(A[J,I]);
"IF" W > NRM "THEN" NRM:= W
"END" I;
TOL2:= (EM[0] * NRM) ** 2; EM[1]:= NRM; R:= N;
"FOR" RM1:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" X:= TAMMAT(1, R - 2, R, R, A, A) + MATTAM(1, R -
2, R, R, A, A); AR:= A[RM1,R]; AI:= - A[R,RM1];
D[R]:= A[R,R];
"IF" X < TOL2 "THEN" BB[RM1]:= AR * AR + AI * AI "ELSE"
"BEGIN" MOD2:= AR * AR + AI * AI; "IF" MOD2 = 0 "THEN"
"BEGIN" A[RM1,R]:= SQRT(X); T:= X "END"
"ELSE"
"BEGIN" X:= X + MOD2; H:= SQRT(MOD2 * X);
T:= X + H; H:= 1 + X / H;
A[R,RM1]:= - AI * H; A[RM1,R]:= AR * H;
"END";
"COMMENT"
1SECTION:3.2.1.2.2.1 (JUNE 1974) PAGE 12
;
J:= 1; JM1:= 0;
"FOR" J1:= 2 "STEP" 1 "UNTIL" R "DO"
"BEGIN" D[J]:= (TAMMAT(1, J, J, R, A, A) +
MATMAT(J1, RM1, J, R, A, A) + MATTAM(1,
JM1, J, R, A, A) - MATMAT(J1, RM1, R, J,
A, A)) / T;
BB[J]:= (MATMAT(1, JM1, J, R, A, A) -
TAMMAT(J1, RM1, J, R, A, A) - MATMAT(1, J,
R, J, A, A) - MATTAM(J1, RM1, J, R, A, A))
/ T; JM1:= J; J:= J1
"END" J1;
Q:= (TAMVEC(1, RM1, R, A, D) - MATVEC(1, RM1,
R, A, BB)) / T / 2;
ELMVECCOL(1, RM1, R, D, A, - Q);
ELMVECROW(1, RM1, R, BB, A, Q); J:= 1;
"FOR" J1:= 2 "STEP" 1 "UNTIL" R "DO"
"BEGIN" AJR:= A[J,R]; ARJ:= A[R,J]; DJ:= D[J];
BBJ:= BB[J];
ELMROWVEC(J, RM1, J, A, D, - AJR);
ELMROWVEC(J, RM1, J, A, BB, ARJ);
ELMROWCOL(J, RM1, J, R, A, A, - DJ);
ELMROW(J, RM1, J, R, A, A, BBJ);
ELMCOLVEC(J1, RM1, J, A, D, - ARJ);
ELMCOLVEC(J1, RM1, J, A, BB, - AJR);
ELMCOL(J1, RM1, J, R, A, A, BBJ);
ELMCOLROW(J1, RM1, J, R, A, A, DJ); J:= J1;
"END" J1;
BB[RM1]:= X;
"END";
R:= RM1;
"END" RM1;
D[1]:= A[1,1];
"END" HSHHRMTRIVAL;
"EOP"
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 1
AUTHOR : C.G. VAN DER LAAN.
CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN.
INSTITUTE : MATHEMATICAL CENTRE.
RECEIVED: 731016.
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THE PROCEDURES HSHCOMHES AND BAKCOMHES.
HSHCOMHES TRANSFORMS A COMPLEX MATRIX BY MEANS OF HOUSEHOLDER'S
TRANSFORMATION FOLLOWED BY A COMPLEX DIAGONAL TRANSFORMATION INTO
A SIMILAR UNITARY UPPER-HESSENBERG MATRIX WITH A REAL NONNEGATIVE
SUBDIAGONAL.
BAKCOMHES PERFORMS THE CORRESPONDING BACK TRANSFORMATION.
KEYWORDS:
COMPLEX EIGENPROBLEM,
REDUCTION HESSENBERG FORM,
HOUSEHOLDER'S TRANSFORMATION.
SUBSECTION: HSHCOMHES.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" HSHCOMHES(AR, AI, N, EM, B, TR, TI, DEL); "VALUE" N;
"INTEGER" N; "ARRAY" AR, AI, EM, B, TR, TI, DEL;
"CODE" 34366;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI: <ARRAY IDENTIFIER>;
"ARRAY" AR,AI[1:N,1:N];
ENTRY:
THE REAL PART AND THE IMAGINARY PART OF THE MATRIX TO BE
TRANSFORMED MUST BE GIVEN IN THE ARRAYS AR AND AI,
RESPECTIVELY;
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE UPPER
TRIANGLE OF THE RESULTING UPPER-HESSENBERG MATRIX ARE
DELIVERED IN THE CORRESPONDING PARTS OF THE ARRAYS AR
AND AI, RESPECTIVELY; DATA FOR THE HOUSEHOLDER BACK-
TRANSFORMATION ARE DELIVERED IN THE STRICT LOWER
TRIANGLES OF THE ARRAYS AR AND AI;
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 2
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE GIVEN MATRIX;
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY:
EM[0]: THE MACHINE PRECISION;
EM[1]: AN ESTIMATE OF THE NORM OF THE COMPLEX MATRIX;
(OR, E.G. THE SUM OF THE INFINITY NORMS OF THE REAL
(PART AND IMAGINARY PART OF THE MATRIX);
B: <ARRAY IDENTIFIER>;
"ARRAY"B[1:N-1];
EXIT:
THE REAL NONNEGATIVE SUBDIAGONAL OF THE RESULTING
UPPER-HESSENBERG MATRIX;
TR,TI: <ARRAY IDENTIFIER>;
"ARRAY" TR,TI[1:N];
EXIT:
THE REAL PART AND THE IMAGINARY PART OF THE DIAGONAL
ELEMENTS OF A DIAGONAL SIMILARITY TRANSFORMATION ARE
DELIVERED IN THE ARRAYS TR AND TI, RESPECTIVELY; BY THIS
INFORMATION THE COMPLEX UPPER-HESSENBERG MATRIX IS
TRANSFORMED INTO A UPPER-HESSENBERG MATRIX WITH A REAL
SUBDIAGONAL;
DEL: <ARRAY IDENTIFIER>;
"ARRAY"DEL[1:N-2];
EXIT:
INFORMATION CONCERNING THE SEQUENCE OF HOUSEHOLDER
MATRICES.
PROCEDURES USED:
HSHCOMCOL = CP34355,
MATMAT = CP34013,
ELMROWCOL = CP34028,
HSHCOMPRD = CP34356,
CARPOL = CP34344,
COMMUL = CP34341,
COMCOLCST = CP34352,
COMROWCST = CP34353.
RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED.
LANGUAGE: ALGOL 60.
METHOD AND PERFORMANCE: SEE BAKCOMHES (THIS SECTION).
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 3
SUBSECTION: BAKCOMHES.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" BAKCOMHES(AR, AI, TR, TI, DEL, VR, VI, N, N1, N2);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2;
"ARRAY" AR, AI, TR, TI, DEL, VR, VI;
"CODE" 34367;
THE MEANING OF THE FORMAL PARAMETERS IS:
AR,AI,TR,TI,DEL:
<ARRAY IDENTIFIER>;
"ARRAY" AR,AI[1:N,1:N];
"ARRAY" TR,TI[1:N];
"ARRAY"DEL[1:N-2];
ENTRY: THE DATA FOR THE BACKTRANSFORMATION AS PRODUCED
BY HSHCOMHES;
VR,VI: <ARRAY IDENTIFIER>;
"ARRAY" VR,VI[1:N,N1:N2];
ENTRY:
THE BACK TRANSFORMATION IS PERFORMED ON THE EIGENVECTORS
WITH THE REAL PARTS GIVEN IN ARRAY VR AND THE IMAGINARY
PARTS GIVEN IN ARRAY VI;
EXIT:
THE REAL PARTS AND IMAGINARY PARTS OF THE RESULTING
EIGENVECTORS ARE DELIVERED IN THE COLUMNS OF THE ARRAYS
VR AND VI, RESPECTIVELY;
N: <ARITHMETIC EXPRESSION>;
THE ORDER OF THE MATRIX OF WHICH THE EIGENVECTORS ARE
CALCULATED;
N1,N2: <ARITHMETIC EXPRESSION>;
THE EIGENVECTORS CORRESPONDING TO THE EIGENVALUES WITH
INDICES N1,...,N2 ARE TO BE TRANSFORMED;
PROCEDURES USED:
COMROWCST = CP34353,
HSHCOMPRD = CP34356.
RUNNING TIME: PROPORTIONAL TO (N2-N1) * N**2.
LANGUAGE : ALGOL 60.
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 4
THE FOLLOWING HOLDS FOR BOTH PROCEDURES:
METHOD AND PERFORMANCE:
HSHCOMHES:
HOUSEHOLDER'S TRANSFORMATION (FOR COMPLEX MATRICES) IS A UNITARY
SIMILARITY TRANSFORMATION, WHICH TRANSFORMS A COMPLEX MATRIX INTO A
SIMILAR UPPER-HESSENBERG MATRIX (SEE WILKINSON, 1965, P. 347-349).
LET M BE A GIVEN COMPLEX MATRIX OF ORDER N, P THE TRANSFORMING
MATRIX AND H THE RESULTING UPPER-HESSENBERG MATRIX. SINCE P IS
UNITARY, WE THEN HAVE H = P''MP, WHERE '' STANDS FOR CONJUGATING
AND TRANSPOSING. THE MATRIX P IS THE PRODUCT OF N-2 HOUSEHOLDER
MATRICES, THESE BEING UNITARY HERMITEAN MATRICES OF
THE FORM I - UU''/T, WHERE T IS A SCALAR (>0), AND U A COMPLEX
VECTOR. THE R-TH HOUSEHOLDER MATRIX, R=1,...,N-2, IS CHOSEN IN SUCH
A WAY THAT THE FIRST R ELEMENTS OF U VANISH, AND THE DESIRED ZEROS
ARE INTRODUCED IN THE LAST N-R-1 ELEMENTS OF THE R-TH COLUMN OF THE
MATRIX M. HOWEVER, IF THE EUCLIDEAN NORM OF THE LAST N-R-1 ELEMENTS
OF COLUMN R OF THE MATRIX M IS SMALLER THAN THE MACHINE PRECISION
TIMES A NORM OF THE MATRIX THEN THE R-TH TRANSFORMATION IS SKIPPED
(I.E. THE R-TH HOUSEHOLDER MATRIX IS REPLACED BY I). THE COMPLEX
DIAGONAL SIMILARITY TRANSFORMATION D TRANSFORMS THE UPPER-
HESSENBERG MATRIX H INTO AN UPPER-HESSENBERG MATRIX HR, WITH REAL
NONNEGATIVE ELEMENTS. THE DIAGONAL OF D IS CHOSEN IN SUCH A WAY
THAT SUBDIAGONAL ELEMENTS OF H ARE TRANSFORMED INTO THEIR ABSOLUTE
VALUES (SEE MUELLER, 1966).
BAKCOMHES:
THE BACK TRANSFORMATION TRANSFORMS A COMPLEX VECTOR X INTO THE
COMPLEX VECTOR PDX. IF X IS AN EIGENVECTOR OF H THEN PDX IS THE
CORRESPONDING EIGENVECTOR OF M. STARTING FROM THE VECTOR V=DX, THE
VECTOR PDX IS OBTAINED BY SUCCESSIVELY REPLACING V BY THE R-TH
HOUSEHOLDER MATRIX TIMES V, FOR R=N-2,...,1. THE RESULTING VECTOR
THEN EQUALS PDX.
REFERENCES:
MUELLER, D.J. (1966),
HOUSEHOLDER'S METHOD FOR COMPLEX MATRICES AND EIGENSYSTEMS OF
HERMITIAN MATRICES,
NUMER.MATH., 8, P.72-92;
WILKINSON, J.H. (1965),
THE ALGEBRAIC EIGENVALUE PROBLEM,
CLARENDON PRESS, OXFORD;
EXAMPLE OF USE:
HSHCOMHES IS USED IN THE PROCEDURES EIGVALCOM AND EIGCOM.
BAKCOMHES IS USED IN THE PROCEDURE EIGCOM.
(SEE SECTION 3.3.2.2.2.).
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 5
SOURCE TEXT(S) :
0"CODE" 34366;
"PROCEDURE" HSHCOMHES(AR, AI, N, EM, B, TR, TI, DEL); "VALUE" N;
"INTEGER" N; "ARRAY" AR, AI, EM, B, TR, TI, DEL;
"BEGIN" "INTEGER" R, RM1, I, J, NM1;
"REAL" TOL, T, XR, XI;
NM1:= N - 1; TOL:= (EM[0] * EM[1]) ** 2; RM1:= 1;
"FOR" R:= 2 "STEP" 1 "UNTIL" NM1 "DO"
"BEGIN" "IF" HSHCOMCOL(R, N, RM1, AR, AI, TOL, B[RM1],
TR[R], TI[R], T) "THEN"
"BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" XR:= (MATMAT(R, N, I, RM1, AI, AI) -
MATMAT(R, N, I, RM1, AR, AR)) / T;
XI:= ( - MATMAT(R, N, I, RM1, AR, AI) -
MATMAT(R, N, I, RM1, AI, AR)) / T;
ELMROWCOL(R, N, I, RM1, AR, AR, XR);
ELMROWCOL(R, N, I, RM1, AR, AI, XI);
ELMROWCOL(R, N, I, RM1, AI, AR, XI);
ELMROWCOL(R, N, I, RM1, AI, AI, - XR)
"END";
HSHCOMPRD(R, N, R, N, RM1, AR, AI, AR, AI, T);
"END";
DEL[RM1]:= T; RM1:= R
"END" FORR;
"IF" N > 1 "THEN" CARPOL(AR[N,NM1], AI[N,NM1], B[NM1],
TR[N], TI[N]); RM1:= 1; TR[1]:= 1; TI[1]:= 0;
"FOR" R:= 2 "STEP" 1 "UNTIL" N "DO"
"BEGIN" COMMUL(TR[RM1], TI[RM1], TR[R], TI[R], TR[R],
TI[R]); COMCOLCST(1, RM1, R, AR, AI, TR[R], TI[R]);
COMROWCST(R + 1, N, R, AR, AI, TR[R], - TI[R]);
RM1:= R
"END";
"END" HSHCOMHES
1SECTION:3.2.1.2.2.2 (JUNE 1974) PAGE 6
;
"EOP"
0"CODE" 34367;
"PROCEDURE" BAKCOMHES(AR, AI, TR, TI, DEL, VR, VI, N, N1, N2);
"VALUE" N, N1, N2; "INTEGER" N, N1, N2;
"ARRAY" AR, AI, TR, TI, DEL, VR, VI;
"BEGIN" "INTEGER" I, R, RM1;
"REAL" H;
"FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" COMROWCST(N1, N2, I, VR,
VI, TR[I], TI[I]); R:= N - 1;
"FOR" RM1:= N - 2 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" H:= DEL[RM1];
"IF" H > 0 "THEN" HSHCOMPRD(R, N, N1, N2, RM1, VR, VI,
AR, AI, H); R:= RM1
"END"
"END" BAKCOMHES;
"EOP"
1SECTION:3.2.2.1.1 (DECEMBER 1975) PAGE 1
AUTHOR : D.T.WINTER
INSTITUTE : MATHEMATICAL CENTRE
RECEIVED : 731217
BRIEF DESCRIPTION :
THIS SECTION CONTAINS THREE PROCEDURES :
1. HSHREABID.
THIS PROCEDURE TRANSFORMS A GIVEN MATRIX TO BIDIAGONAL FORM,
BY PREMULTIPLYING AND POSTMULTIPLYING THE GIVEN MATRIX WITH
ORTHOGONAL MATRICES.
2. PSTTFMMAT.
THIS PROCEDURE CALCULATES THE POSTMULTIPLYING MATRIX FROM THE
DATA GENERATED BY HSHREABID.
3. PRETFMMAT.
THIS PROCEDURE CALCULATES THE PREMULTIPLYING MATRIX FROM THE
DATA GENERATED BY HSHREABID.
KEYWORDS :
HOUSEHOLDER'S TRANSFORMATION
BIDIAGONALISATION
SUBSECTION : HSHREABID
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" HSHREABID(A, M, N, D, B, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, D, B, EM;
"CODE" 34260;
THE MEANING OF THE FORMAL PARAMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY" A[1:M,1:N];
ENTRY: THE GIVEN MATRIX;
EXIT: DATA CONCERNING THE PREMULTIPLYING AND POSTMULTIPLYING
MATRICES;
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF THE GIVEN MATRIX;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF THE GIVEN MATRIX,
N SHOULD SATISFY N <= M;
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
EXIT: THE DIAGONAL OF THE BIDIAGONAL MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY"B[1:N];
EXIT: THE SUPERDIAGONAL OF THE BIDIAGONAL MATRIX IS DELIVERED
IN B[1:N-1];
EM: <ARRAY IDENTIFIER>;
"ARRAY"EM[0:1];
ENTRY: EM[0]: THE MACHINE-PRECISION;
EXIT: EM[1]: THE INFINITY NORM OF THE GIVEN MATRIX.
1SECTION:3.2.2.1.1 (JUNE 1974) PAGE 2
PROCEDURES USED :
TAMMAT = CP34014
MATTAM = CP34015
ELMCOL = CP34023
ELMROW = CP34024
RUNNING TIME :
RUNNING TIME IS ROUGHLY PROPORTIONAL TO (M + N) * N * N
METHOD AND PERFORMANCE :
LET US ASSUME A GIVEN MATRIX A[ 1:M , 1:N ], WITH M >= N.
FIRSTLY WE PREMULTIPLY A WITH A HOUSEHOLDER MATRIX, CHOSEN IN SUCH
A WAY THAT THE FIRST COLUMN OF THE RESULTING MATRIX A' IS ZERO WITH
THE EXCEPTION OF THE FIRST ELEMENT. SECONDLY WE POSTMULTIPLY A'
WITH A HOUSEHOLDER MATRIX SO THAT THE FIRST ROW OF THE RESULTING
MATRIX IS ZERO WITH THE EXCEPTION OF THE FIRST TWO ELEMENTS.
NOW WE REMOVE THE FIRST ROW AND COLUMN, AND REPEAT THIS PROCESS
UNTIL THE MATRIX IS TOTALLY TRANSFORMED TO BIDIAGONAL FORM.
THIS PROCEDURE IS A REWRITING OF A PART OF THE PROCEDURE SVD
PUBLISHED BY G.H.GOLUB AND C.REINSCH[1]. HOWEVER IN CONTRAST TO
THEIR PROCEDURE, HERE WE SKIP A TRANSFORMATION IF THE COLUMN OR ROW
ON WHICH OUR ATTENTION IS FOCUSSED IS ALREADY (NEARLY) IN THE
DESIRED FORM, I.E. IF THE SUM OF THE SQUARES OF THE ELEMENTS THAT
OUGHT TO BE ZERO IS SMALLER THAN A CERTAIN CONSTANT, IN SVD THE
TRANSFORMATION IS SKIPPED ONLY IF THE NORM OF THE FULL ROW OR
COLUMN IS SMALL ENOUGH. OUR WAY SEEMS TO GIVE BETTER RESULTS, AS
SOME ILL-DEFINED TRANSFORMATIONS ARE SKIPPED. MOREOVER, IF A
TRANSFORMATION IS SKIPPED, WE DO NOT STORE A ZERO IN THE
DIAGONAL OR SUPERDIAGONAL, BUT WE STORE THE VALUE THAT WOULD
HAVE BEEN FOUND IF THE COLUMN OR ROW WAS IN THE DESIRED FORM
ALREADY.
LANGUAGE : ALGOL-60
1SECTION:3.2.2.1.1 (DECEMBER 1975) PAGE 3
SUBSECTION : PSTTFMMAT
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" PSTTFMMAT(A, N, V, B);
"VALUE" N; "INTEGER" N; "ARRAY" A, V, B;
"CODE" 34261;
THE MEANING OF THE FORMAL PARAMETERS IS:
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:N,1:N];
THE DATA CONCERNING THE POSTMULTIPLYING MATRIX, AS GENERATED
BY HSHREABID;
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS AND ROWS OF A;
V: <ARRAY IDENTIFIER>;
"ARRAY"V[1:N,1:N];
EXIT: THE POSTMULTIPLYING MATRIX;
B: <ARRAY IDENTIFIER>;
"ARRAY"B[1:N];
THE SUPERDIAGONAL AS GENERATED BY HSHREABID.
PROCEDURES USED :
MATMAT = CP34013
ELMCOL = CP34023
RUNNING TIME :
THE RUNNING TIME IS ABOUT PROPORTIONAL TO N ** 3
LANGUAGE : ALGOL 60
1SECTION:3.2.2.1.1 (JUNE 1974) PAGE 4
SUBSECTION : PRETFMMAT
CALLING SEQUENCE :
THE HEADING OF THE PROCEDURE IS :
"PROCEDURE" PRETFMMAT(A, M, N, D);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, D;
"CODE" 34262;
THE MEANING OF THE FORMAL PARAMETERS IS :
A: <ARRAY IDENTIFIER>;
"ARRAY"A[1:M,1:N];
ENTRY: THE DATA CONCERNING THE PREMULTIPLYING MATRIX AS
GENERATED BY HSHREABID
EXIT : THE PREMULTIPLYING MATRIX.
M: <ARITHMETIC EXPRESSION>;
THE NUMBER OF ROWS OF A.
N: <ARITHMETIC EXPRESSION>;
THE NUMBER OF COLUMNS OF A, N SHOULD SATISFY N <= M.
D: <ARRAY IDENTIFIER>;
"ARRAY"D[1:N];
THE DIAGONAL AS GENERATED BY HSHREABID.
PROCEDURES USED :
TAMMAT = CP34014
ELMCOL = CP34023
RUNNING TIME :
THE RUNNING TIME IS ABOUT PROPORTIONAL TO M * N * N
LANGUAGE : ALGOL-60
REFERENCES :
[1] WILKINSON, J.H. AND C.REINSCH
HANDBOOK FOR AUTOMATIC COMPUTATION, VOL. 2
LINEAR ALGEBRA
HEIDELBERG (1971)
EXAMPLE OF USE :
FOR AN EXAMPLE OF USE ONE IS REFERRED TO SECTION 3.5.1.2
1SECTION:3.2.2.1.1 (JUNE 1974) PAGE 5
SOURCE TEXT(S) :
0"CODE" 34260;
"PROCEDURE" HSHREABID(A, M, N, D, B, EM);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, D, B, EM;
"BEGIN" "INTEGER" I, J, I1;
"REAL" NORM, MACHTOL, W, S, F, G, H;
NORM:= 0;
"FOR" I:= 1 "STEP" 1 "UNTIL" M "DO"
"BEGIN" W:= 0;
"FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" W:= ABS(A[I,J]) + W;
"IF" W > NORM "THEN" NORM:= W
"END";
MACHTOL:= EM[0] * NORM; EM[1]:= NORM;
"FOR" I:= 1 "STEP" 1 "UNTIL" N "DO"
"BEGIN" I1:= I + 1; S:= TAMMAT(I1, M, I, I, A, A);
"IF" S < MACHTOL "THEN" D[I]:= A[I,I] "ELSE"
"BEGIN" F:= A[I,I]; S:= F * F + S;
D[I]:= G:= "IF" F < 0 "THEN" SQRT(S) "ELSE" - SQRT(S);
H:= F * G - S; A[I,I]:= F - G;
"FOR" J:= I1 "STEP" 1 "UNTIL" N "DO"
ELMCOL(I, M, J, I, A, A, TAMMAT(I, M, I, J, A, A) / H)
"END";
"IF" I < N "THEN"
"BEGIN" S:= MATTAM(I1 + 1, N, I, I, A, A);
"IF" S < MACHTOL "THEN" B[I]:= A[I,I1] "ELSE"
"BEGIN" F:= A[I,I1]; S:= F * F + S;
B[I]:= G:= "IF" F < 0 "THEN" SQRT(S) "ELSE" - SQRT(S);
H:= F * G - S; A[I,I1]:= F - G;
"FOR" J:= I1 "STEP" 1 "UNTIL" M "DO"
ELMROW(I1, N, J, I, A, A, MATTAM(I1, N, I, J, A, A) /
H)
"END"
"END"
"END"
"END" HSHREABID
1SECTION:3.2.2.1.1 (JUNE 1974) PAGE 6
;
"EOP"
0"CODE" 34261;
"PROCEDURE" PSTTFMMAT(A, N, V, B);
"VALUE" N; "INTEGER" N; "ARRAY" A, V, B;
"BEGIN" "INTEGER" I, I1, J;
"REAL" H;
I1:= N; V[N,N]:= 1;
"FOR" I:= N - 1 "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" H:= B[I] * A[I,I1]; "IF" H < 0 "THEN"
"BEGIN" "FOR" J:= I1 "STEP" 1 "UNTIL" N "DO" V[J,I]:= A[I,J] /
H;
"FOR" J:= I1 "STEP" 1 "UNTIL" N "DO"
ELMCOL(I1, N, J, I, V, V, MATMAT(I1, N, I, J, A, V))
"END";
"FOR" J:= I1 "STEP" 1 "UNTIL" N "DO" V[I,J]:= V[J,I]:= 0;
V[I,I]:= 1; I1:= I
"END"
"END" PSTTFMMAT;
"EOP"
0"CODE" 34262;
"PROCEDURE" PRETFMMAT(A, M, N, D);
"VALUE" M, N; "INTEGER" M, N; "ARRAY" A, D;
"BEGIN" "INTEGER" I, I1, J;
"REAL" G, H;
"FOR" I:= N "STEP" - 1 "UNTIL" 1 "DO"
"BEGIN" I1:= I + 1; G:= D[I]; H:= G * A[I,I];
"FOR" J:= I1 "STEP" 1 "UNTIL" N "DO" A[I,J]:= 0;
"IF" H < 0 "THEN"
"BEGIN" "FOR" J:= I1 "STEP" 1 "UNTIL" N "DO"
ELMCOL(I, M, J, I, A, A, TAMMAT(I1, M, I, J, A, A) / H);
"FOR" J:= I "STEP" 1 "UNTIL" M "DO" A[J,I]:= A[J,I] / G
"END"
"ELSE"
"FOR" J:= I "STEP" 1 "UNTIL" M "DO" A[J,I]:= 0;
A[I,I]:= A[I,I] + 1
"END"
"END" PRETFMMAT;
"EOP"