NUMAL Section 3.1.1.1.1.1.3
BEGIN SECTION : 3.1.1.1.1.1.3 (May, 1974)
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.
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).
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.
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
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).
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.
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
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.
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"
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;
"CODE" 34301;
"CODE" 34061;
"CODE" 34232;
"CODE" 34243;