Refined Solution to a Linear System in a UDU- or LDL-Factored Symmetric Matrix

Calling Sequence

CALL DSYRFS	(UPLO, N, NRHS, DA, LDA, DAF, LDAF, IPIVOT, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, DWORK2, INFO)
CALL SSYRFS	(UPLO, N, NRHS, SA, LDA, SAF, LDAF, IPIVOT, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, SWORK2, INFO)
CALL ZSYRFS	(UPLO, N, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CSYRFS	(UPLO, N, NRHS, CA, LDA, CAF, LDAF, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dsyrfs	(char uplo, int n, int nrhs, double *da, int lda, double *daf, int ldaf, int *ipivot, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void ssyrfs	(char uplo, int n, int nrhs, float *sa, int lda, float *saf, int ldaf, int *ipivot, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zsyrfs	(char uplo, int n, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, int *ipivot, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void csyrfs	(char uplo, int n, int nrhs, complex *ca, int lda, complex *caf, int ldaf, int *ipivot, complex *cb, int ldb, complex *cx, int ldx, float *sferr, float *sberr, int *info)

Arguments

UPLO	Indicates whether xA contains the upper or lower triangle of the matrix. The legal values for UPLO are listed below. Any values not listed below are illegal.
	'U' or 'u'	xA contains the upper triangle.
	'L' or 'l'	xA contains the lower triangle.
N	Order of the matrix A. N 0.
NRHS	Number of right-hand sides, equal to the number of columns of the matrices B and X. NRHS 0.
xA	Upper or lower triangle of the matrix A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
xAF	UDU or LDL factorization of the matrix A as computed by xSYTRF.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF max(1, N).
IPIVOT	Pivot indices as computed by xSYTRF.
xB	The N×NRHS right-hand side matrix B.
LDB	Leading dimension of the array B as specified in a dimension or type statement. LDB max(1, N).
xX	On entry, the N×NRHS solution matrix X. On exit, the refined N×NRHS solution matrix X.
LDX	Leading dimension of the array X as specified in a dimension or type statement. LDX max(1, N).
xFERR	On exit, the estimated forward error bound for each solution vector X(*, j) for 1 j NRHS. If X' is the true solution corresponding to X(*, j) then FERR(j) is an upper bound on the magnitude of the largest element in X(*, j) - X' divided by the magnitude of the largest element in X(*, j).
xBERR	On exit, BERR(j) is the smallest relative change in any element of A or B(*, j) that makes X(*, j) an exact solution to AX(*, j) = B(*, j) for 1 j NRHS.
xWORK	Scratch array with a dimension of 3 × N.
xWORK2	Scratch array with a dimension of N.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDAF, LDB, LDWORK, LDX, N, NRHS

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (LDAF = N)

PARAMETER (LDB = N)

PARAMETER (LDWORK = 1)

PARAMETER (LDX = LDB)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION A(LDA,N), AF(LDAF,N), B(LDB,NRHS)

DOUBLE PRECISION BERR(NRHS), EPSLON, FERR(NRHS)

DOUBLE PRECISION WORK(LDWORK), X(LDX,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW, IWORK(N)

C

EXTERNAL DCOPY, DSYRFS, DSYTRF, DSYTRS

INTRINSIC ABS, MAX

C

C Initialize the array A to store the coefficient matrix A

C shown below. Initialize the array B to store the right

C hand side vector b shown below.

C

C 0 0 30

C A = 0 2 2 b = 50

C 2 2 0 70

C 0 0 110

C

DATA A / 0.0D0, 8.0D8, 8.0D8, 8.0D8, 0.0D0, 2.0D0, 8.0D8,

$ 8.0D8, 0.0D0, 2.0D0, 2.0D0, 8.0D8, 0.0D0, 0.0D0,

$ 0.0D0, 0.0D0 /

DATA B / 3.0D0, 5.0D0, 7.0D0, 1.1D1 /

C

C Slightly perturb each of the elements on the diagonal, the

C first subdiagonal, and the first superdiagonal of A. After

C this code, A will resemble the matrix shown below.

C

C 2e -e

C A = -e 2+e 2-e

C 2-e 2+e -e

C -e 2e

C

EPSLON = ABS ((((2.0D0 / 3.0D0) + 2.0D0) - 2.0D0) -

$ (2.0D0 / 3.0D0))

A(1,1) = EPSLON + EPSLON

A(N,N) = EPSLON + EPSLON

DO 100, IROW = 2, N - 1

A(IROW,IROW) = A(IROW,IROW) + EPSLON

100 CONTINUE

DO 110, IROW = 1, N - 1

A(IROW,IROW + 1) = A(IROW,IROW + 1) - EPSLON

110 CONTINUE

C

C Slightly perturb each element of B. After this code, B will

C resemble the matrix shown below.

C

C 3-small

C B = 5-small

C 7-small

C 11-small

C

DO 130, ICOL = 1, NRHS

DO 120, IROW = 1, N

B(IROW,ICOL) = B(IROW,ICOL) * (1.0D0 - EPSLON)

120 CONTINUE

130 CONTINUE

C

C Print the initial values of the arrays.

C

PRINT 1000

DO 140, IROW = 1, N

PRINT 1010, (A(ICOL,IROW), ICOL = 1, IROW),

$ (A(IROW,ICOL), ICOL = IROW + 1, N)

140 CONTINUE

PRINT 1020

DO 150, IROW = 1, N

PRINT 1010, (A(IROW,ICOL), ICOL = 1, N)

150 CONTINUE

PRINT 1030

PRINT 1040, B

C

C Make copies to use with DSYRFS.

C

CALL DCOPY (LDA * N, A, 1, AF, 1)

CALL DCOPY (LDB * NRHS, B, 1, X, 1)

C

C LU factor A.

C

CALL DSYTRF ('UPPER TRIANGLE OF A STORED', N, AF, LDAF,

$ IPIVOT, WORK, LDWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 1

END IF

C

C Solve Ax=b and print the solution.

C

CALL DSYTRS ('UPPER TRIANGLE OF A STORED', N, NRHS, AF, LDA,

$ IPIVOT, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, INFO

STOP 2

END IF

PRINT 1070

PRINT 1080, X

PRINT 1090

PRINT 1040, A(1,1) * X(1,1) + A(1,2) * X(2,1)

PRINT 1040, A(1,2) * X(1,1) + A(2,2) * X(2,1) + A(2,3) *

$ X(3,1)

PRINT 1040, A(2,3) * X(2,1) + A(3,3) * X(3,1) + A(3,4) *

$ X(4,1)

PRINT 1040, A(3,4) * X(3,1) + A(4,4) *

$ X(4,1)

C

C Refine the solution to Ax=b and print the refined solution.

C

CALL DSYRFS ('UPPER TRIANGLE OF A STORED', N, NRHS, A, LDA,

$ AF, LDAF, IPIVOT, B, LDB, X, LDX, FERR, BERR,

$ WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1100, INFO

STOP 3

END IF

PRINT 1110

PRINT 1080, X

PRINT 1120

PRINT 1040, A(1,1) * X(1,1) + A(1,2) * X(2,1)

PRINT 1040, A(1,2) * X(1,1) + A(2,2) * X(2,1) + A(2,3) *

$ X(3,1)

PRINT 1040, A(2,3) * X(2,1) + A(3,3) * X(3,1) + A(3,4) *

$ X(4,1)

PRINT 1040, A(3,4) * X(3,1) + A(4,4) *

$ X(4,1)

PRINT 1130, FERR(1)

PRINT 1140, BERR(1)

C

1000 FORMAT (1X, 'A in full form:')

1010 FORMAT (4(2X, F18.16))

1020 FORMAT (/1X, 'A in symmetric form: (* in unused elements)')

1030 FORMAT (/1X, 'b:')

1040 FORMAT (1X, F22.17)

1050 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1060 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

1070 FORMAT (/1X, 'Initial solution to Ax=b:')

1080 FORMAT (1X, E25.17)

1090 FORMAT (/1X, 'Ax with the initial x:')

1100 FORMAT (1X, 'Illegal argument to DSYRFS, INFO = ', I3)

1110 FORMAT (/1X, 'Refined solution to Ax=b:')

1120 FORMAT (/1X, 'Ax with refined x:')

1130 FORMAT (/1X, 'Forward error: ', E14.8)

1140 FORMAT (1X, 'Backward error: ', E14.8)

C

END

Sample Output

A in full form:
0.0000000000000002 -.0000000000000001 0.0000000000000000 0.0000000000000000
-.0000000000000001 2.0000000000000000 2.0000000000000000 0.0000000000000000
0.0000000000000000 2.0000000000000000 2.0000000000000000 -.0000000000000001
0.0000000000000000 0.0000000000000000 -.0000000000000001 0.0000000000000002
A in symmetric form: (* in unused elements)
0.0000000000000002 -.0000000000000001 0.0000000000000000 0.0000000000000000
****************** 2.0000000000000000 2.0000000000000000 0.0000000000000000
****************** ****************** 2.0000000000000000 -.0000000000000001
****************** ****************** ****************** 0.0000000000000002
b:
2.99999999999999956
4.99999999999999911
6.99999999999999911
10.99999999999999822
Initial solution to Ax=b:
0.67553994410557432E+17
0.10808639105689189E+18
-0.10808639105689189E+18
-0.45035996273704960E+16
Ax with the initial x:
3.00000000000000000
0.00000000000000000
0.50000000000000000
10.99999999999999822
Refined solution to Ax=b:
0.67553994410557432E+17
0.10808639105689189E+18
-0.10808639105689189E+18
-0.45035996273704960E+16
Ax with refined x:
3.00000000000000000
0.00000000000000000
0.50000000000000000
10.99999999999999822
Forward error: 0.73614937E-01
Backward error: 0.11564823E-17