Solution to a Linear System in a Symmetric Matrix (Expert Driver)

Calling Sequence

CALL DSYSVX	(FACT, UPLO, N, NRHS, DA, LDA, DAF, LDAF, IPIVOT, DB, LDB, DX, LDX, DRCOND, DFERR, DBERR, DWORK, LDWORK, IWORK2, INFO)
CALL SSYSVX	(FACT, UPLO, N, NRHS, SA, LDA, SAF, LDAF, IPIVOT, SB, LDB, SX, LDX, SRCOND, SFERR, SBERR, SWORK, LDWORK, IWORK2, INFO)
CALL ZSYSVX	(FACT, UPLO, N, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, ZB, LDB, ZX, LDX, DRCOND, DFERR, DBERR, ZWORK, LDWORK, DWORK2, INFO)
CALL CSYSVX	(FACT, UPLO, N, NRHS, CA, LDA, CAF, LDAF, IPIVOT, CB, LDB, CX, LDX, SRCOND, SFERR, SBERR, CWORK, LDWORK, SWORK2, INFO)
void dsysvx	(char fact, 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 *drcond, double *dferr, double *dberr, int *info)
void ssysvx	(char fact, 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 *srcond, float *sferr, float *sberr, int *info)
void zsysvx	(char fact, 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 *drcond, double *dferr, double *dberr, int *info)
void csysvx	(char fact, 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 *srcond, float *sferr, float *sberr, int *info)

Arguments

FACT	Indicates whether or not the factored form of A has been supplied on entry.
	'F' or 'f'	On entry, AF and IPIVOT contain the factored form of A. AF and IPIVOT will not be modified.
	'N' or 'n'	On entry, the matrix A will be copied to AF and factored.
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	On entry, if FACT = 'F' or 'f' then AF contains the UDU or LDL factorization of A, as computed by xSYTRF. Otherwise the entry value of AF is not used. On exit, if FACT = 'N' or 'n' then AF contains the UDU or LDL factorization of A, as computed by xSYTRF. Otherwise AF is unchanged.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF max(1, N).
IPIVOT	On entry, if FACT = 'F' or 'f' then IPIVOT contains pivot indices as computed by xSYTRF. Otherwise the entry values of IPIVOT are not used. On exit, if FACT = 'N' or 'n' then IPIVOT contains the pivot indices as computed by xSYTRF. Otherwise IPIVOT is unchanged.
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 exit, the N×NRHS solution matrix X.
LDX	Leading dimension of the array X as specified in a dimension or type statement. LDX max(1, N).
xRCOND	On exit, an estimate of the reciprocal condition number of the matrix A, where the reciprocal condition number is defined to be 1 / (||A|| × ||A-1||). The reciprocal of the condition number is estimated instead of the condition number itself to avoid overflow or division by zero. If RCOND is less than machine precision (in particular, if RCOND = 0) then A is singular to working precision. In this case, INFO > 0 is returned and the solution and error bounds are not computed.
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 LDWORK.
LDWORK	Leading dimension of the array WORK as specified in a dimension or type statement. LDWORK 3 × N for real subroutines or LDWORK 2 × N for complex subroutines.
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.
	1 INFO N	D(i,i), where i = INFO, is exactly zero and D is therefore singular. The factorization has has been completed, but the solution and error bounds could not be computed.
	INFO = N+1	The block diagonal matrix D is nonsingular, but RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, so the solution and error bounds could not be computed.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDAF, LDB, LDIWRK, LDWORK, LDX

INTEGER N, NRHS

PARAMETER (N = 3)

PARAMETER (NRHS = 1)

PARAMETER (LDA = N)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = LDA)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = N)

C

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

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

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

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

C

EXTERNAL DSYSVX

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 vector b shown below.

C

C 1 2 2 15

C A = 2 1 2 b = 15

C 2 2 1 15

C

DATA A / 1.0D0, 8D8, 8D8, 2.0D0, 1.0D0, 8D8, 2.0D0, 2.0D0,

$ 1.0D0 /

DATA B / 1.5D1, 1.5D1, 1.5D1 /

C

C Print the initial values of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

PRINT 1010, (0.0D0, ICOL = 1, IROW - 1),

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

100 CONTINUE

PRINT 1020

DO 110, IROW = 1, LDA

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

110 CONTINUE

PRINT 1030

PRINT 1040, B

C

C Solve the system Ax=b, then print the result.

C

CALL DSYSVX ('NOT FACTORED A', 'UPPER TRIANGLE OF A STORED',

$ N, NRHS, A, LDA, AF, LDAF, IPIVOT, B, LDB,

$ X, LDX, RCOND, FERR, BERR, WORK, LDWORK,

$ IWORK, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1050

PRINT 1040, X

PRINT 1060, (1.0D0 / RCOND)

PRINT 1070, (IROW, BERR(IROW), IROW = 1, NRHS)

PRINT 1080, (IROW, FERR(IROW), IROW = 1, NRHS)

ELSE

PRINT 1090, INFO

END IF

C

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

1010 FORMAT (3(3X, F4.1))

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

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

1040 FORMAT (1X, 2X, F4.1)

1050 FORMAT (/1X, 'x:')

1060 FORMAT (/1X, 'Estimated condition number: ', F6.3)

1070 FORMAT (/1X, 'Backward error for system #', I1, ': ', E12.6)

1080 FORMAT (1X, 'Forward error for system #', I1, ': ', E12.6)

1090 FORMAT (1X, 'Solve failed with INFO = ', I6)

C

END

Sample Output

A in full form:
1.0 2.0 2.0
0.0 1.0 2.0
0.0 0.0 1.0
A in symmetric form: (* in unused elements)
1.0 2.0 2.0
**** 1.0 2.0
**** **** 1.0
b:
15.0
15.0
15.0
x:
3.0
3.0
3.0
Estimated condition number: 7.000
Backward error for system #1: 0.592119E-16
Forward error for system #1: 0.645410E-14