Refined Solution to a Linear System in a Cholesky-Factored Symmetric Positive Definite Matrix

Calling Sequence

CALL DPORFS	(UPLO, N, NRHS, DA, LDA, DAF, LDAF, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SPORFS	(UPLO, N, NRHS, SA, LDA, SAF, LDAF, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZPORFS	(UPLO, N, NRHS, ZA, LDA, ZAF, LDAF, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CPORFS	(UPLO, N, NRHS, CA, LDA, CAF, LDAF, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dporfs	(char uplo, int n, int nrhs, double *da, int lda, double *daf, int ldaf, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void sporfs	(char uplo, int n, int nrhs, float *sa, int lda, float *saf, int ldaf, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zporfs	(char uplo, int n, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void cporfs	(char uplo, int n, int nrhs, complex *ca, int lda, complex *caf, int ldaf, 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	Cholesky factorization of the matrix A as computed by xPOTRF.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF max(1, N).
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 for real subroutines or 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.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

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

INTEGER N, NRHS

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (LDAF = N)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = LDB)

PARAMETER (NRHS = 1)

C

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

DOUBLE PRECISION B1(LDB,NRHS), BERR(NRHS), FERR(NRHS)

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

INTEGER ICOL, INFO, IROW, IWORK(LDIWRK)

C

EXTERNAL DCOPY, DPORFS, DPOTRF, DPOTRS, DSYMV

INTRINSIC ABS

C

C Initialize the array A to store in symmetric form the

C coefficient matrix A shown below. Initialize the array B

C to store the right hand side vector b shown below.

C

C 10**5 10**-5 10**-6 10**-7 10**4

C A = 10**-5 10**5 10**-6 10**-7 b = 10**4

C 10**-6 10**-6 10**5 10**-7 10**4

C 10**-7 10**-7 10**-7 10**5 10**4

C

DATA A / 1.0D5, 1.0D-5, 1.0D-6, 1.0D-7,

$ 8D180, 1.0D5, 1.0D-6, 1.0D-7,

$ 8D180, 8D180, 1.0D5, 1.0D-7,

$ 8D180, 8D180, 8D180, 1.0D5 /

DATA B / 1.0D4, 1.0D4, 1.0D4, 1.0D4 /

C

C Print the initial value of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

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

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

100 CONTINUE

PRINT 1020

DO 110, IROW = 1, LDA

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

110 CONTINUE

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

PRINT 1030

PRINT 1040, B

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

C

C Cholesky factor A.

C

CALL DPOTRF ('LOWER TRIANGLE OF A STORED', N, AF, LDAF,

$ INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 1

END IF

C

C Solve Ax=b and print the solution.

C

CALL DPOTRS ('LOWER TRIANGLE OF A STORED', N, NRHS, AF,

$ LDAF, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, ABS(INFO)

STOP 3

END IF

PRINT 1070

PRINT 1040, X

CALL DSYMV ('LOWER TRIANGLE OF A STORED', N, 1.0D0, A, LDA,

$ X, 1, 0.0D0, B1, 1)

PRINT 1080

PRINT 1040, B1

C

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

C

CALL DPORFS ('LOWER TRIANGLE OF A STORED', N, NRHS, A, LDA,

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

$ IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1090, ABS(INFO)

STOP 4

END IF

PRINT 1100

PRINT 1040, X

CALL DSYMV ('LOWER TRIANGLE OF A STORED', N, 1.0D0, A, LDA,

$ X, 1, 0.0D0, B1, 1)

PRINT 1110

PRINT 1040, B1

C

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

1010 FORMAT (4(3X, F14.7))

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

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

1040 FORMAT (3X, E25.17)

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

1060 FORMAT (1X, 'Illegal argument to DPOTRS, argument #', I2)

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

1080 FORMAT (/1X, 'Ax with the initial x:')

1090 FORMAT (1X, 'Illegal argument to DPORFS, INFO = ', I2)

1100 FORMAT (/1X, 'Refined solution to Ax=b:')

1110 FORMAT (/1X, 'Ax with refined x:')

C

END

Sample Output

A in full form:

100000.0000000 0.0000100 0.0000010 0.0000001

0.0000100 100000.0000000 0.0000010 0.0000001

0.0000010 0.0000010 100000.0000000 0.0000001

0.0000001 0.0000001 0.0000001 100000.0000000

A in symmetric form: (* in unused elements)

100000.0000000 ************** ************** **************

0.0000100 100000.0000000 ************** **************

0.0000010 0.0000010 100000.0000000 **************

0.0000001 0.0000001 0.0000001 100000.0000000

b:

0.10000000000000000E+05

0.10000000000000000E+05

0.10000000000000000E+05

0.10000000000000000E+05

Initial solution to Ax=b:

0.99999999988899982E-01

0.99999999988899982E-01

0.99999999997899963E-01

0.99999999999699954E-01

Ax with the initial x:

0.99999999999999982E+04

0.99999999999999982E+04

0.99999999999999964E+04

0.99999999999999964E+04

Refined solution to Ax=b:

0.99999999988899996E-01

0.99999999988899996E-01

0.99999999997899991E-01

0.99999999999699968E-01

Ax with refined x:

0.10000000000000000E+05

0.10000000000000000E+05

0.10000000000000000E+05

0.99999999999999982E+04