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

Calling Sequence

CALL DPPRFS	(UPLO, N, NRHS, DA, DAF, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SPPRFS	(UPLO, N, NRHS, SA, SAF, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZPPRFS	(UPLO, N, NRHS, ZA, ZAF, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CPPRFS	(UPLO, N, NRHS, CA, CAF, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dpprfs	(char uplo, int n, int nrhs, double *da, double *daf, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void spprfs	(char uplo, int n, int nrhs, float *sa, float *saf, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zpprfs	(char uplo, int n, int nrhs, doublecomplex *za, doublecomplex *zaf, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void cpprfs	(char uplo, int n, int nrhs, complex *ca, complex *caf, 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 of the matrices B and X. NRHS 0.
xA	Upper or lower triangle of the matrix A. The dimension of xA is (N × N + N) / 2.
xAF	Cholesky factorization of A as computed by xPPTRF. The dimension of xAF is (N × N + N) / 2.
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 * (N + 1)) / 2)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = LDB)

PARAMETER (NRHS = 1)

C

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

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

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

INTEGER INFO, IWORK(LDIWRK)

C

EXTERNAL DCOPY, DPPRFS, DPPTRF, DPPTRS, DSPMV

INTRINSIC ABS

C

C Initialize the array A to store in symmetric form the

C coefficient matrix A shown below. Initialize the array

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

C

C 10**5 10**-5 10**-6 10**-7 3 * 10**4

C A = 10**-5 10**5 10**-6 10**-7 b = 3 * 10**3

C 10**-6 10**-6 10**5 10**-7 3 * 10**2

C 10**-7 10**-7 10**-7 10**5 3 * 10**1

C

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

$ 1.0D-7, 1.0D5, 1.0D-7, 1.0D5 /

DATA B / 3.0D4, 3.0D3, 3.0D2, 3.0D1 /

C

C Print the initial values of the arrays.

C

PRINT 1000

PRINT 1010, A(1), A(2), A(3), A(4)

PRINT 1010, A(2), A(5), A(6), A(7)

PRINT 1010, A(3), A(6), A(8), A(9)

PRINT 1010, A(4), A(7), A(9), A(10)

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

PRINT 1020

PRINT 1030, B

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

C

C Cholesky factor A.

C

CALL DPPTRF ('LOWER TRIANGLE OF A STORED', N, AF, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

C

C Solve Ax=b and print the solution.

C

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

$ X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, ABS(INFO)

STOP 2

END IF

PRINT 1060

PRINT 1030, X

CALL DSPMV ('LOWER TRIANGLE OF A STORED', N, 1.0D0, A, X, 1,

$ 0.0D0, B1, 1)

PRINT 1070

PRINT 1030, B1

C

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

C

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

$ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1080, ABS(INFO)

STOP 3

END IF

PRINT 1090

PRINT 1030, X

CALL DSPMV ('LOWER TRIANGLE OF A STORED', N, 1.0D0, A, X, 1,

$ 0.0D0, B1, 1)

PRINT 1100

PRINT 1030, B1

C

1000 FORMAT (1X, 'A:')

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

1020 FORMAT (/1X, 'b:')

1030 FORMAT (3X, E25.17)

1040 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1050 FORMAT (1X, 'Illegal argument to DPPTRS, argument #', I2)

1060 FORMAT (/1X, 'Initial solution to Ax=b:')

1070 FORMAT (/1X, 'Ax with the initial x:')

1080 FORMAT (1X, 'Illegal argument to DPPRFS, INFO = ', I2)

1090 FORMAT (/1X, 'Refined solution to Ax=b:')

1100 FORMAT (/1X, 'Ax with refined x:')

C

END

Sample Output

A:

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

b:

0.30000000000000000E+05

0.30000000000000000E+04

0.30000000000000000E+03

0.30000000000000000E+02

Initial solution to Ax=b:

0.29999999999696958E+00

0.29999999969969694E-01

0.29999999966996991E-02

0.29999999966700002E-03

Ax with the initial x:

0.29999999999999985E+05

0.29999999999999995E+04

0.29999999999999989E+03

0.30000000000000004E+02

Refined solution to Ax=b:

0.29999999999696969E+00

0.29999999969969698E-01

0.29999999966996999E-02

0.29999999966700002E-03

Ax with refined x:

0.29999999999999996E+05

0.30000000000000000E+04

0.30000000000000000E+03

0.30000000000000004E+02