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

Calling Sequence

CALL DPBRFS	(UPLO, N, NDIAG, NRHS, DA, LDA, DAF, LDAF, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SPBRFS	(UPLO, N, NDIAG, NRHS, SA, LDA, SAF, LDAF, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZPBRFS	(UPLO, N, NDIAG, NRHS, ZA, LDA, ZAF, LDAF, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CPBRFS	(UPLO, N, NDIAG, NRHS, CA, LDA, CAF, LDAF, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dpbrfs	(char uplo, int n, int ndiag, 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 spbrfs	(char uplo, int n, int ndiag, 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 zpbrfs	(char uplo, int n, int ndiag, 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 cpbrfs	(char uplo, int n, int ndiag, 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.
NDIAG	Number of superdiagonals or subdiagonals of the matrix A. N-1 NDIAG 0 but if N = 0 then NDIAG = 0.
	If UPLO = 'U' or 'u', NDIAG is the number of superdiagonals.
	If UPLO = 'L' or 'l', NDIAG is the number of subdiagonals.
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 xPBTRF.
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, NDIAG

PARAMETER (N = 4)

PARAMETER (NDIAG = 1)

PARAMETER (LDA = NDIAG + 1)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = N)

PARAMETER (NRHS = 1)

C

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

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

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

INTEGER ICOL, INFO, IROW, IWORK(LDIWRK)

C

EXTERNAL DCOPY, DSBMV, DPBTRF, DPBTRS, DPBRFS

INTRINSIC ABS, DBLE, MAX

C

C Initialize the array B to store the right hand side

C vector b shown.

C

C 4

C b = 4

C 4

C 4

C

DATA B / N*4.0D0 /

C

C Initialize the array A to contain the matrix A shown below

C where e=epsilon=small.

C

C e e*e

C A = e*e e e*e

C e*e e e*e

C e*e e

C

A(1,1) = SQRT(-1.0D0)

EPSLON = ABS((((2.0D0 / 3.0D0) + 8.0D0) - 8.0D0) -

$ (2.0D0 / 3.0D0))

DO 100, ICOL = 1, N

A(NDIAG + 1,ICOL) = EPSLON

100 CONTINUE

DO 110, ICOL = 2, N

A(1,ICOL) = EPSLON * EPSLON

110 CONTINUE

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

C

C Print the A array.

C

PRINT 1000

PRINT 1010, A(2,1), A(1,2), 0.0D0, 0.0D0

PRINT 1010, A(1,2), A(2,2), A(1,3), 0.0D0

PRINT 1010, 0.0D0, A(1,3), A(2,3), A(1,4)

PRINT 1010, 0.0D0, 0.0D0, A(1,4), A(2,4)

PRINT 1020

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

C

C Make a copy of B and then print B.

C

CALL DCOPY (N, B, 1, X, 1)

PRINT 1030

PRINT 1040, B

C

C Factor A.

C

CALL DPBTRF ('UPPER TRIANGLE OF A STORED', N, NDIAG,

$ 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 DPBTRS ('UPPER TRIANGLE OF A STORED', N, NDIAG, NRHS,

$ AF, LDAF, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, INFO

STOP 2

END IF

PRINT 1070

PRINT 1080, X

CALL DSBMV ('UPPER TRIANGLE OF A STORED', N, NDIAG, 1.0D0,

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

PRINT 1090

PRINT 1040, B1

C

C Refine the initial solution and print the new solution.

C

CALL DPBRFS ('UPPER TRIANGLE OF A STORED', N, NDIAG, NRHS,

$ A, LDA, AF, LDAF, 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

CALL DSBMV ('UPPER TRIANGLE OF A STORED', N, NDIAG, 1.0D0,

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

PRINT 1120

PRINT 1040, B1

C

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

1010 FORMAT (4(2X, E15.8))

1020 FORMAT (/1X, 'A in banded form: ("NaN" in unused elements)')

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

1040 FORMAT (1X, F19.15)

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

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

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

1080 FORMAT (1X, E24.16)

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

1100 FORMAT (1X, 'Error improving solution in DPBRFS, INFO = ',

$ I5)

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

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

C

END

Sample Output

A in full form:

0.55511151E-15 0.30814879E-30 0.00000000E+00 0.00000000E+00

0.30814879E-30 0.55511151E-15 0.30814879E-30 0.00000000E+00

0.00000000E+00 0.30814879E-30 0.55511151E-15 0.30814879E-30

0.00000000E+00 0.00000000E+00 0.30814879E-30 0.55511151E-15

A in banded form: ("NaN" in unused elements)

NaN 0.30814879E-30 0.30814879E-30 0.30814879E-30

0.55511151E-15 0.55511151E-15 0.55511151E-15 0.55511151E-15

b:

4.000000000000000

4.000000000000000

4.000000000000000

4.000000000000000

Initial solution for Ax=b:

0.7205759403792790E+16

0.7205759403792784E+16

0.7205759403792784E+16

0.7205759403792788E+16

Ax with initial x:

4.000000000000000

3.999999999999999

3.999999999999999

3.999999999999999

Refined solution for Ax=b:

0.7205759403792789E+16

0.7205759403792786E+16

0.7205759403792786E+16

0.7205759403792790E+16

Ax with improved x:

4.000000000000000

4.000000000000000

4.000000000000000

4.000000000000000