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

Calling Sequence

CALL DSPRFS	(UPLO, N, NRHS, DA, DAF, IPIVOT, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SSPRFS	(UPLO, N, NRHS, SA, SAF, IPIVOT, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZSPRFS	(UPLO, N, NRHS, ZA, ZAF, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CSPRFS	(UPLO, N, NRHS, CA, CAF, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dsprfs	(char uplo, int n, int nrhs, double *da, double *daf, int *ipivot, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void ssprfs	(char uplo, int n, int nrhs, float *sa, float *saf, int *ipivot, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zsprfs	(char uplo, int n, int nrhs, doublecomplex *za, doublecomplex *zaf, int *ipivot, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void csprfs	(char uplo, int n, int nrhs, complex *ca, complex *caf, 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. The dimension of xA is (N × N + N) / 2.
xAF	UDU or LDL factorization of the matrix A as computed by xSPTRF.
	The dimension of xAF is (N × N + N) / 2.
IPIVOT	Pivot indices as computed by xSPTRF.
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 LDB, LDIWRK, LDWORK, LDX, LENGTA, N, NRHS

PARAMETER (N = 4)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = LDB)

PARAMETER (LENGTA = (N * N + N) / 2)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION A(LENGTA), AF(LENGTA), B(LDB,NRHS)

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

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

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

C

EXTERNAL DCOPY, DSPRFS, DSPTRF, DSPTRS

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, 0.0D0, 2.0D0, 0.0D0, 2.0D0, 2.0D0,

$ 0.0D0, 0.0D0, 0.0D0, 0.0D0 /

DATA B / 3.0D1, 5.0D1, 7.0D1, 1.1D2 /

C

C Slightly perturb some of the entries of A. After this

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

C after adding epsilon.

C

C 2e -e

C A = -e 2+e 2-e

C 2-e 2+e -e

C -e -e

C

EPSLON = ABS ((((2.0D0 / 3.0D0) + 4.0D0) - 4.0D0) -

$ (2.0D0 / 3.0D0))

A(1) = A(1) + 2.0D0 * EPSLON

A(2) = A(2) - EPSLON

A(3) = A(3) + EPSLON

A(5) = A(5) - EPSLON

A(6) = A(6) + EPSLON

A(9) = A(9) - EPSLON

A(10) = A(10) - EPSLON

PRINT 1000

PRINT 1010, A(1), A(2), A(4), A(7)

PRINT 1010, A(2), A(3), A(5), A(8)

PRINT 1010, A(4), A(5), A(6), A(9)

PRINT 1010, A(7), A(8), A(9), A(10)

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

C

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

C resemble the matrix shown below. Print B after scaling.

C

C 30-small

C B = 50-small

C 70-small

C 110-small

C

DO 110, ICOL = 1, NRHS

DO 100, IROW = 1, N

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

100 CONTINUE

110 CONTINUE

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

PRINT 1020

PRINT 1030, B

C

C LDL factor A.

C

CALL DSPTRF ('UPPER TRIANGLE OF A STORED', N, AF, IPIVOT,

$ INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

C

C Solve Ax=b and print the solution.

C

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

$ IPIVOT, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 2

END IF

PRINT 1060

PRINT 1070, X

PRINT 1080

PRINT 1030, A(1) * X(1,1) + A(2) * X(2,1)

PRINT 1030, A(2) * X(1,1) + A(3) * X(2,1) + A(5) * X(3,1)

PRINT 1030, A(5) * X(2,1) + A(6) * X(3,1) + A(9) * X(4,1)

PRINT 1030, A(9) * X(3,1) + A(10) * X(4,1)

C

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

C

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

$ IPIVOT, B, LDB, X, LDX, FERR, BERR, WORK,

$ IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1090, ABS(INFO)

STOP 3

END IF

PRINT 1100

PRINT 1070, X

PRINT 1110

PRINT 1030, A(1) * X(1,1) + A(2) * X(2,1)

PRINT 1030, A(2) * X(1,1) + A(3) * X(2,1) + A(5) * X(3,1)

PRINT 1030, A(5) * X(2,1) + A(6) * X(3,1) + A(9) * X(4,1)

PRINT 1030, A(9) * X(3,1) + A(10) * X(4,1)

PRINT 1120, FERR(1)

PRINT 1130, BERR(1)

C

1000 FORMAT (1X, 'A:')

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

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

1030 FORMAT (1X, F21.17)

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

1050 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

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

1070 FORMAT (1X, E25.17)

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

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

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

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

1120 FORMAT (/1X, 'Forward error: ', E14.8)

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

C

END

Sample Output

A:
0.0000000000000007 -.0000000000000003 0.0000000000000000 0.0000000000000000
-.0000000000000003 2.0000000000000004 1.9999999999999996 0.0000000000000000
0.0000000000000000 1.9999999999999996 2.0000000000000004 - .0000000000000003
0.0000000000000000 0.0000000000000000 -.0000000000000003 - .0000000000000003
b:
29.99999999999998934
49.99999999999998579
69.99999999999997158
109.99999999999995737
Initial solution to Ax=b:
0.70596967131753704E+17
0.51121941716097496E+17
-0.51121941716097480E+17
-0.27914203095773875E+18
Ax with the initial x:
30.00000000000000000
48.00000000000000000
92.97297297297293994
109.99999999999995737
Refined solution to Ax=b:
0.77650084444450784E+17
0.65228176341491688E+17
-0.65228176341491680E+17
-0.26503579633234458E+18
Ax with refined x:
29.99999999999998579
48.00000000000000000
40.27465303140976971
109.99999999999995737
Forward error: 0.57571539E+00
Backward error: 0.90932126E-16