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

Calling Sequence

CALL ZHPRFS	(UPLO, N, NRHS, ZA, ZAF, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CHPRFS	(UPLO, N, NRHS, CA, CAF, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void zhprfs	(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 chprfs	(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 xHPTRF.
	The dimension of xAF is (N × N + N) / 2.
IPIVOT	Pivot indices as computed by xHPTRF.
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 2 × N.
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

DOUBLE PRECISION ZERO

INTEGER LDB, LDB1, LDWORK, LDWRK2, LDX, LENGTA

INTEGER N, NRHS

PARAMETER (N = 3)

PARAMETER (LDB = N)

PARAMETER (LDB1 = N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (LDWRK2 = N)

PARAMETER (LDX = LDB)

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

PARAMETER (NRHS = 1)

PARAMETER (ZERO = 0.0D0)

C

DOUBLE PRECISION ANORM, BERR(N), FERR(N), RCOND

DOUBLE PRECISION WORK2(LDWRK2)

COMPLEX*16 A(LENGTA), AF(LENGTA), B(LDB,NRHS)

COMPLEX*16 B1(LDB,NRHS), WORK(LDWORK), X(LDX,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW

C

EXTERNAL ZCOPY, ZHPCON, ZHPMV, ZHPRFS, ZHPTRF

EXTERNAL ZHPTRS

INTRINSIC ABS, CMPLX, CONJG, DBLE, MAX

C

C Initialize the array A to store the coefficient array A

C shown below. Initialize the array B to store the right

C hand side vector b shown below.

C

C (1e+16, 0) (1e+16, 1) (3e+16, 3e-16)

C A = (1e+16, -1) (1e+16, 0) (3e+26, 3e-16)

C (3e+16, 3e-16) (3e+26, -3e-16) (7e-16, 0)

C

C (2e-26, 2e+16)

C b = (2e+26, 2e-16)

C (2, 2)

C

DATA A / (1.0D16,8D8), (1.0D16,1.0D0), (1.0D16,8D8),

$ (3.0D16,3.0D-16), (3.0D26,3.0D-16), (7.0D-16,8D8) /

DATA B / (2.0D-26,2.0D16), (2.0D26,2.0D-16), (2.0D0,2.0D0) /

C

C Print the initial value of the arrays.

C

PRINT 1000

PRINT 1010, CMPLX(DBLE(A(1)),ZERO), A(2), A(4)

PRINT 1010, CONJG(A(2)), CMPLX(DBLE(A(3)),ZERO), A(5)

PRINT 1010, CONJG(A(4)), CONJG(A(5)), CMPLX(DBLE(A(6)),ZERO)

DO 110, ICOL = 1, NRHS

PRINT 1020, ICOL

DO 100, IROW = 1, N

PRINT 1100, B(IROW,ICOL)

100 CONTINUE

110 CONTINUE

C

C UDU factor A.

C

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

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

$ INFO)

IF (INFO .LT. 0) THEN

PRINT 1030, ABS(INFO)

STOP 1

ELSE IF (INFO .GT. 0) THEN

PRINT 1040

STOP 2

END IF

C

C Estimate the condition number. Print a warning if it is

C unreasonably high.

C

ANORM = 3.0D26

CALL ZHPCON ('UPPER TRIANGULAR FACTOR OF A', N, AF, IPIVOT,

$ ANORM, RCOND, WORK, INFO)

IF (INFO .LT. 0) THEN

PRINT 1050, ABS(INFO)

STOP 3

END IF

PRINT 1060, RCOND

IF ((1.0D0 + RCOND) .EQ. 1.0D0) THEN

PRINT 1070

END IF

C

C Compute and print the solution.

C

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

CALL ZHPTRS ('UPPER TRIANGULAR FACTOR OF A', N, NRHS, AF,

$ IPIVOT, X, LDX, INFO)

IF (INFO .LT. 0) THEN

PRINT 1080, ABS(INFO)

STOP 4

END IF

DO 210, ICOL = 1, NRHS

PRINT 1090, ICOL

DO 200, IROW = 1, N

PRINT 1100, X(IROW,ICOL)

200 CONTINUE

210 CONTINUE

DO 220, ICOL = 1, NRHS

CALL ZHPMV ('UPPER TRIANGLE OF A STORED', N,

$ (1.0D0,0.0D0), A, X(1,ICOL), 1, (0.0D0,0.0D0),

$ B1(1,ICOL), 1)

220 CONTINUE

DO 240, ICOL = 1, NRHS

PRINT 1110, ICOL

DO 230, IROW = 1, N

PRINT 1100, B1(IROW,ICOL)

230 CONTINUE

240 CONTINUE

C

C Refine the solution and print the refined solution with the

C error bounds.

C

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

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

$ WORK2, INFO)

IF (INFO .LT. 0) THEN

PRINT 1120, ABS(INFO)

STOP 5

END IF

DO 310, ICOL = 1, NRHS

PRINT 1130, ICOL

DO 300, IROW = 1, N

PRINT 1100, X(IROW,ICOL)

300 CONTINUE

310 CONTINUE

DO 320, ICOL = 1, NRHS

CALL ZHPMV ('UPPER TRIANGLE OF A STORED', N,

$ (1.0D0,0.0D0), A, X(1,ICOL), 1, (0.0D0,0.0D0),

$ B1(1,ICOL), 1)

320 CONTINUE

DO 340, ICOL = 1, NRHS

PRINT 1140, ICOL

DO 330, IROW = 1, N

PRINT 1100, B1(IROW,ICOL)

330 CONTINUE

340 CONTINUE

DO 350, IROW = 1, NRHS

PRINT 1150, IROW, BERR(IROW)

PRINT 1160, IROW, FERR(IROW)

350 CONTINUE

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (1X, 10(: 1X, '(', E8.2, ', ', E8.2, ')'))

1020 FORMAT (/1X, 'b for system #', I1)

1030 FORMAT (1X, 'Illegal argument to ZHPTRF, argument #', I2)

1040 FORMAT (1X, 'A is singular to working precision.')

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

1060 FORMAT (/1X,

$ 'Estimated reciprocal condition number of A: ',

$ E9.3)

1070 FORMAT (1X, 'A may be singular to working precision.')

1080 FORMAT (1X, 'Illegal argument to ZHPTRS, argument #', I2)

1090 FORMAT (/1X, 'Initial xb for system #', I1)

1100 FORMAT (3X, '(', E20.14, ', ', E20.14, ')')

1110 FORMAT (/1X, 'Ax with initial x for system #', I1)

1120 FORMAT (1X, 'Illegal argument to ZHPRFS, argument #', I2)

1130 FORMAT (/1X, 'Refined xb for system #', I1)

1140 FORMAT (/1X, 'Ax with refined x for system #', I1)

1150 FORMAT (/1X, 'Backward error for system #', I1, ': ', E12.6)

1160 FORMAT (1X, 'Forward error for system #', I1, ': ', E12.6)

C

END

Sample Output

A:

(0.10E+17, 0.00E+00) (0.10E+17, 0.10E+01) (0.30E+17, 0.30E-15)

(0.10E+17, -.10E+01) (0.10E+17, 0.00E+00) (0.30E+27, 0.30E-15)

(0.30E+17, -.30E-15) (0.30E+27, -.30E-15) (0.70E-15, 0.00E+00)

b for system #1

(0.20000000000000E-25, 0.20000000000000E+17)

(0.20000000000000E+27, 0.20000000000000E-15)

(0.20000000000000E+01, 0.20000000000000E+01)

Estimated reciprocal condition number of A: 0.333E-10

Initial xb for system #1

(-.20000000004000E+01, 0.20000000004000E+01)

(0.20000000004000E-09, -.20000000004000E-09)

(0.66666666673333E+00, -.66666666673333E-10)

Ax with initial x for system #1

(0.40000000000000E+01, 0.20000000000000E+17)

(0.20000000000000E+27, 0.40000000000000E+01)

(-.80000000000000E+01, 0.80000000000000E+01)

Refined xb for system #1

(-.20000000004000E+01, 0.20000000004000E+01)

(0.20000000004000E-09, -.20000000004000E-09)

(0.66666666673333E+00, -.66666666673333E-10)

Ax with refined x for system #1

(0.40000000000000E+01, 0.20000000000000E+17)

(0.20000000000000E+27, 0.40000000000000E+01)

(-.80000000000000E+01, 0.80000000000000E+01)

Backward error for system #1: 0.666667E-16

Forward error for system #1: 0.143228E-14