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

Calling Sequence

CALL ZHERFS	(UPLO, N, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CHERFS	(UPLO, N, NRHS, CA, LDA, CAF, LDAF, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void zherfs	(char uplo, int n, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, int *ipivot, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void cherfs	(char uplo, int n, int nrhs, complex *ca, int lda, complex *caf, int ldaf, 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.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
xAF	UDU or LDL factorization of the matrix A as computed by xHETRF.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF max(1, N).
IPIVOT	Pivot indices as computed by xHETRF.
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 LDA, LDAF, LDB, LDB1, LDWORK, LDWRK2, LDX

INTEGER N, NRHS

PARAMETER (N = 3)

PARAMETER (LDA = N)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = N)

PARAMETER (LDB1 = N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (LDWRK2 = N)

PARAMETER (LDX = LDB)

PARAMETER (NRHS = 1)

PARAMETER (ZERO = 0.0D0)

C

DOUBLE PRECISION BERR(N), FERR(N), WORK2(LDWRK2)

COMPLEX*16 A(LDA,N), AF(LDA,N), B(LDB,NRHS)

COMPLEX*16 B1(LDB,NRHS), INFNTY, WORK(LDWORK)

COMPLEX*16 X(LDX,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW

C

EXTERNAL ZCOPY, ZHEMM, ZHERFS, ZHETRF, ZHETRS

INTRINSIC ABS, DCMPLX, DCONJG, DBLE, SQRT, MAX

C

C Initialize the array A to store the coefficient array A

C 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 Initialize the array B to store the right-hand side vector b

C shown below.

C

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

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

C (2, 2)

C

DATA A / (1.0D16,0.0D0), (-8D300,-8D300), (-8D300,-8D300),

$ (1.0D16,1.0D0), (1.0D16,0.0D0), (-8D300,-8D300),

$ (3.0D16,3.0D-16), (3.0D26,3.0D-16),

$ (7.0D-16,0.0D0) /

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

INFNTY = DCMPLX(SQRT(-1.0D0), SQRT(-1.0D0))

DO 110, IROW = 1, N

DO 100, ICOL = 1, IROW - 1

A(IROW,ICOL) = INFNTY

100 CONTINUE

110 CONTINUE

PRINT 1000

DO 120, IROW = 1, N

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

$ DCMPLX(DBLE(A(IROW,IROW)), ZERO),

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

120 CONTINUE

PRINT 1020

DO 130, IROW = 1, N

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

130 CONTINUE

DO 150, ICOL = 1, NRHS

PRINT 1030, ICOL

DO 140, IROW = 1, N

PRINT 1080, B(IROW,ICOL)

140 CONTINUE

150 CONTINUE

C

C Factor the matrix.

C

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

CALL ZHETRF ('UPPER TRIANGLE OF A STORED', N, AF, LDAF,

$ IPIVOT, WORK, LDWORK, INFO)

IF (INFO .LT. 0) THEN

PRINT 1040, ABS(INFO)

STOP 1

ELSE IF (INFO .GT. 0) THEN

PRINT 1050

STOP 2

END IF

C

C Compute and print the solution.

C

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

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

$ LDAF, IPIVOT, X, LDX, INFO)

IF (INFO .LT. 0) THEN

PRINT 1060, ABS(INFO)

STOP 3

END IF

DO 210, ICOL = 1, NRHS

PRINT 1070, ICOL

DO 200, IROW = 1, N

PRINT 1080, X(IROW,ICOL)

200 CONTINUE

210 CONTINUE

CALL ZHEMM ('LEFT SIDE: AX', 'UPPER TRIANGLE OF A STORED',

$ N, NRHS, DCMPLX(1.0D0,ZERO), A, LDA, X, LDX,

$ DCMPLX(ZERO,ZERO), B1, LDB1)

DO 230, ICOL = 1, NRHS

PRINT 1090, ICOL

DO 220, IROW = 1, N

PRINT 1080, B1(IROW,ICOL)

220 CONTINUE

230 CONTINUE

C

C Refine the solution and print the refined solution with the

C error bounds.

C

CALL ZHERFS ('UPPER TRIANGLE OF A STORED', N, NRHS, A, LDA,

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

$ WORK, WORK2, INFO)

DO 310, ICOL = 1, NRHS

PRINT 1100, ICOL

DO 300, IROW = 1, N

PRINT 1080, X(IROW,ICOL)

300 CONTINUE

310 CONTINUE

CALL ZHEMM ('LEFT SIDE: AX', 'UPPER TRIANGLE OF A STORED',

$ N, NRHS, DCMPLX(1.0D0,ZERO), A, LDA, X, LDX,

$ DCMPLX(ZERO,ZERO), B1, LDB1)

DO 330, ICOL = 1, NRHS

PRINT 1110, ICOL

DO 320, IROW = 1, N

PRINT 1080, B1(IROW,ICOL)

320 CONTINUE

330 CONTINUE

DO 340, IROW = 1, NRHS

PRINT 1120, IROW, BERR(IROW)

PRINT 1130, IROW, FERR(IROW)

340 CONTINUE

C

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

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

1020 FORMAT (/1X, 'A in Hermitian form: ',

$ ' ("NaN" in unused entries)')

1030 FORMAT (/1X, 'b for system #', I1)

1040 FORMAT (1X, 'Illegal argument to ZHETRF, argument #', I2)

1050 FORMAT (1X, 'A is singular to working precision.')

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

1070 FORMAT (/1X, 'Initial x for system #', I1)

1080 FORMAT (3X, '(', E20.14, ', ', E20.14, ')')

1090 FORMAT (/1X, 'Ax with initial x for system #', I1)

1100 FORMAT (/1X, 'Refined x for system #', I1)

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

1120 FORMAT (/1X, 'Backward error for system #', I1, ': ', E12.6)

1130 FORMAT (1X, 'Forward error for system #', I1, ': ', E12.6)

C

END

Sample Output

A in full form:

(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)

A in Hermitian form: ("NaN" in unused entries)

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

(NaN , NaN ) (0.10E+17, 0.00E+00) (0.30E+27, 0.30E-15)

(NaN , NaN ) (NaN , NaN ) (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)

Initial x 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 x 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