Refined Solution to a Linear System in a UDU- or LDL-Factored Symmetric Positive Definite Tridiagonal Matrix

Calling Sequence

CALL DPTRFS	(N, NRHS, DDIAG, DOFFD, DDIAGF, DOFFDF, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, INFO)
CALL SPTRFS	(N, NRHS, SDIAG, SOFFD, SDIAGF, SOFFDF, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, INFO)
CALL ZPTRFS	(UPLO, N, NRHS, DDIAG, ZOFFD, DDIAGF, ZOFFDF, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CPTRFS	(UPLO, N, NRHS, SDIAG, COFFD, SDIAGF, COFFDF, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dptrfs	(int n, int nrhs, double *ddiag, double *doffd, double *ddiagf, double *doffdf, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void sptrfs	(int n, int nrhs, float *sdiag, float *soffd, float *sdiagf, float *soffdf, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zptrfs	(char uplo, int n, int nrhs, double *ddiag, doublecomplex *zoffd, double *ddiagf, doublecomplex *zoffdf, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void cptrfs	(char uplo, int n, int nrhs, float *sdiag, complex *coffd, float *sdiagf, complex *coffdf, complex *cb, int ldb, complex *cx, int ldx, float *sferr, float *sberr, int *info)

Arguments

UPLO	(For complex matrices)
	Indicates whether OFFDF contains the superdiagonal or subdiagonal of the matrix A and the form of the factorization. The legal values for UPLO are listed below. Any values not listed are illegal.
	'U' or 'u'	xOFFDF contains the superdiagonal and the factorization is UDU.
	'L' or 'l'	xOFFDF contains the subdiagonal and the factorization is LDL.
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.
xDIAG	The N diagonal elements of A.
xOFFD	The N-1 off-diagonal elements of A.
xDIAGF	The N diagonal elements of the matrix D from the UDU or LDL factorization of A, as computed by xPTTRF.
xOFFDF	The N-1 off-diagonal elements of the matrix U or L from the UDU or LDL factorization of A, as computed by xPTTRF.
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 for real subroutines or N for complex subroutines.
xWORK2	Scratch array with a dimension of N for complex subroutines.
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, LDWORK, LDX, N, NRHS

PARAMETER (N = 4)

PARAMETER (LDB = N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (LDX = N)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION B(LDB,NRHS), BERR(NRHS), DIAG(N)

DOUBLE PRECISION DIAGF(N), DLOW(N-1), DLOWF(N-1), EPSLON

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

INTEGER ICOL, INFO, IROW

C

EXTERNAL DCOPY, DPTCON, DPTRFS, DPTTRF, DPTTRS

INTRINSIC ABS, MAX

C

C Initialize the arrays DLOW, DIAG, and DUP1 to store

C the first subdiagonal, the diagonal, and the first

C superdiagonal of the symmetric coefficient matrix A

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

C hand side matrix 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 DLOW / 0.0D0, 2.0D0, 0.0D0 /

DATA DIAG / 0.0D0, 2.0D0, 2.0D0, 0.0D0 /

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

C

C Add a small value to each of the elements on the diagonal

C and subtract the same value from the first sub- and

C super-diagonal of A. After this loop, A will resemble the

C matrix shown below. Print A after adding epsilon.

C

C -e -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))

DO 100, IROW = 1, N - 1

DLOW(IROW) = DLOW(IROW) - EPSLON

DIAG(IROW) = DIAG(IROW) + EPSLON

100 CONTINUE

DIAG(N) = DIAG(N) + EPSLON

CALL DCOPY (N - 1, DLOW, 1, DLOWF, 1)

CALL DCOPY (N, DIAG, 1, DIAGF, 1)

C

PRINT 1000

DO 110, IROW = 1, N

PRINT 1010, (0.0D0, ICOL = 1, IROW - 2),

$ (DLOW(ICOL + 1), ICOL = ABS(IROW - 2), IROW - 2),

$ DIAG(IROW),

$ (DLOW(IROW), ICOL = 1, MIN(1, N - IROW)),

$ (0.0D0, ICOL = IROW + 2, N)

110 CONTINUE

C

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

C resemble the matrix shown below. Print B afterwards.

C

C 30+e

C B = 50+e

C 70+e

C 110+e

C

DO 130, ICOL = 1, NRHS

DO 120, IROW = 1, N

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

120 CONTINUE

130 CONTINUE

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

PRINT 1020

PRINT 1030, B

C

C LDL factor A.

C

CALL DPTTRF (N, DIAGF, DLOWF, INFO)

IF (INFO .LT. 0) THEN

PRINT 1040, ABS(INFO)

STOP 1

ELSE IF (INFO .GT. 0) THEN

PRINT 1050, INFO

STOP 2

END IF

C

C Solve Ax=b and print the solution.

C

CALL DPTTRS (N, NRHS, DIAGF, DLOWF, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, INFO

STOP 3

END IF

PRINT 1070

PRINT 1080, X

PRINT 1090

PRINT 1030, DIAG(1) * X(1,1) + DLOW(1) *

$ X(2,1)

PRINT 1030, DLOW(1) * X(1,1) + DIAG(2) * X(2,1) + DLOW(2) *

$ X(3,1)

PRINT 1030, DLOW(2) * X(2,1) + DIAG(3) * X(3,1) + DLOW(3) *

$ X(4,1)

PRINT 1030, DLOW(3) * X(3,1) + DIAG(4) * X(4,1)

C

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

C

CALL DPTRFS (N, NRHS, DIAG, DLOW, DIAGF, DLOWF, B, LDB,

$ X, LDX, FERR, BERR, WORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1100, ABS(INFO)

STOP 4

END IF

PRINT 1110

PRINT 1080, X

PRINT 1120

PRINT 1030, DIAG(1) * X(1,1) + DLOW(1) *

$ X(2,1)

PRINT 1030, DLOW(1) * X(1,1) + DIAG(2) * X(2,1) + DLOW(2) *

$ X(3,1)

PRINT 1030, DLOW(2) * X(2,1) + DIAG(3) * X(3,1) + DLOW(3) *

$ X(4,1)

PRINT 1030, DLOW(3) * X(3,1) + DIAG(4) * X(4,1)

PRINT 1130, FERR(1)

PRINT 1140, 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, 'Illegal argument to DPTRFS, argument #', I2)

1050 FORMAT (1X, 'A is not positive definite, INFO = ', I2)

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

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

1080 FORMAT (1X, E25.17)

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

1100 FORMAT (1X, 'Illegal argument to DPTRFS, INFO = ', I2)

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

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

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

1140 FORMAT (1X, 'Backward error: ', E14.8)

C

END

Sample Output

A:
0.0000000000000003 -.0000000000000003 0.0000000000000000 0.0000000000000000
-.0000000000000003 2.0000000000000004 1.9999999999999996 0.0000000000000000
0.0000000000000000 1.9999999999999996 2.0000000000000004 -.0000000000000003
0.0000000000000000 0.0000000000000000 -.0000000000000003 0.0000000000000003
b:
29.99999999999998934
49.99999999999998579
69.99999999999997158
109.99999999999995737
Initial solution to Ax=b:
-0.10007999171934384E+17
-0.10007999171934427E+18
0.10007999171934432E+18
0.43034396439318054E+18
Ax with the initial x:
29.99999999999999289
32.00000000000000000
16.66666666666674246
109.99999999999994316
Refined solution to Ax=b:
-0.57379195252423992E+17
-0.14745118779983389E+18
0.14745118779983392E+18
0.47771516047367027E+18
Ax with refined x:
29.99999999999999645
-64.00000000000000000
32.88888888888891415
110.00000000000000000
Forward error: 0.47796400E+01
Backward error: 0.17862806E-15