Error Bounds and Estimates for the Solution of a Linear System in a Triangular Matrix in Packed Storage

Calling Sequence

CALL DTPRFS	(UPLO, TRANSA, DIAG, N, NRHS, DA, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL STPRFS	(UPLO, TRANSA, DIAG, N, NRHS, SA, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZTPRFS	(UPLO, TRANSA, DIAG, N, NRHS, ZA, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CTPRFS	(UPLO, TRANSA, DIAG, N, NRHS, CA, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dtprfs	(char uplo, char trans, char diag, int n, int nrhs, double *da, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void stprfs	(char uplo, char trans, char diag, int n, int nrhs, float *sa, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void ztprfs	(char uplo, char trans, char diag, int n, int nrhs, doublecomplex *za, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void ctprfs	(char uplo, char trans, char diag, int n, int nrhs, complex *ca, 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.
TRANSA	Indicates the form of the system of equations. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	'N' or 'n'	No transpose, use AX = B.
	'T' or 't'	Transpose, use ATX = B.
	'C' or 'c'	Conjugate transpose, use AHX = B.
	Note that AT and AH are the same for real matrices.
DIAG	Indicates whether or not A is unit triangular. The legal values for DIAG are listed below. Any values not listed below are illegal.
	'N' or 'n'	A is not unit triangular.
	'U' or 'u'	A is unit triangular.
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 triangular matrix A. The dimension of xA is (N × N + N) / 2.
	If DIAG = 'U' or 'u', the diagonal elements of A are assumed to be 1 and are not used.
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	The 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

DOUBLE PRECISION ZERO

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)

PARAMETER (ZERO = 0.0D0)

C

DOUBLE PRECISION A(LENGTA), B(LDB,NRHS), BERR(NRHS), EPSLON

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

INTEGER I, INFO, IWORK(LDIWRK)

C

EXTERNAL DCOPY, DTPMV, DTPRFS, DTPTRS

INTRINSIC ABS, DBLE

C

C Initialize the array A to store in banded form the matrix A

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

C side vector b shown below.

C

C 1 4

C A = 1 0 b = 4

C 0 1 0 4

C 0 0 1 0 4

C

DATA A / 1.0D0, 1.0D0, 0.0D0, 0.0D0, 0.0D0, 1.0D0, 0.0D0,

$ 0.0D0, 1.0D0, 0.0D0 /

DATA B / N*4.0D0 /

C

C Add a small value to the elements of A on and below the

C diagonal. After this loop, A will contain something

C similar to:

C

C 1+e

C A = 1+e e

C e 1+e e

C e e 1+e e

C

EPSLON = ((((2.0D0 / 3.0D0) + 1.6D1) - 1.6D1) -

$ (2.0D0 / 3.0D0))

DO 100, I = 1, LENGTA

A(I) = A(I) + EPSLON

100 CONTINUE

C

C Print the initial value of A.

C

PRINT 1000

PRINT 1010, A(1), ZERO, ZERO, ZERO

PRINT 1010, A(2), A(5), ZERO, ZERO

PRINT 1010, A(3), A(6), A(8), ZERO

PRINT 1010, A(4), A(7), A(9), A(10)

C

C Make a copy of B and then print B.

C

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

PRINT 1020

PRINT 1030, B

C

C Solve Ax=b and print the solution.

C

CALL DTPTRS ('LOWER TRIANGULAR A', 'NO TRANSPOSE A',

$ 'NO UNIT DIAGONAL A', N, NRHS, A, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

PRINT 1050

PRINT 1060, X

C

C Estimate the error bounds for the computed solution.

C

CALL DTPRFS ('LOWER TRIANGULAR A', 'NO TRANSPOSE A',

$ 'NO UNIT DIAGONAL A', N, NRHS, A, B, LDB,

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

IF (INFO .NE. 0) THEN

PRINT 1070, ABS(INFO)

STOP 2

END IF

CALL DTPMV ('LOWER TRIANGULAR A', 'NO TRANSPOSE A',

$ 'NO UNIT DIAGONAL A', N, A, X, 1)

PRINT 1080

PRINT 1030, X

PRINT 1090, (I, BERR(I), I = 1, NRHS)

PRINT 1100, (I, FERR(I), I = 1, NRHS)

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (4(2X, F17.15))

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

1030 FORMAT (1X, F21.18)

1040 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

1050 FORMAT (/1X, 'Initial solution for Ax=b:')

1060 FORMAT (1X, E16.8)

1070 FORMAT (1X, 'Illegal argument to DTPRFS, argument #', I5)

1080 FORMAT (/1X, 'Ax:')

1090 FORMAT (/1X, 'Backward error bound for system #', I2, ' =',

$ E14.7)

1100 FORMAT (1X, 'Forward error bound for system #', I2, ' = ',

$ E14.7)

C

END

Sample Output

A:
1.000000000000001 0.000000000000000 0.000000000000000 0.000000000000000
1.000000000000001 0.000000000000001 0.000000000000000 0.000000000000000
0.000000000000001 1.000000000000001 0.000000000000001 0.000000000000000
0.000000000000001 0.000000000000001 1.000000000000001 0.000000000000001
b:
4.000000000000000000
4.000000000000000000
4.000000000000000000
4.000000000000000000
Initial solution for Ax=b:
0.40000000E+01
0.00000000E+00
0.32753452E+16
-0.26819715E+31
Ax:
4.000000000000000000
4.000000000000000000
4.000000000000000000
4.000000000000004441
Backward error bound for system # 1 = 0.6779273E-30
Forward error bound for system # 1 = 0.1818182E+01