Refined Solution to a Linear System in an LU-Factored General Matrix in Banded Storage

Calling Sequence

CALL DGBRFS	(TRANSA, N, NSUB, NSUPER, NRHS, DA, LDA, DAF, LDAF, IPIVOT, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SGBRFS	(TRANSA, N, NSUB, NSUPER, NRHS, SA, LDA, SAF, LDAF, IPIVOT, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZGBRFS	(TRANSA, N, NSUB, NSUPER, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, ZB, LDB, ZX, LDX, ZFERR, ZBERR, ZWORK, DWORK2, INFO)
CALL CGBRFS	(TRANSA, N, NSUB, NSUPER, NRHS, CA, LDA, CAF, LDAF, IPIVOT, CB, LDB, CX, LDX, CFERR, CBERR, CWORK, SWORK2, INFO)
void dgbrfs	(char trans, int n, int nsub, int nsuper, int nrhs, double *da, int lda, double *daf, int ldaf, int *ipivot, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void sgbrfs	(char trans, int n, int nsub, int nsuper, int nrhs, float *sa, int lda, float *saf, int ldaf, int *ipivot, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zgbrfs	(char trans, int n, int nsub, int nsuper, 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 cgbrfs	(char trans, int n, int nsub, int nsuper, 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

TRANSA	Indicates the form of the linear system whose solution is to be refined. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	'N' or 'n'	No transpose, refine AX = B.
	'T' or 't'	Transpose, refine ATX = B.
	'C' or 'c'	Conjugate transpose, refine AHX = B.
	Note that AT and AH are the same for real matrices.
N	Order of the matrix A. N 0.
NSUB	Number of subdiagonals of A. N-1 NSUB 0 but if N = 0 then NSUB = 0.
NSUPER	Number of superdiagonals of A. N-1 NSUPER 0 but if N = 0 then NSUPER = 0.
NRHS	Number of right-hand sides, equal to the number of columns in the matrices B and X. NRHS 0.
xA	Matrix A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA NSUB + NSUPER + 1.
xAF	LU factorization of the matrix A, as computed by xGBTRF.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF 2 × NSUB + NSUPER + 1.
IPIVOT	Pivot indices as computed by xGBTRF.
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.
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 LDA, LDAF, LDB, LDIWRK, LDWORK, LDX, N

INTEGER NRHS, NSUB, NSUPER

PARAMETER (N = 4)

PARAMETER (NSUB = 1)

PARAMETER (NSUPER = 0)

PARAMETER (LDA = 2*NSUB + 1 + NSUPER)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = N)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION A(LDA,N), AF(LDAF,N), B(LDB,NRHS)

DOUBLE PRECISION B1(LDB,NRHS), BERR(NRHS), EPSLON

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

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

C

EXTERNAL DCOPY, DGBMV, DGBTRF, DGBTRS, DGBRFS

INTRINSIC ABS, DBLE, MAX

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 0 4

C A = 9 0 b = 4

C 0.25 5 4

C 0 70 4

C

DATA A / NSUB*8D8, 0.0D0, 9.0D0, NSUB*8D8, 0.0D0, 2.5D-1,

$ NSUB*8D8, 5.0D0, 0.0D0, NSUB*8D8, 7.0D1, 8D8 /

DATA B / N*4.0D0 /

C

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

C on the subdiagonal. After this loop, A will contain

C something similar to:

C

C e

C A = 9+e e

C 0.25+e 5+e

C 0+e 70+e

C

EPSLON = ((((2.0D0 / 3.0D0) + 8.0D0) - 8.0D0) -

$ (2.0D0 / 3.0D0))

DO 110, ICOL = 1, N

DO 100, IROW = 1, NSUB + 1 + NSUPER

A(NSUB+IROW,ICOL) = A(NSUB+IROW,ICOL) + EPSLON

100 CONTINUE

110 CONTINUE

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

C

C Print the A array.

C

PRINT 1000

PRINT 1010, A(2,1), 0.0D0, 0.0D0, 0.0D0

PRINT 1010, A(3,1), A(2,2), 0.0D0, 0.0D0

PRINT 1010, 0.0D0, A(3,2), A(2,3), 0.0D0

PRINT 1010, 0.0D0, 0.0D0, A(3,3), A(2,4)

PRINT 1020

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

C

C Make a copy of B and then print B.

C

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

PRINT 1030

PRINT 1040, B

C

C LU factor A.

C

CALL DGBTRF (N, N, NSUB, NSUPER, AF, LDAF, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 1

END IF

C

C Solve Ax=b and print the solution.

C

CALL DGBTRS ('NO TRANSPOSE A', N, NSUB, NSUPER, NRHS, AF,

$ LDAF, IPIVOT, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, INFO

STOP 2

END IF

PRINT 1070

PRINT 1080, X

CALL DGBMV ('NO TRANSPOSE A', N, N, NSUB, NSUPER, 1.0D0,

$ A(NSUB+1,1), LDA, X, 1, 0.0D0, B1, 1)

PRINT 1090

PRINT 1040, B1

C

C Refine the initial solution and print the new solution.

C

CALL DGBRFS ('NO TRANSPOSE A', N, NSUB, NSUPER, NRHS,

$ A(NSUB+1,1), LDA, AF, LDAF, IPIVOT, B, LDB,

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

IF (INFO .NE. 0) THEN

PRINT 1100, INFO

STOP 3

END IF

PRINT 1110

PRINT 1080, X

CALL DGBMV ('NO TRANSPOSE A', N, N, NSUB, NSUPER, 1.0D0,

$ A(NSUB+1,1), LDA, X, 1, 0.0D0, B1, 1)

PRINT 1120

PRINT 1040, B1

C

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

1010 FORMAT (4(2X, F18.15))

1020 FORMAT (/1X, 'A in banded form: (* in unused elements)')

1030 FORMAT (/1X, 'b:')

1040 FORMAT (1X, F19.15)

1050 FORMAT (1X, 'Error factoring A in DGBTRF, INFO = ', I5)

1060 FORMAT (1X, 'Error solving Ax=b in DGBTRF, INFO = ', I5)

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

1080 FORMAT (1X, E24.16)

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

1100 FORMAT (1X, 'Error improving solution in DGBRFS, INFO = ',

$ I5)

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

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

C

END

Sample Output

A in full form:
-0.000000000000001 0.000000000000000 0.000000000000000 0.000000000000000
9.000000000000000 -0.000000000000001 0.000000000000000 0.000000000000000
0.000000000000000 0.249999999999999 4.999999999999999 0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000001 70.000000000000000
A in banded form: (* in unused elements)
****************** ****************** ****************** ******************
-0.000000000000001 -0.000000000000001 4.999999999999999 70.000000000000000
9.000000000000000 0.249999999999999 -0.000000000000001 ******************
b:
4.000000000000000
4.000000000000000
4.000000000000000
4.000000000000000
Initial solution for Ax=b:
-0.7205759403792791E+16
-0.1168266793170336E+33
0.5841333965851668E+31
0.4632273902438220E+14
Ax with initial x:
3.999999999999999
8.000000000000000
0.000000000000000
4.000000000000000
Refined solution for Ax=b:
-0.7205759403792793E+16
-0.1168266793170336E+33
0.5841333965851669E+31
0.4632273902438221E+14
Ax with improved x:
4.000000000000000
0.000000000000000
0.000000000000000
4.500000000000000