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

Calling Sequence

CALL DGBTRS	(TRANSA, N, NSUB, NSUPER, NRHS, DA, LDA, IPIVOT, DB, LDB, INFO)
CALL SGBTRS	(TRANSA, N, NSUB, NSUPER, NRHS, SA, LDA, IPIVOT, SB, LDB, INFO)
CALL ZGBTRS	(TRANSA, N, NSUB, NSUPER, NRHS, ZA, LDA, IPIVOT, ZB, LDB, INFO)
CALL CGBTRS	(TRANSA, N, NSUB, NSUPER, NRHS, CA, LDA, IPIVOT, CB, LDB, INFO)
void dgbtrs	(char trans, int n, int nsub, int nsuper, int nrhs, double *da, int lda, int *ipivot, double *db, int ldb, int *info)
void sgbtrs	(char trans, int n, int nsub, int nsuper, int nrhs, float *sa, int lda, int *ipivot, float *sb, int ldb, int *info)
void zgbtrs	(char trans, int n, int nsub, int nsuper, int nrhs, doublecomplex *za, int lda, int *ipivot, doublecomplex *zb, int ldb, int *info)
void cgbtrs	(char trans, int n, int nsub, int nsuper, int nrhs, complex *ca, int lda, int *ipivot, complex *cb, int ldb, int *info)

Arguments

TRANSA	Indicates the form of the linear system to solve. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	'N' or 'n'	No transpose, solve AX = B.
	'T' or 't'	Transpose, solve ATX = B.
	'C' or 'c'	Conjugate transpose, solve AHX = B.
	Note that AT = AH 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 matrix B. NRHS 0.
xA	LU factorization of the matrix A as computed by xGBTRF.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA 2 × NSUB + NSUPER + 1.
IPIVOT	Pivot indices as computed by xGBTRF.
xB	On entry, the N×NRHS right-hand side matrix B. On exit, the N×NRHS solution matrix X.
LDB	Leading dimension of the array B as specified in a dimension or type statement. LDB max(1, 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 LDAB, LDAG, LDB, M, N, NRHS, NSUB, NSUPER

PARAMETER (NSUB = 1)

PARAMETER (NSUPER = 1)

PARAMETER (M = 4)

PARAMETER (N = 4)

PARAMETER (NRHS = 1)

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

PARAMETER (LDAG = M)

PARAMETER (LDB = N)

C

DOUBLE PRECISION AB(LDAB,N), AG(LDAG,N), B(LDB,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW, IROWB, I1, I2

INTEGER M1

C

EXTERNAL DGBTRF, DGBTRS

INTRINSIC MAX0, MIN0

C

C Initialize the array AG to store the 4x4 matrix A with one

C subdiagonal and one superdiagonal shown below. Initialize

C the array B to store the vector b shown below.

C

C 2 -1 5

C AG = -1 2 -1 b = 5

C -1 2 -1 5

C -1 2 5

C

DATA AB / 16 * 8.0D8 /

DATA AG / 2.0D0, -1.0D0, 2*0.0, -1.0D0, 2.0D0, -1.0D0,

$ 0.0D0, 0.0D0, -1.0D0, 2.0D0, -1.0D0, 2*0D0,

$ -1.0D0, 2.0D0 /

DATA B / N*5.0D0 /

C

C Copy the matrix A from the array AG to the array AB. The

C matrix is stored in general storage mode in AG and it will

C be stored in banded storage mode in AB. The code to copy

C from general to banded storage mode was written for LINPACK

C by Cleve Moler.

C

M1 = NSUB + NSUPER + 1

DO 20, ICOL = 1, N

I1 = MAX0 (1, ICOL - NSUPER)

I2 = MIN0 (N, ICOL + NSUB)

DO 10, IROW = I1, I2

IROWB = IROW - ICOL + M1

AB(IROWB,ICOL) = AG(IROW,ICOL)

10 CONTINUE

20 CONTINUE

C

C Print the initial values of the arrays.

C

PRINT 1000

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

PRINT 1020

PRINT 1010, ((AB(IROW,ICOL), ICOL = 1, N), IROW = 1, LDAB)

PRINT 1030

PRINT 1040, B

C

C Factor the matrix in banded form and print the results.

C

CALL DGBTRF (M, N, NSUB, NSUPER, AB, LDAB, IPIVOT, INFO)

IF (INFO .EQ. 0) THEN

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

$ AB, LDAB, IPIVOT, B, LDB, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1050

PRINT 1040, B

ELSE

PRINT 1060, INFO

END IF

ELSE

PRINT 1070, INFO

END IF

C

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

1010 FORMAT (4(3X, F4.1))

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

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

1040 FORMAT (1X, 2X, F4.1)

1050 FORMAT (/1X, 'x:')

1060 FORMAT (1X, 'Error in solving system, INFO = ', I4)

1070 FORMAT (1X, 'Error in factoring system, INFO = ', I4)

C

END

Sample Output

A in full form:
2.0 -1.0 0.0 0.0
-1.0 2.0 -1.0 0.0
0.0 -1.0 2.0 -1.0
0.0 0.0 -1.0 2.0
A in banded form: (* in unused elements)
**** **** **** ****
**** -1.0 -1.0 -1.0
2.0 2.0 2.0 2.0
-1.0 -1.0 -1.0 ****
b:
5.0
5.0
5.0
5.0
x:
10.0
15.0
15.0
10.0