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

Calling Sequence

CALL DGBSL	(DA, LDA, N, NSUB, NSUPER, IPIVOT, DB, JOB)
CALL SGBSL	(SA, LDA, N, NSUB, NSUPER, IPIVOT, SB, JOB)
CALL ZGBSL	(ZA, LDA, N, NSUB, NSUPER, IPIVOT, ZB, JOB)
CALL CGBSL	(CA, LDA, N, NSUB, NSUPER, IPIVOT, CB, JOB)
void dgbsl	(double *da, int lda, int n, int nsub, int nsuper, int *ipivot, double *db, int job)
void sgbsl	(float *sa, int lda, int n, int nsub, int nsuper, int *ipivot, float *sb, int job)
void zgbsl	(doublecomplex *za, int lda, int n, int nsub, int nsuper, int *ipivot, doublecomplex *zb, int job)
void cgbsl	(complex *ca, int lda, int n, int nsub, int nsuper, int *ipivot, complex *cb, int job)

Arguments

xA	LU factorization of the matrix A, as computed by xGBCO or xGBFA.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA 2 × NSUB + NSUPER + 1.
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.
IPIVOT	Pivot vector as computed by xGBCO or xGBFA.
xB	On entry, the right-hand side vector b. On exit, the solution vector x.
JOB	Determines which operation this subroutine will perform:
	0	solve the system Ax = b
	not 0	solve the linear system AHx = b
		Note that ATx = AHx for real matrices.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER IAXEQB, LDA, LDAB, N, NDIAG, NSUB, NSUPER

PARAMETER (IAXEQB = 0)

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (NSUB = 1)

PARAMETER (NSUPER = 1)

PARAMETER (NDIAG = NSUB + 1 + NSUPER)

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

C

DOUBLE PRECISION AB(LDAB,N), AG(LDA,N), B(N), RCOND, WORK(N)

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

C

EXTERNAL DGBCO, DGBSL

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*8D8 /

DATA AG / 2.0D0, -1.0D0, 2*0D0, -1.0D0, 2.0D0, -1.0D0,

$ 2*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 is taken from the

C comment block in the original DGBFA by Cleve Moler.

C

DO 10, ICOL = 1, N

I1 = MAX0 (1, ICOL - NSUPER)

I2 = MIN0 (N, ICOL + NSUB)

DO 10, IROW = I1, I2

IROWB = IROW - ICOL + NDIAG

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 = 2 * NSUB, 2 * NSUB + 1 + NSUPER)

PRINT 1030

PRINT 1040, B

C

C Factor the matrix in banded form.

C

CALL DGBCO (AB, LDA, N, NSUB, NSUPER, IPIVOT, RCOND, WORK)

PRINT 1050, RCOND

IF ((RCOND + 1.0D0) .EQ. 1.0D0) THEN

PRINT 1070

END IF

JOB = IAXEQB

CALL DGBSL (AB, LDA, N, NSUB, NSUPER, IPIVOT, B, JOB)

PRINT 1060

PRINT 1040, B

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 (3X, F4.1)

1050 FORMAT (/1X, 'Reciprocal of the condition number: ', F5.2)

1060 FORMAT (/1X, 'A**(-1) * b:')

1070 FORMAT (1X, 'A may be singular to working precision.')

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
Reciprocal of the condition number: 0.08
A**(-1) * b:
10.0
15.0
15.0
10.0