LU Factorization and Condition Number of a General Matrix in Banded Storage

Calling Sequence

CALL DGBCO	(DA, LDA, N, NSUB, NSUPER, IPIVOT, DRCOND, DWORK)
CALL SGBCO	(SA, LDA, N, NSUB, NSUPER, IPIVOT, SRCOND, SWORK)
CALL ZGBCO	(ZA, LDA, N, NSUB, NSUPER, IPIVOT, DRCOND, ZWORK)
CALL CGBCO	(CA, LDA, N, NSUB, NSUPER, IPIVOT, SRCOND, CWORK)
void dgbco	(double *da, int lda, int n, int nsub, int nsuper, int *ipivot, double *drcond)
void sgbco	(float *sa, int lda, int n, int nsub, int nsuper, int *ipivot, float *srcond)
void zgbco	(doublecomplex *za, int lda, int n, int nsub, int nsuper, int *ipivot, double *drcond)
void cgbco	(complex *ca, int lda, int n, int nsub, int nsuper, int *ipivot, float *srcond)

Arguments

xA	On entry, the matrix A. On exit, an LU factorization of the matrix A.
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	On exit, a vector of pivot indices.
xRCOND	On exit, an estimate of the reciprocal condition number of A. 0.0 RCOND 1.0. As the value of RCOND gets smaller, operations with A such as solving Ax = b may become less stable. If RCOND satisfies RCOND + 1.0 = 1.0 then A may be singular to working precision.
xWORK	Scratch array with a dimension of N.

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