Sun Performance Library Reference Manual: 2

Determinant of an LU-Factored General Matrix in Banded Storage

The subroutines described in this section compute the determinant of a general matrix A in banded storage, which has been LU-factored by xGBCO or xGBFA.

Calling Sequence


CALL DGBDI	(DA, LDA, N, NSUB, NSUPER, IPIVOT, DDET)
CALL SGBDI	(SA, LDA, N, NSUB, NSUPER, IPIVOT, SDET)
CALL ZGBDI	(ZA, LDA, N, NSUB, NSUPER, IPIVOT, ZDET)
CALL CGBDI	(CA, LDA, N, NSUB, NSUPER, IPIVOT, CDET)

void dgbdi	(double da, int lda, int n, int nsub, int nsuper, int ipivot, double *ddet)
void sgbdi	(float sa, int lda, int n, int nsub, int nsuper, int ipivot, float *sdet)
void zgbdi	(doublecomplex za, int lda, int n, int nsub, int nsuper, int ipivot, doublecomplex *zdet)
void cgbdi	(complex ca, int lda, int n, int nsub, int nsuper, int ipivot, complex *cdet)

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 original 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.

xDET

On exit, the determinant of the matrix A. The determinant is stored as b × 10^expon where b is stored in DET(1) and expon is stored in DET(2). 1.0 |DET(1)| < 10.0 or DET(1) = 0.0.

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 original 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.
xDET	On exit, the determinant of the matrix A. The determinant is stored as b × 10^expon where b is stored in DET(1) and expon is stored in DET(2). 1.0 \|DET(1)\| < 10.0 or DET(1) = 0.0.

Sample Program



PROGRAM TEST
IMPLICIT NONE
C
INTEGER LDA, LDAB, LENGAB, N, NDIAG, NSUB, NSUPER
PARAMETER (N = 5)
PARAMETER (NSUB = 4)
PARAMETER (NSUPER = 0)
PARAMETER (NDIAG = NSUB + 1 + NSUPER)
PARAMETER (LDA = N)
PARAMETER (LDAB = 2 * NSUB + 1 + NDIAG)
PARAMETER (LENGAB = LDAB * N)
C
DOUBLE PRECISION A(LDA,N), AB(LDAB, N), DET(2)
INTEGER ICOL, INFO, IPIVOT(N), IROW, IROWB, I1, I2
C
EXTERNAL DGBDI, DGBFA
C
C Initialize the array A to store the 5x5 matrix A with four
C subdiagonals and no superdiagonals shown below.
C
C 1
C 1 -1
C A = 1 -2 1
C 1 -3 3 -1
C 1 -4 6 -4 1
C
DATA A / 1.0D0, 1.0D0, 1.0D0, 1.0D0, 1.0D0,
$ 0.0D0, -1.0D0, -2.0D0, -3.0D0, -4.0D0,
$ 0.0D0, 0.0D0, 1.0D0, 3.0D0, 6.0D0,
$ 0.0D0, 0.0D0, 0.0D0, -1.0D0, -4.0D0,
$ 0.0D0, 0.0D0, 0.0D0, 0.0D0, 1.0D0 /
DATA AB / LENGAB*8D8 /

DO 20, 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) = A(IROW, ICOL)
10 CONTINUE
20 CONTINUE
C
C Print the initial values of the arrays.
C
PRINT 1000
PRINT 1010, ((A(IROW, ICOL), ICOL = 1, N), IROW = 1, N)
PRINT 1020
PRINT 1010, ((AB(IROW, ICOL), ICOL = 1, N),
$ IROW = 1, 2 * NSUB + 1 + NSUPER)
C
C Factor the matrix in banded form.
C
CALL DGBFA (AB, LDAB, N, NSUB, NSUPER, IPIVOT, INFO)
IF (INFO .EQ. 0) THEN
CALL DGBDI (AB, LDAB, N, NSUB, NSUPER, IPIVOT, DET)
PRINT 1030, DET(1) * (1.0D0 ** DET(2))
ELSE
PRINT 1040, INFO
END IF
C
1000 FORMAT (1X, 'A in full form:')
1010 FORMAT (5(3X, F4.1))
1020 FORMAT (/1X, 'A in banded form: (* in unused elements)')
1030 FORMAT (/1X, 'det(A) = ', F4.1)
1040 FORMAT (/1X, 'A appears singular at ', I2)
C
END

Sample Output



A in full form:
1.0 0.0 0.0 0.0 0.0
1.0 -1.0 0.0 0.0 0.0
1.0 -2.0 1.0 0.0 0.0
1.0 -3.0 3.0 -1.0 0.0
1.0 -4.0 6.0 -4.0 1.0

A in banded form: (* in unused elements)
** **
** **
** **
** **
1.0 -1.0 1.0 -1.0 1.0
1.0 -2.0 3.0 -4.0 ****
1.0 -3.0 6.0 ** **
1.0 -4.0 ** **
1.0 ** **

det(A) = 1.0