dpbdi(3P)

NAME

dpbdi - compute the determinant of a symmetric positive definite matrix A in banded storage, which has been Cholesky-factored by xPBCO or xPBFA.

SYNOPSIS

CALL DPBDI (DA, LDA, N, NDIAG, DDET)

CALL SPBDI (SA, LDA, N, NDIAG, SDET)

CALL ZPBDI (ZA, LDA, N, NDIAG, DDET)

CALL CPBDI (CA, LDA, N, NDIAG, SDET)

void dpbdi(double ∗abd, long int lda, long int n, long int m,
double ∗det)

void spbdi(float ∗abd, long int lda, long int n, long int m,
float ∗det)

void zpbdi(doublecomplex ∗abd, long int lda, long int n,
long int m, double ∗det)

void cpbdi(complex ∗abd, long int lda, long int n, long int m,
float ∗det)

ARGUMENTS

xACholesky factorization of the matrix A, as computed by xPBCO or xPBFA.

LDALeading dimension of the array A as specified in a dimension or type statement. LDA >= NDIAG + 1.

NOrder of the original matrix A. N >= 0.

NDIAGNumber of diagonals. N-1 >= NDIAG >= 0 but if N = 0 then NDIAG = 0.

xDETOn 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, N, NDIAG
      PARAMETER        (N = 4)
      PARAMETER        (NDIAG = 1)
      PARAMETER        (LDA = NDIAG + 1)
C
      DOUBLE PRECISION A(LDA,N), DET(2)
      INTEGER           ICOL, INFO, IROW, NDIAG
      EXTERNAL          DPBDI, DPBFA
C
C     Initialize the array A to store in banded storage mode
C     the matrix A shown below.
C
C          2 -1   0   0
C     A = -1   2 -1   0
C          0 -1   2 -1
C          0   0 -1   2
C
      DATA A / 8D8, 2.0D0, -1.0D0, 2.0D0, -1.0D0, 2.0D0, -1.0D0, 2.0D0 /
C
      PRINT 1000
      PRINT 1010, A(2,1), A(1,2)
      PRINT 1010, A(3,1), A(2,2), A(1,3)
      PRINT 1020,         A(3,2), A(2,3), A(1,4)
      PRINT 1030,                 A(3,3), A(2,4)
      PRINT 1040
      PRINT 1010, ((A(IROW,ICOL), ICOL = 1, N), IROW = 1, LDA)
      CALL DPBFA (A, LDA, N, NDIAG, INFO)
      IF (INFO .EQ. 0) THEN
        CALL DPBDI (A, LDA, N, NDIAG, DET)
        PRINT 1050, DET(1) ∗ (10.0D0 ∗∗ DET(2))
      ELSE
        PRINT 1060
      END IF
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(3X, F5.1))
1020 FORMAT (8X, 3(3X, F5.1))
1030 FORMAT (16X, 3(3X, F5.1))
1040 FORMAT (/1X, ’A in banded form: (∗ in unused entries)’)
1050 FORMAT (/1X, ’The determinant of A is ’, F5.1)
1060 FORMAT (/1X, ’A is not positive definite.’)
C
      END

SAMPLE OUTPUT

A in full form:
     2.0    -1.0
    -1.0     2.0    -1.0
            -1.0     2.0    -1.0
                    -1.0     2.0

A in banded form: (∗ in unused entries)
   ∗∗∗∗∗    -1.0    -1.0    -1.0
     2.0     2.0     2.0     2.0

The determinant of A is   5.0