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Determinant of a Cholesky-Factored Symmetric Positive Definite Matrix in Banded Storage

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

Calling Sequence

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 *da, int lda, int n, int ndiag, double *ddet)
void spbdi 
(float *sa, int lda, int n, int ndiag, float *sdet)
void zpbdi 
(doublecomplex *za, int lda, int n, int ndiag, double 
*ddet)
void cpbdi 
(complex *ca, int lda, int n, int ndiag, float *sdet)

Arguments

xA

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

LDA

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

N

Order of the original matrix A. N 0.

NDIAG

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

xDET

On exit, the determinant of the matrix A. The determinant is stored as b × 10expon 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






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