zpbfa(3P)

NAME

zpbfa - compute a Cholesky factorization of a symmetric positive definite matrix A in banded storage. It is typical to follow a call to xPBFA with a call to xPBSL to solve Ax = b or to xPBDI to compute the determinant of A.

SYNOPSIS

CALL DPBFA (DA, LDA, N, NDIAG, INFO)

CALL SPBFA (SA, LDA, N, NDIAG, INFO)

CALL ZPBFA (ZA, LDA, N, NDIAG, INFO)

CALL CPBFA (CA, LDA, N, NDIAG, INFO)

void dpbfa(double ∗abd, long int lda, long int n, long int m,
long int ∗info)

void spbfa(float ∗abd, long int lda, long int n, long int m,
long int ∗info)

void zpbfa(doublecomplex ∗abd, long int lda, long int n,
long int m, long int ∗info)

void cpbfa(complex ∗abd, long int lda, long int n,
long int m, long int ∗info)

ARGUMENTS

xAOn entry, the upper triangle of the matrix A.
On exit, a Cholesky factorization of the matrix A.

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

NOrder of the matrix A. N >= 0.

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

INFOOn exit:
INFO = 0Subroutine completed normally.
INFO ∗ 0Returns a value of k if the leading minor of order k is not positive definite.

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), B(N)
      INTEGER           ICOL, INFO, IROW
C
      EXTERNAL          DPBFA, DPBSL
C
C     Initialize the array A to store in banded storage mode
C     the matrix A shown below. Initialize the array B to
C     store the vector B shown below.
C
C          2 -1   0   0        60
C     A = -1   2 -1   0    b = 60
C          0 -1   2 -1        60
C          0   0 -1   2        60
C
      DATA A / 8D8, 2.0D0, -1.0D0, 2.0D0, -1.0D0, 2.0D0, -1.0D0, 2.0D0 /
      DATA B / 4∗6.0D1 /
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)
      PRINT 1050
      PRINT 1060, B
      CALL DPBFA (A, LDA, N, NDIAG, INFO)
      IF (INFO .EQ. 0) THEN
        CALL DPBSL (A, LDA, N, NDIAG, B)
        PRINT 1070
        PRINT 1060, B
      ELSE
        PRINT 1080
      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, ’b:’)
1060 FORMAT (3X, F5.1)
1070 FORMAT (/1X, ’A∗∗(-1) ∗ b:’)
1080 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

b:
    60.0
    60.0
    60.0
    60.0

A∗∗(-1) ∗ b:
   120.0
   180.0
   180.0
   120.0