NAME

zpbstf - compute a split Cholesky factorization of a complex Hermitian positive definite band matrix A

SYNOPSIS

  SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX AB(LDAB,*)
  INTEGER N, KD, LDAB, INFO

  SUBROUTINE ZPBSTF_64( UPLO, N, KD, AB, LDAB, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX AB(LDAB,*)
  INTEGER*8 N, KD, LDAB, INFO

F95 INTERFACE

  SUBROUTINE PBSTF( UPLO, [N], KD, AB, [LDAB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:,:) :: AB
  INTEGER :: N, KD, LDAB, INFO

  SUBROUTINE PBSTF_64( UPLO, [N], KD, AB, [LDAB], [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:,:) :: AB
  INTEGER(8) :: N, KD, LDAB, INFO

C INTERFACE

#include <sunperf.h>

void zpbstf(char uplo, int n, int *kd, doublecomplex *ab, int ldab, int *info);

void zpbstf_64(char uplo, long n, long *kd, doublecomplex *ab, long ldab, long *info);

PURPOSE

zpbstf computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A.

This routine is designed to be used in conjunction with CHBGST.

The factorization has the form A = S**H*S where S is a band matrix of the same bandwidth as A and the following structure:

  S = ( U    )

      ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

ARGUMENTS

UPLO (input)

 = 'U':  Upper triangle of A is stored;

 = 'L':  Lower triangle of A is stored.

N (input)
The order of the matrix A. N > = 0.
KD (input/output)
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD > = 0.
AB (input/output)
On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd) < =i < =j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j < =i < =min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky factorization A = S**H*S. See Further Details.
LDAB (input)
The leading dimension of the array AB. LDAB > = KD+1.

INFO (output)

 = 0: successful exit

 < 0: if INFO  = -i, the i-th argument had an illegal value

 > 0: if INFO  = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.

FURTHER DETAILS

The band storage scheme is illustrated by the following example, when N = 7, KD = 2:

S = ( s11 s12 s13 )

    (      s22  s23  s24                )

    (           s33  s34                )

    (                s44                )

    (           s53  s54  s55           )

    (                s64  s65  s66      )

    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry: on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53' s64' s75'
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54' s65' s76'
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

If UPLO = 'L', the array AB holds:

on entry: on exit:

a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * *

Array elements marked * are not used by the routine; s12' denotes conjg(s12); the diagonal elements of S are real.