ZPBSTF (3lapack)

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

SYNOPSIS

  SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, KD, LDAB, N

      COMPLEX*16     AB( LDAB, * )

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 ZHBGST.

  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) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U', or the
          number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
          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)
          INTEGER The leading dimension of the array AB.  LDAB >= KD+1.

  INFO    (output) INTEGER
          = 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.