zpbstf(3S)

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      UPLO

         INTEGER        INFO, KD, LDAB, N

         COMPLEX*16     AB( LDAB, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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