CPBSTF  - compute a split Cholesky factorization of a com­
       plex Hermitian positive definite band matrix A


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        INFO, KD, LDAB, N

           COMPLEX        AB( LDAB, * )


PURPOSE

       CPBSTF computes a split Cholesky factorization of  a  com­
       plex Hermitian positive definite band matrix A.  This rou­
       tine 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) 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 array, dimension (LDAB,N)
               On  entry, the upper or lower triangle of the Her­
               mitian 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


       INFO    (output) INTEGER
               = 0: successful exit
               < 0: if INFO = -i, the i-th argument had an  ille­
               gal 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.


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