DPBSTF  - compute a split Cholesky factorization of a real
       symmetric positive definite band matrix A


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        INFO, KD, LDAB, N

           DOUBLE         PRECISION AB( LDAB, * )


PURPOSE

       DPBSTF computes a split Cholesky factorization of  a  real
       symmetric  positive  definite band matrix A.  This routine
       is designed to be used in conjunction with DSBGST.

       The factorization has the form  A = S**T*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) DOUBLE PRECISION array, dimension
               (LDAB,N)
               On  entry, the upper or lower triangle of the sym­
               metric 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**T*S.  See  Further

               dimension of the array AB.  LDAB >= KD+1.

       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.


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