SPBSTF(3)                  MathKeisan LAPACK routine                 SPBSTF(3)



NAME
       SPBSTF  -  a  split Cholesky factorization of a real symmetric positive
       definite band matrix A

SYNOPSIS
       SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, KD, LDAB, N

           REAL           AB( LDAB, * )

PURPOSE
       SPBSTF computes a split Cholesky factorization of a real symmetric pos-
       itive definite band matrix A.

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

       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  tri-
       angular 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) REAL array, dimension (LDAB,N)
               On entry, the upper or lower triangle  of  the  symmetric  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 fac-
               torization A = S**T*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.




 LAPACK routine (version 3.1)    November 2006                       SPBSTF(3)