CPBSTF(3)      LAPACK routine of NEC Numeric Library Collection      CPBSTF(3)



NAME
       CPBSTF

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



PURPOSE
            CPBSTF computes a split Cholesky factorization of a complex
            Hermitian positive definite band matrix A.

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

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

           KD        (input)
                     KD is 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)
                     AB is COMPLEX 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)
                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           INFO      (output)
                     INFO is 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**H s64**H s75**H
              *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
             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**H s23**H s34**H s54  s65  s76   *
             a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *

             Array elements marked * are not used by the routine; s12**H denotes
             conjg(s12); the diagonal elements of S are real.



LAPACK routine                  31 October 2017                      CPBSTF(3)