ZHETRF  - compute the factorization of a complex Hermitian
       matrix A using the Bunch-Kaufman diagonal pivoting method


SYNOPSIS

       SUBROUTINE ZHETRF( UPLO, N, A,  LDA,  IPIV,  WORK,  LWORK,
                          INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDA, LWORK, N

           INTEGER        IPIV( * )

           COMPLEX*16     A( LDA, * ), WORK( * )


PURPOSE

       ZHETRF  computes  the factorization of a complex Hermitian
       matrix A using the Bunch-Kaufman diagonal pivoting method.
       The form of the factorization is

          A = U*D*U**H  or  A = L*D*L**H

       where  U (or L) is a product of permutation and unit upper
       (lower) triangular matrices, and D is Hermitian and  block
       diagonal with 1-by-1 and 2-by-2 diagonal blocks.

       This  is  the  blocked  version  of the algorithm, calling
       Level 3 BLAS.


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.

       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
               On  entry, the Hermitian matrix A.  If UPLO = 'U',
               the leading N-by-N upper triangular part of A con­
               tains  the  upper triangular part of the matrix A,
               and the strictly lower triangular part of A is not
               referenced.   If  UPLO  =  'L', the leading N-by-N
               lower triangular part of A contains the lower tri­
               angular  part  of  the  matrix A, and the strictly
               upper triangular part of A is not referenced.

               On exit, the block diagonal matrix D and the  mul­
               tipliers  used  to  obtain  the factor U or L (see
               below for further details).

               The leading dimension of  the  array  A.   LDA  >=
               max(1,N).

       IPIV    (output) INTEGER array, dimension (N)
               Details  of  the interchanges and the block struc­
               ture of D.  If IPIV(k) > 0, then rows and  columns
               k  and  IPIV(k)  were interchanged and D(k,k) is a
               1-by-1 diagonal block.  If UPLO = 'U' and  IPIV(k)
               =  IPIV(k-1)  <  0,  then rows and columns k-1 and
               -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a
               2-by-2  diagonal block.  If UPLO = 'L' and IPIV(k)
               = IPIV(k+1) < 0, then rows  and  columns  k+1  and
               -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a
               2-by-2 diagonal block.

       WORK    (workspace/output) COMPLEX*16 array, dimension
               (LWORK)
               On  exit, if INFO = 0, WORK(1) returns the optimal
               LWORK.

       LWORK   (input) INTEGER
               The length of WORK.  LWORK >=1.  For best  perfor­
               mance  LWORK  >=  N*NB, where NB is the block size
               returned by ILAENV.

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               >  0:   if  INFO = i, D(i,i) is exactly zero.  The
               factorization has been completed,  but  the  block
               diagonal  matrix  D is exactly singular, and divi­
               sion by zero will occur if it is used to  solve  a
               system of equations.


FURTHER DETAILS

       If UPLO = 'U', then A = U*D*U', where
          U = P(n)*U(n)* ... *P(k)U(k)* ...,
       i.e., U is a product of terms P(k)*U(k), where k decreases
       from n to 1 in steps of 1 or 2, and D is a block  diagonal
       matrix  with 1-by-1 and 2-by-2 diagonal blocks D(k).  P(k)
       is a permutation matrix as defined by IPIV(k), and U(k) is
       a  unit upper triangular matrix, such that if the diagonal
       block D(k) is of order s (s = 1 or 2), then

                  (   I    v    0   )   k-s
          U(k) =  (   0    I    0   )   s
                  (   0    0    I   )   n-k
                     k-s   s   n-k

       If s  =  1,  D(k)  overwrites  A(k,k),  and  v  overwrites
       A(1:k-1,k).   If  s  = 2, the upper triangle of D(k) over­
       writes A(k-1,k-1), A(k-1,k), and A(k,k), and v  overwrites


       If UPLO = 'L', then A = L*D*L', where
          L = P(1)*L(1)* ... *P(k)*L(k)* ...,
       i.e., L is a product of terms P(k)*L(k), where k increases
       from 1 to n in steps of 1 or 2, and D is a block  diagonal
       matrix  with 1-by-1 and 2-by-2 diagonal blocks D(k).  P(k)
       is a permutation matrix as defined by IPIV(k), and L(k) is
       a  unit lower triangular matrix, such that if the diagonal
       block D(k) is of order s (s = 1 or 2), then

                  (   I    0     0   )  k-1
          L(k) =  (   0    I     0   )  s
                  (   0    v     I   )  n-k-s+1
                     k-1   s  n-k-s+1

       If s  =  1,  D(k)  overwrites  A(k,k),  and  v  overwrites
       A(k+1:n,k).   If  s  = 2, the lower triangle of D(k) over­
       writes A(k,k), A(k+1,k), and A(k+1,k+1), and v  overwrites
       A(k+2:n,k:k+1).


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