ZSYTRF  - compute the factorization of a complex symmetric
       matrix A using the Bunch-Kaufman diagonal pivoting method


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        INFO, LDA, LWORK, N

           INTEGER        IPIV( * )

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


PURPOSE

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

          A = U*D*U**T  or  A = L*D*L**T

       where  U (or L) is a product of permutation and unit upper
       (lower) triangular matrices, and D is symmetric and  block
       diagonal with 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 symmetric 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.

               If LWORK = -1, then a workspace query is  assumed;
               the  routine  only  calculates the optimal size of
               the WORK array, returns this value  as  the  first
               entry  of  the  WORK  array,  and no error message
               related to LWORK is issued by XERBLA.

       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

                     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
       A(1:k-2,k-1:k).

       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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