zsytrf(3S)

NAME
     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( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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 contains 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 triangular 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 multipliers used to
             obtain the factor U or L (see below for further details).

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

     IPIV    (output) INTEGER array, dimension (N)
             Details of the interchanges and the block structure 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 performance 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 illegal 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 division 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) overwrites 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) overwrites A(k,k), A(k+1,k), and A(k+1,k+1),
     and v overwrites A(k+2:n,k:k+1).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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