ZSYTRF (3lapack)

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( LWORK )

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.

  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).