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LAPACK version 3.0
csytrf(3)

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

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

      CHARACTER	     UPLO

      INTEGER	     INFO, LDA, LWORK, N

      INTEGER	     IPIV( * )

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

DESCRIPTION
  CSYTRF 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 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 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).