CXML
        	    
LAPACK version 3.0
zhesvx(3)

PURPOSE
  ZHESVX - use the diagonal pivoting factorization to compute the solution to
  a complex system of linear equations A * X = B,

SYNTAX
  SUBROUTINE ZHESVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
		     LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO )

      CHARACTER	     FACT, UPLO

      INTEGER	     INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

      DOUBLE	     PRECISION RCOND

      INTEGER	     IPIV( * )

      DOUBLE	     PRECISION BERR( * ), FERR( * ), RWORK( * )

      COMPLEX*16     A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
		     LDX, * )

DESCRIPTION
  ZHESVX uses the diagonal pivoting factorization to compute the solution to
  a complex system of linear equations A * X = B, where A is an N-by-N
  Hermitian matrix and X and B are N-by-NRHS matrices.

  Error bounds on the solution and a condition estimate are also provided.

DESCRIPTION
  The following steps are performed:

  1. If FACT = 'N', the diagonal pivoting method is used to factor A.
     The form of the factorization is
	A = U * D * U**H,  if UPLO = 'U', or
	A = L * D * L**H,  if UPLO = 'L',
     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.

  2. If some D(i,i)=0, so that D is exactly singular, then the routine
     returns with INFO = i. Otherwise, the factored form of A is used
     to estimate the condition number of the matrix A.	If the
     reciprocal of the condition number is less than machine precision,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  3. The system of equations is solved for X using the factored form
     of A.

  4. Iterative refinement is applied to improve the computed solution
     matrix and calculate error bounds and backward error estimates
     for it.

ARGUMENTS
  FACT	  (input) CHARACTER*1
	  Specifies whether or not the factored form of A has been supplied
	  on entry.  = 'F':  On entry, AF and IPIV contain the factored form
	  of A.	 A, AF and IPIV will not be modified.  = 'N':  The matrix A
	  will be copied to AF and factored.

  UPLO	  (input) CHARACTER*1
	  = 'U':  Upper triangle of A is stored;
	  = 'L':  Lower triangle of A is stored.

  N	  (input) INTEGER
	  The number of linear equations, i.e., the order of the matrix A.  N
	  >= 0.

  NRHS	  (input) INTEGER
	  The number of right hand sides, i.e., the number of columns of the
	  matrices B and X.  NRHS >= 0.

  A	  (input) COMPLEX*16 array, dimension (LDA,N)
	  The Hermitian 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.

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

  AF	  (input or output) COMPLEX*16 array, dimension (LDAF,N)
	  If FACT = 'F', then AF is an input argument and on entry contains
	  the block diagonal matrix D and the multipliers used to obtain the
	  factor U or L from the factorization A = U*D*U**H or A = L*D*L**H
	  as computed by ZHETRF.

	  If FACT = 'N', then AF is an output argument and on exit returns
	  the block diagonal matrix D and the multipliers used to obtain the
	  factor U or L from the factorization A = U*D*U**H or A = L*D*L**H.

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

  IPIV	  (input or output) INTEGER array, dimension (N)
	  If FACT = 'F', then IPIV is an input argument and on entry contains
	  details of the interchanges and the block structure of D, as
	  determined by ZHETRF.	 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.

	  If FACT = 'N', then IPIV is an output argument and on exit contains
	  details of the interchanges and the block structure of D, as
	  determined by ZHETRF.

  B	  (input) COMPLEX*16 array, dimension (LDB,NRHS)
	  The N-by-NRHS right hand side matrix B.

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

  X	  (output) COMPLEX*16 array, dimension (LDX,NRHS)
	  If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

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

  RCOND	  (output) DOUBLE PRECISION
	  The estimate of the reciprocal condition number of the matrix A.
	  If RCOND is less than the machine precision (in particular, if
	  RCOND = 0), the matrix is singular to working precision.  This
	  condition is indicated by a return code of INFO > 0.

  FERR	  (output) DOUBLE PRECISION array, dimension (NRHS)
	  The estimated forward error bound for each solution vector X(j)
	  (the j-th column of the solution matrix X).  If XTRUE is the true
	  solution corresponding to X(j), FERR(j) is an estimated upper bound
	  for the magnitude of the largest element in (X(j) - XTRUE) divided
	  by the magnitude of the largest element in X(j).  The estimate is
	  as reliable as the estimate for RCOND, and is almost always a
	  slight overestimate of the true error.

  BERR	  (output) DOUBLE PRECISION array, dimension (NRHS)
	  The componentwise relative backward error of each solution vector
	  X(j) (i.e., the smallest relative change in any element of A or B
	  that makes X(j) an exact solution).

  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 >= 2*N, and for best performance LWORK
	  >= N*NB, where NB is the optimal blocksize for ZHETRF.

	  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.

  RWORK	  (workspace) DOUBLE PRECISION array, dimension (N)

  INFO	  (output) INTEGER
	  = 0: successful exit
	  < 0: if INFO = -i, the i-th argument had an illegal value
	  > 0: if INFO = i, and i is
	  <= N:	 D(i,i) is exactly zero.  The factorization has been
	  completed but the factor D is exactly singular, so the solution and
	  error bounds could not be computed. RCOND = 0 is returned.  = N+1:
	  D is nonsingular, but RCOND is less than machine precision, meaning
	  that the matrix is singular to working precision.  Nevertheless,
	  the solution and error bounds are computed because there are a
	  number of situations where the computed solution can be more
	  accurate than the value of RCOND would suggest.