CHESVX(l)								 )								 CHESVX(l)

NAME

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

SYNOPSIS

       SUBROUTINE CHESVX( 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

	   REAL 	  RCOND

	   INTEGER	  IPIV( * )

	   REAL 	  BERR( * ), FERR( * ), RWORK( * )

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

PURPOSE

       CHESVX 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 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 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 CHETRF.

	       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 CHETRF.  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 CHETRF.

       B       (input) COMPLEX 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 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) REAL
	       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) REAL 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) REAL 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 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 CHETRF.

	       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) REAL 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.

LAPACK version 3.0						   15 June 2000 							 CHESVX(l)