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CPOSVX(3F)							    CPOSVX(3F)

NAME
     CPOSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to
     compute the solution to a complex system of linear equations  A * X = B,

SYNOPSIS
     SUBROUTINE CPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
			LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

	 CHARACTER	EQUED, FACT, UPLO

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

	 REAL		RCOND

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

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

PURPOSE
     CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
     compute the solution to a complex system of linear equations
	A * X = B, where A is an N-by-N Hermitian positive definite 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 = 'E', real scaling factors are computed to equilibrate
	the system:
	   diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
	Whether or not the system will be equilibrated depends on the
	scaling of the matrix A, but if equilibration is used, A is
	overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

     2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
	factor the matrix A (after equilibration if FACT = 'E') as
	   A = U**H* U,	 if UPLO = 'U', or
	   A = L * L**H,  if UPLO = 'L',
	where U is an upper triangular matrix and L is a lower triangular
	matrix.

     3. 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, steps 4-6 are skipped.

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

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CPOSVX(3F)							    CPOSVX(3F)

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

     6. If equilibration was used, the matrix X is premultiplied by
	diag(S) so that it solves the original system before
	equilibration.

ARGUMENTS
     FACT    (input) CHARACTER*1
	     Specifies whether or not the factored form of the matrix A is
	     supplied on entry, and if not, whether the matrix A should be
	     equilibrated before it is factored.  = 'F':  On entry, AF
	     contains the factored form of A.  If EQUED = 'Y', the matrix A
	     has been equilibrated with scaling factors given by S.  A and AF
	     will not be modified.  = 'N':  The matrix A will be copied to AF
	     and factored.
	     = 'E':  The matrix A will be equilibrated if necessary, then
	     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/output) COMPLEX array, dimension (LDA,N)
	     On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED
	     = 'Y', then A must contain the equilibrated matrix
	     diag(S)*A*diag(S).	 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.	 A is
	     not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
	     'N' on exit.

	     On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
	     diag(S)*A*diag(S).

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

									Page 2

CPOSVX(3F)							    CPOSVX(3F)

     AF	     (input or output) COMPLEX array, dimension (LDAF,N)
	     If FACT = 'F', then AF is an input argument and on entry contains
	     the triangular factor U or L from the Cholesky factorization A =
	     U**H*U or A = L*L**H, in the same storage format as A.  If EQUED
	     .ne. 'N', then AF is the factored form of the equilibrated matrix
	     diag(S)*A*diag(S).

	     If FACT = 'N', then AF is an output argument and on exit returns
	     the triangular factor U or L from the Cholesky factorization A =
	     U**H*U or A = L*L**H of the original matrix A.

	     If FACT = 'E', then AF is an output argument and on exit returns
	     the triangular factor U or L from the Cholesky factorization A =
	     U**H*U or A = L*L**H of the equilibrated matrix A (see the
	     description of A for the form of the equilibrated matrix).

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

     EQUED   (input or output) CHARACTER*1
	     Specifies the form of equilibration that was done.	 = 'N':	 No
	     equilibration (always true if FACT = 'N').
	     = 'Y':  Equilibration was done, i.e., A has been replaced by
	     diag(S) * A * diag(S).  EQUED is an input argument if FACT = 'F';
	     otherwise, it is an output argument.

     S	     (input or output) REAL array, dimension (N)
	     The scale factors for A; not accessed if EQUED = 'N'.  S is an
	     input argument if FACT = 'F'; otherwise, S is an output argument.
	     If FACT = 'F' and EQUED = 'Y', each element of S must be
	     positive.

     B	     (input/output) COMPLEX array, dimension (LDB,NRHS)
	     On entry, the N-by-NRHS righthand side matrix B.  On exit, if
	     EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten
	     by diag(S) * 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, the N-by-NRHS solution matrix X to the original
	     system of equations.  Note that if EQUED = 'Y', A and B are
	     modified on exit, and the solution to the equilibrated system is
	     inv(diag(S))*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
	     after equilibration (if done).  If RCOND is less than the machine

									Page 3

CPOSVX(3F)							    CPOSVX(3F)

	     precision (in particular, if RCOND = 0), the matrix is singular
	     to working precision.  This condition is indicated by a return
	     code of INFO > 0, and the solution and error bounds are not
	     computed.

     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) COMPLEX array, dimension (2*N)

     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: the leading minor of order i of A is not positive definite,
	     so the factorization could not be completed, and the solution and
	     error bounds could not be computed.  = N+1: RCOND is less than
	     machine precision.	 The factorization has been completed, but the
	     matrix is singular to working precision, and the solution and
	     error bounds have not been computed.

									Page 4

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