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cposvxx.f(3)				      LAPACK				     cposvxx.f(3)

NAME
       cposvxx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine cposvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX,
	   RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK,
	   RWORK, INFO)
	    CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Function/Subroutine Documentation
   subroutine cposvxx (characterFACT, characterUPLO, integerN, integerNRHS, complex, dimension(
       lda, * )A, integerLDA, complex, dimension( ldaf, * )AF, integerLDAF, characterEQUED, real,
       dimension( * )S, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X,
       integerLDX, realRCOND, realRPVGRW, real, dimension( * )BERR, integerN_ERR_BNDS, real,
       dimension( nrhs, * )ERR_BNDS_NORM, real, dimension( nrhs, * )ERR_BNDS_COMP,
       integerNPARAMS, real, dimension( * )PARAMS, complex, dimension( * )WORK, real, dimension(
       * )RWORK, integerINFO)
	CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

       Purpose:

	       CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
	       to compute the solution to a complex system of linear equations
	       A * X = B, where A is an N-by-N symmetric positive definite matrix
	       and X and B are N-by-NRHS matrices.

	       If requested, both normwise and maximum componentwise error bounds
	       are returned. CPOSVXX will return a solution with a tiny
	       guaranteed error (O(eps) where eps is the working machine
	       precision) unless the matrix is very ill-conditioned, in which
	       case a warning is returned. Relevant condition numbers also are
	       calculated and returned.

	       CPOSVXX accepts user-provided factorizations and equilibration
	       factors; see the definitions of the FACT and EQUED options.
	       Solving with refinement and using a factorization from a previous
	       CPOSVXX call will also produce a solution with either O(eps)
	       errors or warnings, but we cannot make that claim for general
	       user-provided factorizations and equilibration factors if they
	       differ from what CPOSVXX would itself produce.

       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**T* U,	if UPLO = 'U', or
		  A = L * L**T,  if UPLO = 'L',
	       where U is an upper triangular matrix and L is a lower triangular
	       matrix.

	       3. If the leading i-by-i principal minor is not positive definite,
	       then the routine returns with INFO = i. Otherwise, the factored
	       form of A is used to estimate the condition number of the matrix
	       A (see argument RCOND).	If the reciprocal of the condition number
	       is less than machine precision, the routine still goes on to solve
	       for X and compute error bounds as described below.

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

	       5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
	       the routine will use iterative refinement to try to get a small
	       error and error bounds.	Refinement calculates the residual to at
	       least twice the working precision.

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

		Some optional parameters are bundled in the PARAMS array.  These
		settings determine how refinement is performed, but often the
		defaults are acceptable.  If the defaults are acceptable, users
		can pass NPARAMS = 0 which prevents the source code from accessing
		the PARAMS argument.

       Parameters:
	   FACT

		     FACT is 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 is not 'N', the matrix A has been
			  equilibrated with scaling factors given by S.
			  A and AF are not 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

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

	   N

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

	   NRHS

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

	   A

		     A is COMPLEX array, dimension (LDA,N)
		On entry, the symmetric 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

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

	   AF

		     AF is 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**T*U or A = L*L**T, 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**T*U or A = L*L**T 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**T*U or A = L*L**T of the equilibrated
		matrix A (see the description of A for the form of the
		equilibrated matrix).

	   LDAF

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

	   EQUED

		     EQUED is CHARACTER*1
		Specifies the form of equilibration that was done.
		  = 'N':  No equilibration (always true if FACT = 'N').
		  = 'Y':  Both row and column equilibration, 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

		     S is REAL array, dimension (N)
		The row scale factors for A.  If EQUED = 'Y', A is multiplied on
		the left and right by diag(S).	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.  If S is output, each
		element of S is a power of the radix. If S is input, each element
		of S should be a power of the radix to ensure a reliable solution
		and error estimates. Scaling by powers of the radix does not cause
		rounding errors unless the result underflows or overflows.
		Rounding errors during scaling lead to refining with a matrix that
		is not equivalent to the input matrix, producing error estimates
		that may not be reliable.

	   B

		     B is COMPLEX array, dimension (LDB,NRHS)
		On entry, the N-by-NRHS right hand side matrix B.
		On exit,
		if EQUED = 'N', B is not modified;
		if EQUED = 'Y', B is overwritten by diag(S)*B;

	   LDB

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

	   X

		     X is COMPLEX array, dimension (LDX,NRHS)
		If INFO = 0, the N-by-NRHS solution matrix X to the original
		system of equations.  Note that A and B are modified on exit if
		EQUED .ne. 'N', and the solution to the equilibrated system is
		inv(diag(S))*X.

	   LDX

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

	   RCOND

		     RCOND is REAL
		Reciprocal scaled condition number.  This is an estimate of the
		reciprocal Skeel condition number of the matrix A after
		equilibration (if done).  If this is less than the machine
		precision (in particular, if it is zero), the matrix is singular
		to working precision.  Note that the error may still be small even
		if this number is very small and the matrix appears ill-
		conditioned.

	   RPVGRW

		     RPVGRW is REAL
		Reciprocal pivot growth.  On exit, this contains the reciprocal
		pivot growth factor norm(A)/norm(U). The "max absolute element"
		norm is used.  If this is much less than 1, then the stability of
		the LU factorization of the (equilibrated) matrix A could be poor.
		This also means that the solution X, estimated condition numbers,
		and error bounds could be unreliable. If factorization fails with
		0<INFO<=N, then this contains the reciprocal pivot growth factor
		for the leading INFO columns of A.

	   BERR

		     BERR is REAL array, dimension (NRHS)
		Componentwise relative backward error.	This is 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).

	   N_ERR_BNDS

		     N_ERR_BNDS is INTEGER
		Number of error bounds to return for each right hand side
		and each type (normwise or componentwise).  See ERR_BNDS_NORM and
		ERR_BNDS_COMP below.

	   ERR_BNDS_NORM

		     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
		For each right-hand side, this array contains information about
		various error bounds and condition numbers corresponding to the
		normwise relative error, which is defined as follows:

		Normwise relative error in the ith solution vector:
			max_j (abs(XTRUE(j,i) - X(j,i)))
		       ------------------------------
			     max_j abs(X(j,i))

		The array is indexed by the type of error information as described
		below. There currently are up to three pieces of information
		returned.

		The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
		right-hand side.

		The second index in ERR_BNDS_NORM(:,err) contains the following
		three fields:
		err = 1 "Trust/don't trust" boolean. Trust the answer if the
			 reciprocal condition number is less than the threshold
			 sqrt(n) * slamch('Epsilon').

		err = 2 "Guaranteed" error bound: The estimated forward error,
			 almost certainly within a factor of 10 of the true error
			 so long as the next entry is greater than the threshold
			 sqrt(n) * slamch('Epsilon'). This error bound should only
			 be trusted if the previous boolean is true.

		err = 3  Reciprocal condition number: Estimated normwise
			 reciprocal condition number.  Compared with the threshold
			 sqrt(n) * slamch('Epsilon') to determine if the error
			 estimate is "guaranteed". These reciprocal condition
			 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
			 appropriately scaled matrix Z.
			 Let Z = S*A, where S scales each row by a power of the
			 radix so all absolute row sums of Z are approximately 1.

		See Lapack Working Note 165 for further details and extra
		cautions.

	   ERR_BNDS_COMP

		     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
		For each right-hand side, this array contains information about
		various error bounds and condition numbers corresponding to the
		componentwise relative error, which is defined as follows:

		Componentwise relative error in the ith solution vector:
			       abs(XTRUE(j,i) - X(j,i))
			max_j ----------------------
				    abs(X(j,i))

		The array is indexed by the right-hand side i (on which the
		componentwise relative error depends), and the type of error
		information as described below. There currently are up to three
		pieces of information returned for each right-hand side. If
		componentwise accuracy is not requested (PARAMS(3) = 0.0), then
		ERR_BNDS_COMP is not accessed.	If N_ERR_BNDS .LT. 3, then at most
		the first (:,N_ERR_BNDS) entries are returned.

		The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
		right-hand side.

		The second index in ERR_BNDS_COMP(:,err) contains the following
		three fields:
		err = 1 "Trust/don't trust" boolean. Trust the answer if the
			 reciprocal condition number is less than the threshold
			 sqrt(n) * slamch('Epsilon').

		err = 2 "Guaranteed" error bound: The estimated forward error,
			 almost certainly within a factor of 10 of the true error
			 so long as the next entry is greater than the threshold
			 sqrt(n) * slamch('Epsilon'). This error bound should only
			 be trusted if the previous boolean is true.

		err = 3  Reciprocal condition number: Estimated componentwise
			 reciprocal condition number.  Compared with the threshold
			 sqrt(n) * slamch('Epsilon') to determine if the error
			 estimate is "guaranteed". These reciprocal condition
			 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
			 appropriately scaled matrix Z.
			 Let Z = S*(A*diag(x)), where x is the solution for the
			 current right-hand side and S scales each row of
			 A*diag(x) by a power of the radix so all absolute row
			 sums of Z are approximately 1.

		See Lapack Working Note 165 for further details and extra
		cautions.

	   NPARAMS

		     NPARAMS is INTEGER
		Specifies the number of parameters set in PARAMS.  If .LE. 0, the
		PARAMS array is never referenced and default values are used.

	   PARAMS

		     PARAMS is REAL array, dimension NPARAMS
		Specifies algorithm parameters.  If an entry is .LT. 0.0, then
		that entry will be filled with default value used for that
		parameter.  Only positions up to NPARAMS are accessed; defaults
		are used for higher-numbered parameters.

		  PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
		       refinement or not.
		    Default: 1.0
		       = 0.0 : No refinement is performed, and no error bounds are
			       computed.
		       = 1.0 : Use the double-precision refinement algorithm,
			       possibly with doubled-single computations if the
			       compilation environment does not support DOUBLE
			       PRECISION.
			 (other values are reserved for future use)

		  PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
		       computations allowed for refinement.
		    Default: 10
		    Aggressive: Set to 100 to permit convergence using approximate
				factorizations or factorizations other than LU. If
				the factorization uses a technique other than
				Gaussian elimination, the guarantees in
				err_bnds_norm and err_bnds_comp may no longer be
				trustworthy.

		  PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
		       will attempt to find a solution with small componentwise
		       relative error in the double-precision algorithm.  Positive
		       is true, 0.0 is false.
		    Default: 1.0 (attempt componentwise convergence)

	   WORK

		     WORK is COMPLEX array, dimension (2*N)

	   RWORK

		     RWORK is REAL array, dimension (2*N)

	   INFO

		     INFO is INTEGER
		  = 0:	Successful exit. The solution to every right-hand side is
		    guaranteed.
		  < 0:	If INFO = -i, the i-th argument had an illegal value
		  > 0 and <= N:  U(INFO,INFO) is exactly zero.	The factorization
		    has been completed, but the factor U is exactly singular, so
		    the solution and error bounds could not be computed. RCOND = 0
		    is returned.
		  = N+J: The solution corresponding to the Jth right-hand side is
		    not guaranteed. The solutions corresponding to other right-
		    hand sides K with K > J may not be guaranteed as well, but
		    only the first such right-hand side is reported. If a small
		    componentwise error is not requested (PARAMS(3) = 0.0) then
		    the Jth right-hand side is the first with a normwise error
		    bound that is not guaranteed (the smallest J such
		    that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
		    the Jth right-hand side is the first with either a normwise or
		    componentwise error bound that is not guaranteed (the smallest
		    J such that either ERR_BNDS_NORM(J,1) = 0.0 or
		    ERR_BNDS_COMP(J,1) = 0.0). See the definition of
		    ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
		    about all of the right-hand sides check ERR_BNDS_NORM or
		    ERR_BNDS_COMP.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Definition at line 494 of file cposvxx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			     cposvxx.f(3)
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