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sgerfsx.f(3)				      LAPACK				     sgerfsx.f(3)

NAME
       sgerfsx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine sgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX,
	   RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
	   INFO)
	   SGERFSX

Function/Subroutine Documentation
   subroutine sgerfsx (characterTRANS, characterEQUED, integerN, integerNRHS, real, dimension(
       lda, * )A, integerLDA, real, dimension( ldaf, * )AF, integerLDAF, integer, dimension( *
       )IPIV, real, dimension( * )R, real, dimension( * )C, real, dimension( ldb, * )B,
       integerLDB, real, dimension( ldx , * )X, integerLDX, realRCOND, real, dimension( * )BERR,
       integerN_ERR_BNDS, real, dimension( nrhs, * )ERR_BNDS_NORM, real, dimension( nrhs, *
       )ERR_BNDS_COMP, integerNPARAMS, real, dimension( * )PARAMS, real, dimension( * )WORK,
       integer, dimension( * )IWORK, integerINFO)
       SGERFSX

       Purpose:

	       SGERFSX improves the computed solution to a system of linear
	       equations and provides error bounds and backward error estimates
	       for the solution.  In addition to normwise error bound, the code
	       provides maximum componentwise error bound if possible.	See
	       comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
	       error bounds.

	       The original system of linear equations may have been equilibrated
	       before calling this routine, as described by arguments EQUED, R
	       and C below. In this case, the solution and error bounds returned
	       are for the original unequilibrated system.

		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:
	   TRANS

		     TRANS is CHARACTER*1
		Specifies the form of the system of equations:
		  = 'N':  A * X = B	(No transpose)
		  = 'T':  A**T * X = B	(Transpose)
		  = 'C':  A**H * X = B	(Conjugate transpose = Transpose)

	   EQUED

		     EQUED is CHARACTER*1
		Specifies the form of equilibration that was done to A
		before calling this routine. This is needed to compute
		the solution and error bounds correctly.
		  = 'N':  No equilibration
		  = 'R':  Row equilibration, i.e., A has been premultiplied by
			  diag(R).
		  = 'C':  Column equilibration, i.e., A has been postmultiplied
			  by diag(C).
		  = 'B':  Both row and column equilibration, i.e., A has been
			  replaced by diag(R) * A * diag(C).
			  The right hand side B has been changed accordingly.

	   N

		     N is INTEGER
		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 REAL array, dimension (LDA,N)
		The original N-by-N matrix A.

	   LDA

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

	   AF

		     AF is REAL array, dimension (LDAF,N)
		The factors L and U from the factorization A = P*L*U
		as computed by SGETRF.

	   LDAF

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

	   IPIV

		     IPIV is INTEGER array, dimension (N)
		The pivot indices from SGETRF; for 1<=i<=N, row i of the
		matrix was interchanged with row IPIV(i).

	   R

		     R is REAL array, dimension (N)
		The row scale factors for A.  If EQUED = 'R' or 'B', A is
		multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
		is not accessed.
		If R is accessed, each element of R 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.

	   C

		     C is REAL array, dimension (N)
		The column scale factors for A.  If EQUED = 'C' or 'B', A is
		multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
		is not accessed.
		If C is accessed, each element of C 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 REAL array, dimension (LDB,NRHS)
		The right hand side matrix B.

	   LDB

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

	   X

		     X is REAL array, dimension (LDX,NRHS)
		On entry, the solution matrix X, as computed by SGETRS.
		On exit, the improved solution matrix 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.

	   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 REAL array, dimension (4*N)

	   IWORK

		     IWORK is INTEGER array, dimension (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:
	   November 2011

       Definition at line 412 of file sgerfsx.f.

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

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