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version 1.10, 2011/11/22 10:35:58 version 1.11, 2012/07/31 11:06:37
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   *> \brief <b> DSGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement)
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSGESV + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsgesv.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsgesv.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsgesv.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
   *                          SWORK, ITER, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       REAL               SWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( N, * ),
   *      $                   X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSGESV computes the solution to a real system of linear equations
   *>    A * X = B,
   *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
   *>
   *> DSGESV first attempts to factorize the matrix in SINGLE PRECISION
   *> and use this factorization within an iterative refinement procedure
   *> to produce a solution with DOUBLE PRECISION normwise backward error
   *> quality (see below). If the approach fails the method switches to a
   *> DOUBLE PRECISION factorization and solve.
   *>
   *> The iterative refinement is not going to be a winning strategy if
   *> the ratio SINGLE PRECISION performance over DOUBLE PRECISION
   *> performance is too small. A reasonable strategy should take the
   *> number of right-hand sides and the size of the matrix into account.
   *> This might be done with a call to ILAENV in the future. Up to now, we
   *> always try iterative refinement.
   *>
   *> The iterative refinement process is stopped if
   *>     ITER > ITERMAX
   *> or for all the RHS we have:
   *>     RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
   *> where
   *>     o ITER is the number of the current iteration in the iterative
   *>       refinement process
   *>     o RNRM is the infinity-norm of the residual
   *>     o XNRM is the infinity-norm of the solution
   *>     o ANRM is the infinity-operator-norm of the matrix A
   *>     o EPS is the machine epsilon returned by DLAMCH('Epsilon')
   *> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
   *> respectively.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of linear equations, i.e., the order of the
   *>          matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>          The number of right hand sides, i.e., the number of columns
   *>          of the matrix B.  NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array,
   *>          dimension (LDA,N)
   *>          On entry, the N-by-N coefficient matrix A.
   *>          On exit, if iterative refinement has been successfully used
   *>          (INFO.EQ.0 and ITER.GE.0, see description below), then A is
   *>          unchanged, if double precision factorization has been used
   *>          (INFO.EQ.0 and ITER.LT.0, see description below), then the
   *>          array A contains the factors L and U from the factorization
   *>          A = P*L*U; the unit diagonal elements of L are not stored.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          The pivot indices that define the permutation matrix P;
   *>          row i of the matrix was interchanged with row IPIV(i).
   *>          Corresponds either to the single precision factorization
   *>          (if INFO.EQ.0 and ITER.GE.0) or the double precision
   *>          factorization (if INFO.EQ.0 and ITER.LT.0).
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
   *>          The N-by-NRHS right hand side matrix B.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] X
   *> \verbatim
   *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
   *>          If INFO = 0, the N-by-NRHS solution matrix X.
   *> \endverbatim
   *>
   *> \param[in] LDX
   *> \verbatim
   *>          LDX is INTEGER
   *>          The leading dimension of the array X.  LDX >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (N,NRHS)
   *>          This array is used to hold the residual vectors.
   *> \endverbatim
   *>
   *> \param[out] SWORK
   *> \verbatim
   *>          SWORK is REAL array, dimension (N*(N+NRHS))
   *>          This array is used to use the single precision matrix and the
   *>          right-hand sides or solutions in single precision.
   *> \endverbatim
   *>
   *> \param[out] ITER
   *> \verbatim
   *>          ITER is INTEGER
   *>          < 0: iterative refinement has failed, double precision
   *>               factorization has been performed
   *>               -1 : the routine fell back to full precision for
   *>                    implementation- or machine-specific reasons
   *>               -2 : narrowing the precision induced an overflow,
   *>                    the routine fell back to full precision
   *>               -3 : failure of SGETRF
   *>               -31: stop the iterative refinement after the 30th
   *>                    iterations
   *>          > 0: iterative refinement has been sucessfully used.
   *>               Returns the number of iterations
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *>          > 0:  if INFO = i, U(i,i) computed in DOUBLE PRECISION is
   *>                exactly zero.  The factorization has been completed,
   *>                but the factor U is exactly singular, so the solution
   *>                could not be computed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEsolve
   *
   *  =====================================================================
       SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,        SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
      $                   SWORK, ITER, INFO )       $                   SWORK, ITER, INFO )
 *  *
 *  -- LAPACK PROTOTYPE driver routine (version 3.3.1) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     ..  
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS        INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
 *     ..  *     ..
Line 17 Line 210
      $                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSGESV computes the solution to a real system of linear equations  
 *     A * X = B,  
 *  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.  
 *  
 *  DSGESV first attempts to factorize the matrix in SINGLE PRECISION  
 *  and use this factorization within an iterative refinement procedure  
 *  to produce a solution with DOUBLE PRECISION normwise backward error  
 *  quality (see below). If the approach fails the method switches to a  
 *  DOUBLE PRECISION factorization and solve.  
 *  
 *  The iterative refinement is not going to be a winning strategy if  
 *  the ratio SINGLE PRECISION performance over DOUBLE PRECISION  
 *  performance is too small. A reasonable strategy should take the  
 *  number of right-hand sides and the size of the matrix into account.  
 *  This might be done with a call to ILAENV in the future. Up to now, we  
 *  always try iterative refinement.  
 *  
 *  The iterative refinement process is stopped if  
 *      ITER > ITERMAX  
 *  or for all the RHS we have:  
 *      RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX  
 *  where  
 *      o ITER is the number of the current iteration in the iterative  
 *        refinement process  
 *      o RNRM is the infinity-norm of the residual  
 *      o XNRM is the infinity-norm of the solution  
 *      o ANRM is the infinity-operator-norm of the matrix A  
 *      o EPS is the machine epsilon returned by DLAMCH('Epsilon')  
 *  The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00  
 *  respectively.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  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 matrix B.  NRHS >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION array,  
 *          dimension (LDA,N)  
 *          On entry, the N-by-N coefficient matrix A.  
 *          On exit, if iterative refinement has been successfully used  
 *          (INFO.EQ.0 and ITER.GE.0, see description below), then A is  
 *          unchanged, if double precision factorization has been used  
 *          (INFO.EQ.0 and ITER.LT.0, see description below), then the  
 *          array A contains the factors L and U from the factorization  
 *          A = P*L*U; the unit diagonal elements of L are not stored.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  IPIV    (output) INTEGER array, dimension (N)  
 *          The pivot indices that define the permutation matrix P;  
 *          row i of the matrix was interchanged with row IPIV(i).  
 *          Corresponds either to the single precision factorization  
 *          (if INFO.EQ.0 and ITER.GE.0) or the double precision  
 *          factorization (if INFO.EQ.0 and ITER.LT.0).  
 *  
 *  B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *          If INFO = 0, the N-by-NRHS solution matrix X.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (N,NRHS)  
 *          This array is used to hold the residual vectors.  
 *  
 *  SWORK   (workspace) REAL array, dimension (N*(N+NRHS))  
 *          This array is used to use the single precision matrix and the  
 *          right-hand sides or solutions in single precision.  
 *  
 *  ITER    (output) INTEGER  
 *          < 0: iterative refinement has failed, double precision  
 *               factorization has been performed  
 *               -1 : the routine fell back to full precision for  
 *                    implementation- or machine-specific reasons  
 *               -2 : narrowing the precision induced an overflow,  
 *                    the routine fell back to full precision  
 *               -3 : failure of SGETRF  
 *               -31: stop the iterative refinement after the 30th  
 *                    iterations  
 *          > 0: iterative refinement has been sucessfully used.  
 *               Returns the number of iterations  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *          > 0:  if INFO = i, U(i,i) computed in DOUBLE PRECISION is  
 *                exactly zero.  The factorization has been completed,  
 *                but the factor U is exactly singular, so the solution  
 *                could not be computed.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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