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version 1.8, 2011/07/22 07:38:09 version 1.9, 2011/11/21 20:43:02
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   *> \brief <b> DPOSVX computes the solution to system of linear equations A * X = B for PO matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DPOSVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
   *                          S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
   *                          IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          EQUED, FACT, UPLO
   *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   *      $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
   *      $                   X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
   *> compute the solution to a real 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.
   *>
   *> Error bounds on the solution and a condition estimate are also
   *> provided.
   *> \endverbatim
   *
   *> \par Description:
   *  =================
   *>
   *> \verbatim
   *>
   *> 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.  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.
   *>
   *> 4. The system of equations is solved for X using the factored form
   *>    of A.
   *>
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] FACT
   *> \verbatim
   *>          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 = '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.
   *> \endverbatim
   *>
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \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 matrices B and X.  NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION 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).
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] AF
   *> \verbatim
   *>          AF is or output) DOUBLE PRECISION 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).
   *> \endverbatim
   *>
   *> \param[in] LDAF
   *> \verbatim
   *>          LDAF is INTEGER
   *>          The leading dimension of the array AF.  LDAF >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] EQUED
   *> \verbatim
   *>          EQUED is 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.
   *> \endverbatim
   *>
   *> \param[in,out] S
   *> \verbatim
   *>          S is or output) DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION 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.
   *> \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 or INFO = N+1, 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.
   *> \endverbatim
   *>
   *> \param[in] LDX
   *> \verbatim
   *>          LDX is INTEGER
   *>          The leading dimension of the array X.  LDX >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>          The estimate of the reciprocal condition number of the matrix
   *>          A after equilibration (if done).  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.
   *> \endverbatim
   *>
   *> \param[out] FERR
   *> \verbatim
   *>          FERR is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] BERR
   *> \verbatim
   *>          BERR is DOUBLE PRECISION 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).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (3*N)
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (N)
   *> \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, 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 has not
   *>                       been computed. RCOND = 0 is returned.
   *>                = N+1: U 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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doublePOsolve
   *
   *  =====================================================================
       SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,        SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
      $                   S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,       $                   S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK 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 ..
       CHARACTER          EQUED, FACT, UPLO        CHARACTER          EQUED, FACT, UPLO
Line 19 Line 324
      $                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to  
 *  compute the solution to a real 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.  
 *  
 *  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**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.  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.  
 *  
 *  4. The system of equations is solved for X using the factored form  
 *     of A.  
 *  
 *  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) DOUBLE PRECISION 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     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  AF      (input or output) DOUBLE PRECISION 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    (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) DOUBLE PRECISION 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) DOUBLE PRECISION 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     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *          If INFO = 0 or INFO = N+1, 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) DOUBLE PRECISION  
 *          The estimate of the reciprocal condition number of the matrix  
 *          A after equilibration (if done).  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N)  
 *  
 *  IWORK   (workspace) INTEGER 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 has not  
 *                       been computed. RCOND = 0 is returned.  
 *                = N+1: U 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.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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