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version 1.1.1.1, 2010/01/26 15:22:46 version 1.11, 2012/07/31 11:06:37
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   *> \brief <b> DSYSVX computes the solution to system of linear equations A * X = B for SY matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DSYSVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
   *                          LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
   *                          IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          FACT, UPLO
   *       INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * ), IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   *      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DSYSVX uses the diagonal pivoting factorization to compute the
   *> solution to a real system of linear equations A * X = B,
   *> where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A.
   *>    The form of the factorization is
   *>       A = U * D * U**T,  if UPLO = 'U', or
   *>       A = L * D * L**T,  if UPLO = 'L',
   *>    where U (or L) is a product of permutation and unit upper (lower)
   *>    triangular matrices, and D is symmetric and block diagonal with
   *>    1-by-1 and 2-by-2 diagonal blocks.
   *>
   *> 2. If some D(i,i)=0, so that D is exactly singular, 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.
   *>
   *> 3. The system of equations is solved for X using the factored form
   *>    of A.
   *>
   *> 4. Iterative refinement is applied to improve the computed solution
   *>    matrix and calculate error bounds and backward error estimates
   *>    for it.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] FACT
   *> \verbatim
   *>          FACT is CHARACTER*1
   *>          Specifies whether or not the factored form of A has been
   *>          supplied on entry.
   *>          = 'F':  On entry, AF and IPIV contain the factored form of
   *>                  A.  AF and IPIV will not be modified.
   *>          = 'N':  The matrix A will be 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] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          The symmetric matrix A.  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.
   *> \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 DOUBLE PRECISION array, dimension (LDAF,N)
   *>          If FACT = 'F', then AF is an input argument and on entry
   *>          contains the block diagonal matrix D and the multipliers used
   *>          to obtain the factor U or L from the factorization
   *>          A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
   *>
   *>          If FACT = 'N', then AF is an output argument and on exit
   *>          returns the block diagonal matrix D and the multipliers used
   *>          to obtain the factor U or L from the factorization
   *>          A = U*D*U**T or A = L*D*L**T.
   *> \endverbatim
   *>
   *> \param[in] LDAF
   *> \verbatim
   *>          LDAF is INTEGER
   *>          The leading dimension of the array AF.  LDAF >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          If FACT = 'F', then IPIV is an input argument and on entry
   *>          contains details of the interchanges and the block structure
   *>          of D, as determined by DSYTRF.
   *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
   *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
   *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
   *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
   *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
   *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
   *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
   *>
   *>          If FACT = 'N', then IPIV is an output argument and on exit
   *>          contains details of the interchanges and the block structure
   *>          of D, as determined by DSYTRF.
   *> \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 or INFO = N+1, 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] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>          The estimate of the reciprocal condition number of the matrix
   *>          A.  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 (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The length of WORK.  LWORK >= max(1,3*N), and for best
   *>          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
   *>          NB is the optimal blocksize for DSYTRF.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \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:  D(i,i) is exactly zero.  The factorization
   *>                       has been completed but the factor D is exactly
   *>                       singular, so the solution and error bounds could
   *>                       not be computed. RCOND = 0 is returned.
   *>                = N+1: D 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 April 2012
   *
   *> \ingroup doubleSYsolve
   *
   *  =====================================================================
       SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,        SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
      $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,       $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.4.1) --
 *  -- 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..--
 *     November 2006  *     April 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          FACT, UPLO        CHARACTER          FACT, UPLO
Line 18 Line 300
      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )       $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DSYSVX uses the diagonal pivoting factorization to compute the  
 *  solution to a real system of linear equations A * X = B,  
 *  where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A.  
 *     The form of the factorization is  
 *        A = U * D * U**T,  if UPLO = 'U', or  
 *        A = L * D * L**T,  if UPLO = 'L',  
 *     where U (or L) is a product of permutation and unit upper (lower)  
 *     triangular matrices, and D is symmetric and block diagonal with  
 *     1-by-1 and 2-by-2 diagonal blocks.  
 *  
 *  2. If some D(i,i)=0, so that D is exactly singular, 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.  
 *  
 *  3. The system of equations is solved for X using the factored form  
 *     of A.  
 *  
 *  4. Iterative refinement is applied to improve the computed solution  
 *     matrix and calculate error bounds and backward error estimates  
 *     for it.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  FACT    (input) CHARACTER*1  
 *          Specifies whether or not the factored form of A has been  
 *          supplied on entry.  
 *          = 'F':  On entry, AF and IPIV contain the factored form of  
 *                  A.  AF and IPIV will not be modified.  
 *          = 'N':  The matrix A will be 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) DOUBLE PRECISION array, dimension (LDA,N)  
 *          The symmetric matrix A.  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.  
 *  
 *  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 block diagonal matrix D and the multipliers used  
 *          to obtain the factor U or L from the factorization  
 *          A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.  
 *  
 *          If FACT = 'N', then AF is an output argument and on exit  
 *          returns the block diagonal matrix D and the multipliers used  
 *          to obtain the factor U or L from the factorization  
 *          A = U*D*U**T or A = L*D*L**T.  
 *  
 *  LDAF    (input) INTEGER  
 *          The leading dimension of the array AF.  LDAF >= max(1,N).  
 *  
 *  IPIV    (input or output) INTEGER array, dimension (N)  
 *          If FACT = 'F', then IPIV is an input argument and on entry  
 *          contains details of the interchanges and the block structure  
 *          of D, as determined by DSYTRF.  
 *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were  
 *          interchanged and D(k,k) is a 1-by-1 diagonal block.  
 *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and  
 *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)  
 *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =  
 *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were  
 *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.  
 *  
 *          If FACT = 'N', then IPIV is an output argument and on exit  
 *          contains details of the interchanges and the block structure  
 *          of D, as determined by DSYTRF.  
 *  
 *  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 or INFO = N+1, the N-by-NRHS solution matrix 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.  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/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The length of WORK.  LWORK >= max(1,3*N), and for best  
 *          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where  
 *          NB is the optimal blocksize for DSYTRF.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  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:  D(i,i) is exactly zero.  The factorization  
 *                       has been completed but the factor D is exactly  
 *                       singular, so the solution and error bounds could  
 *                       not be computed. RCOND = 0 is returned.  
 *                = N+1: D 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 ..
Line 254 Line 369
 *  *
       IF( NOFACT ) THEN        IF( NOFACT ) THEN
 *  *
 *        Compute the factorization A = U*D*U' or A = L*D*L'.  *        Compute the factorization A = U*D*U**T or A = L*D*L**T.
 *  *
          CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )           CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
          CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )           CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
Removed from v.1.1.1.1  
changed lines
  Added in v.1.11

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