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version 1.6, 2010/08/13 21:04:14 version 1.19, 2018/05/29 07:18:33
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   *> \brief <b> ZPTSVX computes the solution to system of linear equations A * X = B for PT matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZPTSVX + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zptsvx.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zptsvx.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zptsvx.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
   *                          RCOND, FERR, BERR, WORK, RWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          FACT
   *       INTEGER            INFO, LDB, LDX, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
   *      $                   RWORK( * )
   *       COMPLEX*16         B( LDB, * ), E( * ), EF( * ), WORK( * ),
   *      $                   X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZPTSVX uses the factorization A = L*D*L**H to compute the solution
   *> to a complex system of linear equations A*X = B, where A is an
   *> N-by-N Hermitian positive definite tridiagonal 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 matrix A is factored as A = L*D*L**H, where L
   *>    is a unit lower bidiagonal matrix and D is diagonal.  The
   *>    factorization can also be regarded as having the form
   *>    A = U**H*D*U.
   *>
   *> 2. 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.
   *>
   *> 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 the matrix
   *>          A is supplied on entry.
   *>          = 'F':  On entry, DF and EF contain the factored form of A.
   *>                  D, E, DF, and EF will not be modified.
   *>          = 'N':  The matrix A will be copied to DF and EF and
   *>                  factored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          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] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The n diagonal elements of the tridiagonal matrix A.
   *> \endverbatim
   *>
   *> \param[in] E
   *> \verbatim
   *>          E is COMPLEX*16 array, dimension (N-1)
   *>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
   *> \endverbatim
   *>
   *> \param[in,out] DF
   *> \verbatim
   *>          DF is DOUBLE PRECISION array, dimension (N)
   *>          If FACT = 'F', then DF is an input argument and on entry
   *>          contains the n diagonal elements of the diagonal matrix D
   *>          from the L*D*L**H factorization of A.
   *>          If FACT = 'N', then DF is an output argument and on exit
   *>          contains the n diagonal elements of the diagonal matrix D
   *>          from the L*D*L**H factorization of A.
   *> \endverbatim
   *>
   *> \param[in,out] EF
   *> \verbatim
   *>          EF is COMPLEX*16 array, dimension (N-1)
   *>          If FACT = 'F', then EF is an input argument and on entry
   *>          contains the (n-1) subdiagonal elements of the unit
   *>          bidiagonal factor L from the L*D*L**H factorization of A.
   *>          If FACT = 'N', then EF is an output argument and on exit
   *>          contains the (n-1) subdiagonal elements of the unit
   *>          bidiagonal factor L from the L*D*L**H factorization of A.
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is COMPLEX*16 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 COMPLEX*16 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 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 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).
   *> \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 COMPLEX*16 array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION 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 December 2016
   *
   *> \ingroup complex16PTsolve
   *
   *  =====================================================================
       SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,        SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
      $                   RCOND, FERR, BERR, WORK, RWORK, INFO )       $                   RCOND, FERR, BERR, WORK, RWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK driver routine (version 3.7.0) --
 *  -- 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..--
 *     November 2006  *     December 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          FACT        CHARACTER          FACT
Line 18 Line 251
      $                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZPTSVX uses the factorization A = L*D*L**H to compute the solution  
 *  to a complex system of linear equations A*X = B, where A is an  
 *  N-by-N Hermitian positive definite tridiagonal 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 matrix A is factored as A = L*D*L**H, where L  
 *     is a unit lower bidiagonal matrix and D is diagonal.  The  
 *     factorization can also be regarded as having the form  
 *     A = U**H*D*U.  
 *  
 *  2. 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.  
 *  
 *  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 the matrix  
 *          A is supplied on entry.  
 *          = 'F':  On entry, DF and EF contain the factored form of A.  
 *                  D, E, DF, and EF will not be modified.  
 *          = 'N':  The matrix A will be copied to DF and EF and  
 *                  factored.  
 *  
 *  N       (input) INTEGER  
 *          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.  
 *  
 *  D       (input) DOUBLE PRECISION array, dimension (N)  
 *          The n diagonal elements of the tridiagonal matrix A.  
 *  
 *  E       (input) COMPLEX*16 array, dimension (N-1)  
 *          The (n-1) subdiagonal elements of the tridiagonal matrix A.  
 *  
 *  DF      (input or output) DOUBLE PRECISION array, dimension (N)  
 *          If FACT = 'F', then DF is an input argument and on entry  
 *          contains the n diagonal elements of the diagonal matrix D  
 *          from the L*D*L**H factorization of A.  
 *          If FACT = 'N', then DF is an output argument and on exit  
 *          contains the n diagonal elements of the diagonal matrix D  
 *          from the L*D*L**H factorization of A.  
 *  
 *  EF      (input or output) COMPLEX*16 array, dimension (N-1)  
 *          If FACT = 'F', then EF is an input argument and on entry  
 *          contains the (n-1) subdiagonal elements of the unit  
 *          bidiagonal factor L from the L*D*L**H factorization of A.  
 *          If FACT = 'N', then EF is an output argument and on exit  
 *          contains the (n-1) subdiagonal elements of the unit  
 *          bidiagonal factor L from the L*D*L**H factorization of A.  
 *  
 *  B       (input) COMPLEX*16 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) COMPLEX*16 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 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 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).  
 *  
 *  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) COMPLEX*16 array, dimension (N)  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION 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 ..
Line 191 Line 297
 *  *
       IF( NOFACT ) THEN        IF( NOFACT ) THEN
 *  *
 *        Compute the L*D*L' (or U'*D*U) factorization of A.  *        Compute the L*D*L**H (or U**H*D*U) factorization of A.
 *  *
          CALL DCOPY( N, D, 1, DF, 1 )           CALL DCOPY( N, D, 1, DF, 1 )
          IF( N.GT.1 )           IF( N.GT.1 )
Removed from v.1.6  
changed lines
  Added in v.1.19

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