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version 1.2, 2010/04/21 13:45:23 version 1.19, 2023/08/07 08:39:05
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   *> \brief \b DPTRFS
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DPTRFS + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptrfs.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptrfs.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
   *                          BERR, WORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDB, LDX, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
   *      $                   E( * ), EF( * ), FERR( * ), WORK( * ),
   *      $                   X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DPTRFS improves the computed solution to a system of linear
   *> equations when the coefficient matrix is symmetric positive definite
   *> and tridiagonal, and provides error bounds and backward error
   *> estimates for the solution.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \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 matrix B.  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 DOUBLE PRECISION array, dimension (N-1)
   *>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
   *> \endverbatim
   *>
   *> \param[in] DF
   *> \verbatim
   *>          DF is DOUBLE PRECISION array, dimension (N)
   *>          The n diagonal elements of the diagonal matrix D from the
   *>          factorization computed by DPTTRF.
   *> \endverbatim
   *>
   *> \param[in] EF
   *> \verbatim
   *>          EF is DOUBLE PRECISION array, dimension (N-1)
   *>          The (n-1) subdiagonal elements of the unit bidiagonal factor
   *>          L from the factorization computed by DPTTRF.
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
   *>          The 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[in,out] X
   *> \verbatim
   *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
   *>          On entry, the solution matrix X, as computed by DPTTRS.
   *>          On exit, the improved 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] 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 DOUBLE PRECISION array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *> \endverbatim
   *
   *> \par Internal Parameters:
   *  =========================
   *>
   *> \verbatim
   *>  ITMAX is the maximum number of steps of iterative refinement.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doublePTcomputational
   *
   *  =====================================================================
       SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,        SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
      $                   BERR, WORK, INFO )       $                   BERR, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine --
 *  -- 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  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDB, LDX, N, NRHS        INTEGER            INFO, LDB, LDX, N, NRHS
Line 15 Line 174
      $                   X( LDX, * )       $                   X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DPTRFS improves the computed solution to a system of linear  
 *  equations when the coefficient matrix is symmetric positive definite  
 *  and tridiagonal, and provides error bounds and backward error  
 *  estimates for the solution.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  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 matrix B.  NRHS >= 0.  
 *  
 *  D       (input) DOUBLE PRECISION array, dimension (N)  
 *          The n diagonal elements of the tridiagonal matrix A.  
 *  
 *  E       (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (n-1) subdiagonal elements of the tridiagonal matrix A.  
 *  
 *  DF      (input) DOUBLE PRECISION array, dimension (N)  
 *          The n diagonal elements of the diagonal matrix D from the  
 *          factorization computed by DPTTRF.  
 *  
 *  EF      (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (n-1) subdiagonal elements of the unit bidiagonal factor  
 *          L from the factorization computed by DPTTRF.  
 *  
 *  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)  
 *          The right hand side matrix B.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *          On entry, the solution matrix X, as computed by DPTTRS.  
 *          On exit, the improved solution matrix X.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *  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) DOUBLE PRECISION array, dimension (2*N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *  
 *  Internal Parameters  
 *  ===================  
 *  
 *  ITMAX is the maximum number of steps of iterative refinement.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 263 Line 353
 *           m(i,j) =  abs(A(i,j)), i = j,  *           m(i,j) =  abs(A(i,j)), i = j,
 *           m(i,j) = -abs(A(i,j)), i .ne. j,  *           m(i,j) = -abs(A(i,j)), i .ne. j,
 *  *
 *        and e = [ 1, 1, ..., 1 ]'.  Note M(A) = M(L)*D*M(L)'.  *        and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**T.
 *  *
 *        Solve M(L) * x = e.  *        Solve M(L) * x = e.
 *  *
Line 272 Line 362
             WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )              WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
    60    CONTINUE     60    CONTINUE
 *  *
 *        Solve D * M(L)' * x = b.  *        Solve D * M(L)**T * x = b.
 *  *
          WORK( N ) = WORK( N ) / DF( N )           WORK( N ) = WORK( N ) / DF( N )
          DO 70 I = N - 1, 1, -1           DO 70 I = N - 1, 1, -1
Removed from v.1.2  
changed lines
  Added in v.1.19

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