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version 1.4, 2010/08/06 15:32:36 version 1.18, 2023/08/07 08:39:13
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   *> \brief \b DTRRFS
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DTRRFS + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrrfs.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrrfs.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrrfs.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
   *                          LDX, FERR, BERR, WORK, IWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          DIAG, TRANS, UPLO
   *       INTEGER            INFO, LDA, LDB, LDX, N, NRHS
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
   *      $                   WORK( * ), X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DTRRFS provides error bounds and backward error estimates for the
   *> solution to a system of linear equations with a triangular
   *> coefficient matrix.
   *>
   *> The solution matrix X must be computed by DTRTRS or some other
   *> means before entering this routine.  DTRRFS does not do iterative
   *> refinement because doing so cannot improve the backward error.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  A is upper triangular;
   *>          = 'L':  A is lower triangular.
   *> \endverbatim
   *>
   *> \param[in] TRANS
   *> \verbatim
   *>          TRANS is CHARACTER*1
   *>          Specifies the form of the system of equations:
   *>          = 'N':  A * X = B  (No transpose)
   *>          = 'T':  A**T * X = B  (Transpose)
   *>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
   *> \endverbatim
   *>
   *> \param[in] DIAG
   *> \verbatim
   *>          DIAG is CHARACTER*1
   *>          = 'N':  A is non-unit triangular;
   *>          = 'U':  A is unit triangular.
   *> \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] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
   *>          upper triangular part of the array A contains the upper
   *>          triangular matrix, and the strictly lower triangular part of
   *>          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
   *>          triangular part of the array A contains the lower triangular
   *>          matrix, and the strictly upper triangular part of A is not
   *>          referenced.  If DIAG = 'U', the diagonal elements of A are
   *>          also not referenced and are assumed to be 1.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \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] X
   *> \verbatim
   *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
   *>          The 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 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
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doubleOTHERcomputational
   *
   *  =====================================================================
       SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,        SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
      $                   LDX, FERR, BERR, WORK, IWORK, INFO )       $                   LDX, FERR, BERR, WORK, IWORK, 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  
 *  
 *     Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          DIAG, TRANS, UPLO        CHARACTER          DIAG, TRANS, UPLO
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      $                   WORK( * ), X( LDX, * )       $                   WORK( * ), X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DTRRFS provides error bounds and backward error estimates for the  
 *  solution to a system of linear equations with a triangular  
 *  coefficient matrix.  
 *  
 *  The solution matrix X must be computed by DTRTRS or some other  
 *  means before entering this routine.  DTRRFS does not do iterative  
 *  refinement because doing so cannot improve the backward error.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          = 'U':  A is upper triangular;  
 *          = 'L':  A is lower triangular.  
 *  
 *  TRANS   (input) CHARACTER*1  
 *          Specifies the form of the system of equations:  
 *          = 'N':  A * X = B  (No transpose)  
 *          = 'T':  A**T * X = B  (Transpose)  
 *          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)  
 *  
 *  DIAG    (input) CHARACTER*1  
 *          = 'N':  A is non-unit triangular;  
 *          = 'U':  A is unit triangular.  
 *  
 *  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.  
 *  
 *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)  
 *          The triangular matrix A.  If UPLO = 'U', the leading N-by-N  
 *          upper triangular part of the array A contains the upper  
 *          triangular matrix, and the strictly lower triangular part of  
 *          A is not referenced.  If UPLO = 'L', the leading N-by-N lower  
 *          triangular part of the array A contains the lower triangular  
 *          matrix, and the strictly upper triangular part of A is not  
 *          referenced.  If DIAG = 'U', the diagonal elements of A are  
 *          also not referenced and are assumed to be 1.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  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) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *          The 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 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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       DO 250 J = 1, NRHS        DO 250 J = 1, NRHS
 *  *
 *        Compute residual R = B - op(A) * X,  *        Compute residual R = B - op(A) * X,
 *        where op(A) = A or A', depending on TRANS.  *        where op(A) = A or A**T, depending on TRANS.
 *  *
          CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )           CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
          CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )           CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )
Line 250 Line 343
             END IF              END IF
          ELSE           ELSE
 *  *
 *           Compute abs(A')*abs(X) + abs(B).  *           Compute abs(A**T)*abs(X) + abs(B).
 *  *
             IF( UPPER ) THEN              IF( UPPER ) THEN
                IF( NOUNIT ) THEN                 IF( NOUNIT ) THEN
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          IF( KASE.NE.0 ) THEN           IF( KASE.NE.0 ) THEN
             IF( KASE.EQ.1 ) THEN              IF( KASE.EQ.1 ) THEN
 *  *
 *              Multiply by diag(W)*inv(op(A)').  *              Multiply by diag(W)*inv(op(A)**T).
 *  *
                CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ),                 CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ),
      $                     1 )       $                     1 )
Removed from v.1.4  
changed lines
  Added in v.1.18

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