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version 1.5, 2010/08/07 13:22:15 version 1.19, 2023/08/07 08:38:52
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   *> \brief <b> DGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DGTSVX + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgtsvx.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgtsvx.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgtsvx.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
   *                          DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
   *                          WORK, IWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          FACT, TRANS
   *       INTEGER            INFO, LDB, LDX, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * ), IWORK( * )
   *       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DF( * ),
   *      $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
   *      $                   FERR( * ), WORK( * ), X( LDX, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGTSVX uses the LU factorization to compute the solution to a real
   *> system of linear equations A * X = B or A**T * X = B,
   *> where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
   *>    as A = L * U, where L is a product of permutation and unit lower
   *>    bidiagonal matrices and U is upper triangular with nonzeros in
   *>    only the main diagonal and first two superdiagonals.
   *>
   *> 2. If some U(i,i)=0, so that U 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':  DLF, DF, DUF, DU2, and IPIV contain the factored
   *>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
   *>                  will not be modified.
   *>          = 'N':  The matrix will be copied to DLF, DF, and DUF
   *>                  and factored.
   *> \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] 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] DL
   *> \verbatim
   *>          DL is DOUBLE PRECISION array, dimension (N-1)
   *>          The (n-1) subdiagonal elements of A.
   *> \endverbatim
   *>
   *> \param[in] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The n diagonal elements of A.
   *> \endverbatim
   *>
   *> \param[in] DU
   *> \verbatim
   *>          DU is DOUBLE PRECISION array, dimension (N-1)
   *>          The (n-1) superdiagonal elements of A.
   *> \endverbatim
   *>
   *> \param[in,out] DLF
   *> \verbatim
   *>          DLF is DOUBLE PRECISION array, dimension (N-1)
   *>          If FACT = 'F', then DLF is an input argument and on entry
   *>          contains the (n-1) multipliers that define the matrix L from
   *>          the LU factorization of A as computed by DGTTRF.
   *>
   *>          If FACT = 'N', then DLF is an output argument and on exit
   *>          contains the (n-1) multipliers that define the matrix L from
   *>          the LU factorization of 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 upper triangular
   *>          matrix U from the LU factorization of A.
   *>
   *>          If FACT = 'N', then DF is an output argument and on exit
   *>          contains the n diagonal elements of the upper triangular
   *>          matrix U from the LU factorization of A.
   *> \endverbatim
   *>
   *> \param[in,out] DUF
   *> \verbatim
   *>          DUF is DOUBLE PRECISION array, dimension (N-1)
   *>          If FACT = 'F', then DUF is an input argument and on entry
   *>          contains the (n-1) elements of the first superdiagonal of U.
   *>
   *>          If FACT = 'N', then DUF is an output argument and on exit
   *>          contains the (n-1) elements of the first superdiagonal of U.
   *> \endverbatim
   *>
   *> \param[in,out] DU2
   *> \verbatim
   *>          DU2 is DOUBLE PRECISION array, dimension (N-2)
   *>          If FACT = 'F', then DU2 is an input argument and on entry
   *>          contains the (n-2) elements of the second superdiagonal of
   *>          U.
   *>
   *>          If FACT = 'N', then DU2 is an output argument and on exit
   *>          contains the (n-2) elements of the second superdiagonal of
   *>          U.
   *> \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 the pivot indices from the LU factorization of A as
   *>          computed by DGTTRF.
   *>
   *>          If FACT = 'N', then IPIV is an output argument and on exit
   *>          contains the pivot indices from the LU factorization of A;
   *>          row i of the matrix was interchanged with row IPIV(i).
   *>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
   *>          a row interchange was not required.
   *> \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 (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:  U(i,i) is exactly zero.  The factorization
   *>                       has not been completed unless i = N, but the
   *>                       factor U is exactly singular, so the solution
   *>                       and error bounds could not be 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.
   *
   *> \ingroup doubleGTsolve
   *
   *  =====================================================================
       SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,        SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
      $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,       $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
      $                   WORK, IWORK, INFO )       $                   WORK, IWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK driver 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 ..
       CHARACTER          FACT, TRANS        CHARACTER          FACT, TRANS
Line 19 Line 307
      $                   FERR( * ), WORK( * ), X( LDX, * )       $                   FERR( * ), WORK( * ), X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGTSVX uses the LU factorization to compute the solution to a real  
 *  system of linear equations A * X = B or A**T * X = B,  
 *  where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A  
 *     as A = L * U, where L is a product of permutation and unit lower  
 *     bidiagonal matrices and U is upper triangular with nonzeros in  
 *     only the main diagonal and first two superdiagonals.  
 *  
 *  2. If some U(i,i)=0, so that U 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':  DLF, DF, DUF, DU2, and IPIV contain the factored  
 *                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV  
 *                  will not be modified.  
 *          = 'N':  The matrix will be copied to DLF, DF, and DUF  
 *                  and factored.  
 *  
 *  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)  
 *  
 *  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.  
 *  
 *  DL      (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (n-1) subdiagonal elements of A.  
 *  
 *  D       (input) DOUBLE PRECISION array, dimension (N)  
 *          The n diagonal elements of A.  
 *  
 *  DU      (input) DOUBLE PRECISION array, dimension (N-1)  
 *          The (n-1) superdiagonal elements of A.  
 *  
 *  DLF     (input or output) DOUBLE PRECISION array, dimension (N-1)  
 *          If FACT = 'F', then DLF is an input argument and on entry  
 *          contains the (n-1) multipliers that define the matrix L from  
 *          the LU factorization of A as computed by DGTTRF.  
 *  
 *          If FACT = 'N', then DLF is an output argument and on exit  
 *          contains the (n-1) multipliers that define the matrix L from  
 *          the LU factorization of 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 upper triangular  
 *          matrix U from the LU factorization of A.  
 *  
 *          If FACT = 'N', then DF is an output argument and on exit  
 *          contains the n diagonal elements of the upper triangular  
 *          matrix U from the LU factorization of A.  
 *  
 *  DUF     (input or output) DOUBLE PRECISION array, dimension (N-1)  
 *          If FACT = 'F', then DUF is an input argument and on entry  
 *          contains the (n-1) elements of the first superdiagonal of U.  
 *  
 *          If FACT = 'N', then DUF is an output argument and on exit  
 *          contains the (n-1) elements of the first superdiagonal of U.  
 *  
 *  DU2     (input or output) DOUBLE PRECISION array, dimension (N-2)  
 *          If FACT = 'F', then DU2 is an input argument and on entry  
 *          contains the (n-2) elements of the second superdiagonal of  
 *          U.  
 *  
 *          If FACT = 'N', then DU2 is an output argument and on exit  
 *          contains the (n-2) elements of the second superdiagonal of  
 *          U.  
 *  
 *  IPIV    (input or output) INTEGER array, dimension (N)  
 *          If FACT = 'F', then IPIV is an input argument and on entry  
 *          contains the pivot indices from the LU factorization of A as  
 *          computed by DGTTRF.  
 *  
 *          If FACT = 'N', then IPIV is an output argument and on exit  
 *          contains the pivot indices from the LU factorization of A;  
 *          row i of the matrix was interchanged with row IPIV(i).  
 *          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates  
 *          a row interchange was not required.  
 *  
 *  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) 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:  U(i,i) is exactly zero.  The factorization  
 *                       has not been completed unless i = N, but the  
 *                       factor U is exactly singular, so the solution  
 *                       and error bounds could not be 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.1.5  
changed lines
  Added in v.1.19

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