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## dgerfsx()

subroutine dgerfsx | ( | character | trans,
|  | character | equed,
|  | integer | n,
|  | integer | nrhs,
|  | double precision, dimension( lda, * ) | a,
|  | integer | lda,
|  | double precision, dimension( ldaf, * ) | af,
|  | integer | ldaf,
|  | integer, dimension( * ) | ipiv,
|  | double precision, dimension( * ) | r,
|  | double precision, dimension( * ) | c,
|  | double precision, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | double precision, dimension( ldx , * ) | x,
|  | integer | ldx,
|  | double precision | rcond,
|  | double precision, dimension( * ) | berr,
|  | integer | n_err_bnds,
|  | double precision, dimension( nrhs, * ) | err_bnds_norm,
|  | double precision, dimension( nrhs, * ) | err_bnds_comp,
|  | integer | nparams,
|  | double precision, dimension( * ) | params,
|  | double precision, dimension( * ) | work,
|  | integer, dimension( * ) | iwork,
|  | integer | info )

DGERFSX

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Purpose:

      !>
      !>    DGERFSX improves the computed solution to a system of linear
      !>    equations and provides error bounds and backward error estimates
      !>    for the solution.  In addition to normwise error bound, the code
      !>    provides maximum componentwise error bound if possible.  See
      !>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
      !>    error bounds.
      !>
      !>    The original system of linear equations may have been equilibrated
      !>    before calling this routine, as described by arguments EQUED, R
      !>    and C below. In this case, the solution and error bounds returned
      !>    are for the original unequilibrated system.
      !>

      !>     Some optional parameters are bundled in the PARAMS array.  These
      !>     settings determine how refinement is performed, but often the
      !>     defaults are acceptable.  If the defaults are acceptable, users
      !>     can pass NPARAMS = 0 which prevents the source code from accessing
      !>     the PARAMS argument.
      !>

Parameters

  [in]

  TRANS

      !>          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)
      !>

  [in]

  EQUED

      !>          EQUED is CHARACTER*1
      !>     Specifies the form of equilibration that was done to A
      !>     before calling this routine. This is needed to compute
      !>     the solution and error bounds correctly.
      !>       = 'N':  No equilibration
      !>       = 'R':  Row equilibration, i.e., A has been premultiplied by
      !>               diag(R).
      !>       = 'C':  Column equilibration, i.e., A has been postmultiplied
      !>               by diag(C).
      !>       = 'B':  Both row and column equilibration, i.e., A has been
      !>               replaced by diag(R) * A * diag(C).
      !>               The right hand side B has been changed accordingly.
      !>

  [in]

  N

      !>          N is INTEGER
      !>     The order of the matrix A.  N >= 0.
      !>

  [in]

  NRHS

      !>          NRHS is INTEGER
      !>     The number of right hand sides, i.e., the number of columns
      !>     of the matrices B and X.  NRHS >= 0.
      !>

  [in]

  A

      !>          A is DOUBLE PRECISION array, dimension (LDA,N)
      !>     The original N-by-N matrix A.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>     The leading dimension of the array A.  LDA >= max(1,N).
      !>

  [in]

  AF

      !>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
      !>     The factors L and U from the factorization A = P*L*U
      !>     as computed by DGETRF.
      !>

  [in]

  LDAF

      !>          LDAF is INTEGER
      !>     The leading dimension of the array AF.  LDAF >= max(1,N).
      !>

  [in]

  IPIV

      !>          IPIV is INTEGER array, dimension (N)
      !>     The pivot indices from DGETRF; for 1<=i<=N, row i of the
      !>     matrix was interchanged with row IPIV(i).
      !>

  [in]

  R

      !>          R is DOUBLE PRECISION array, dimension (N)
      !>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
      !>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
      !>     is not accessed.
      !>     If R is accessed, each element of R should be a power of the radix
      !>     to ensure a reliable solution and error estimates. Scaling by
      !>     powers of the radix does not cause rounding errors unless the
      !>     result underflows or overflows. Rounding errors during scaling
      !>     lead to refining with a matrix that is not equivalent to the
      !>     input matrix, producing error estimates that may not be
      !>     reliable.
      !>

  [in]

  C

      !>          C is DOUBLE PRECISION array, dimension (N)
      !>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
      !>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
      !>     is not accessed.
      !>     If C is accessed, each element of C should be a power of the radix
      !>     to ensure a reliable solution and error estimates. Scaling by
      !>     powers of the radix does not cause rounding errors unless the
      !>     result underflows or overflows. Rounding errors during scaling
      !>     lead to refining with a matrix that is not equivalent to the
      !>     input matrix, producing error estimates that may not be
      !>     reliable.
      !>

  [in]

  B

      !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
      !>     The right hand side matrix B.
      !>

  [in]

  LDB

      !>          LDB is INTEGER
      !>     The leading dimension of the array B.  LDB >= max(1,N).
      !>

  [in,out]

  X

      !>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
      !>     On entry, the solution matrix X, as computed by DGETRS.
      !>     On exit, the improved solution matrix X.
      !>

  [in]

  LDX

      !>          LDX is INTEGER
      !>     The leading dimension of the array X.  LDX >= max(1,N).
      !>

  [out]

  RCOND

      !>          RCOND is DOUBLE PRECISION
      !>     Reciprocal scaled condition number.  This is an estimate of the
      !>     reciprocal Skeel condition number of the matrix A after
      !>     equilibration (if done).  If this is less than the machine
      !>     precision (in particular, if it is zero), the matrix is singular
      !>     to working precision.  Note that the error may still be small even
      !>     if this number is very small and the matrix appears ill-
      !>     conditioned.
      !>

  [out]

  BERR

      !>          BERR is DOUBLE PRECISION array, dimension (NRHS)
      !>     Componentwise relative backward error.  This is 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).
      !>

  [in]

  N_ERR_BNDS

      !>          N_ERR_BNDS is INTEGER
      !>     Number of error bounds to return for each right hand side
      !>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
      !>     ERR_BNDS_COMP below.
      !>

  [out]

  ERR_BNDS_NORM

      !>          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
      !>     For each right-hand side, this array contains information about
      !>     various error bounds and condition numbers corresponding to the
      !>     normwise relative error, which is defined as follows:
      !>
      !>     Normwise relative error in the ith solution vector:
      !>             max_j (abs(XTRUE(j,i) - X(j,i)))
      !>            ------------------------------
      !>                  max_j abs(X(j,i))
      !>
      !>     The array is indexed by the type of error information as described
      !>     below. There currently are up to three pieces of information
      !>     returned.
      !>
      !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
      !>     right-hand side.
      !>
      !>     The second index in ERR_BNDS_NORM(:,err) contains the following
      !>     three fields:
      !>     err = 1  boolean. Trust the answer if the
      !>              reciprocal condition number is less than the threshold
      !>              sqrt(n) * dlamch('Epsilon').
      !>
      !>     err = 2  error bound: The estimated forward error,
      !>              almost certainly within a factor of 10 of the true error
      !>              so long as the next entry is greater than the threshold
      !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
      !>              be trusted if the previous boolean is true.
      !>
      !>     err = 3  Reciprocal condition number: Estimated normwise
      !>              reciprocal condition number.  Compared with the threshold
      !>              sqrt(n) * dlamch('Epsilon') to determine if the error
      !>              estimate is . These reciprocal condition
      !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
      !>              appropriately scaled matrix Z.
      !>              Let Z = S*A, where S scales each row by a power of the
      !>              radix so all absolute row sums of Z are approximately 1.
      !>
      !>     See Lapack Working Note 165 for further details and extra
      !>     cautions.
      !>

  [out]

  ERR_BNDS_COMP

      !>          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
      !>     For each right-hand side, this array contains information about
      !>     various error bounds and condition numbers corresponding to the
      !>     componentwise relative error, which is defined as follows:
      !>
      !>     Componentwise relative error in the ith solution vector:
      !>                    abs(XTRUE(j,i) - X(j,i))
      !>             max_j ----------------------
      !>                         abs(X(j,i))
      !>
      !>     The array is indexed by the right-hand side i (on which the
      !>     componentwise relative error depends), and the type of error
      !>     information as described below. There currently are up to three
      !>     pieces of information returned for each right-hand side. If
      !>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
      !>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
      !>     the first (:,N_ERR_BNDS) entries are returned.
      !>
      !>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
      !>     right-hand side.
      !>
      !>     The second index in ERR_BNDS_COMP(:,err) contains the following
      !>     three fields:
      !>     err = 1  boolean. Trust the answer if the
      !>              reciprocal condition number is less than the threshold
      !>              sqrt(n) * dlamch('Epsilon').
      !>
      !>     err = 2  error bound: The estimated forward error,
      !>              almost certainly within a factor of 10 of the true error
      !>              so long as the next entry is greater than the threshold
      !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
      !>              be trusted if the previous boolean is true.
      !>
      !>     err = 3  Reciprocal condition number: Estimated componentwise
      !>              reciprocal condition number.  Compared with the threshold
      !>              sqrt(n) * dlamch('Epsilon') to determine if the error
      !>              estimate is . These reciprocal condition
      !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
      !>              appropriately scaled matrix Z.
      !>              Let Z = S*(A*diag(x)), where x is the solution for the
      !>              current right-hand side and S scales each row of
      !>              A*diag(x) by a power of the radix so all absolute row
      !>              sums of Z are approximately 1.
      !>
      !>     See Lapack Working Note 165 for further details and extra
      !>     cautions.
      !>

  [in]

  NPARAMS

      !>          NPARAMS is INTEGER
      !>     Specifies the number of parameters set in PARAMS.  If <= 0, the
      !>     PARAMS array is never referenced and default values are used.
      !>

  [in,out]

  PARAMS

      !>          PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
      !>     Specifies algorithm parameters.  If an entry is < 0.0, then
      !>     that entry will be filled with default value used for that
      !>     parameter.  Only positions up to NPARAMS are accessed; defaults
      !>     are used for higher-numbered parameters.
      !>
      !>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
      !>            refinement or not.
      !>         Default: 1.0D+0
      !>            = 0.0:  No refinement is performed, and no error bounds are
      !>                    computed.
      !>            = 1.0:  Use the double-precision refinement algorithm,
      !>                    possibly with doubled-single computations if the
      !>                    compilation environment does not support DOUBLE
      !>                    PRECISION.
      !>              (other values are reserved for future use)
      !>
      !>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
      !>            computations allowed for refinement.
      !>         Default: 10
      !>         Aggressive: Set to 100 to permit convergence using approximate
      !>                     factorizations or factorizations other than LU. If
      !>                     the factorization uses a technique other than
      !>                     Gaussian elimination, the guarantees in
      !>                     err_bnds_norm and err_bnds_comp may no longer be
      !>                     trustworthy.
      !>
      !>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
      !>            will attempt to find a solution with small componentwise
      !>            relative error in the double-precision algorithm.  Positive
      !>            is true, 0.0 is false.
      !>         Default: 1.0 (attempt componentwise convergence)
      !>

  [out]

  WORK

      !>          WORK is DOUBLE PRECISION array, dimension (4*N)
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (N)
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>       = 0:  Successful exit. The solution to every right-hand side is
      !>         guaranteed.
      !>       < 0:  If INFO = -i, the i-th argument had an illegal value
      !>       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
      !>         has been completed, but the factor U is exactly singular, so
      !>         the solution and error bounds could not be computed. RCOND = 0
      !>         is returned.
      !>       = N+J: The solution corresponding to the Jth right-hand side is
      !>         not guaranteed. The solutions corresponding to other right-
      !>         hand sides K with K > J may not be guaranteed as well, but
      !>         only the first such right-hand side is reported. If a small
      !>         componentwise error is not requested (PARAMS(3) = 0.0) then
      !>         the Jth right-hand side is the first with a normwise error
      !>         bound that is not guaranteed (the smallest J such
      !>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
      !>         the Jth right-hand side is the first with either a normwise or
      !>         componentwise error bound that is not guaranteed (the smallest
      !>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
      !>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
      !>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
      !>         about all of the right-hand sides check ERR_BNDS_NORM or
      !>         ERR_BNDS_COMP.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.