cla_gerfsx_extended.f - Man Page

subroutine cla_gerfsx_extended (prec_type, trans_type, n, nrhs, a, lda, af, ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, errs_n, errs_c, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
CLA_GERFSX_EXTENDED

CLA_GERFSX_EXTENDED  

Purpose:

 CLA_GERFSX_EXTENDED improves the computed solution to a system of
 linear equations by performing extra-precise iterative refinement
 and provides error bounds and backward error estimates for the solution.
 This subroutine is called by CGERFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERRS_N
 and ERRS_C for details of the error bounds. Note that this
 subroutine is only responsible for setting the second fields of
 ERRS_N and ERRS_C.

Parameters

PREC_TYPE

          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and P
          = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X' or 'E':  Extra

TRANS_TYPE

          TRANS_TYPE is INTEGER
     Specifies the transposition operation on A.
     The value is defined by ILATRANS(T) where T is a CHARACTER and T
          = 'N':  No transpose
          = 'T':  Transpose
          = 'C':  Conjugate transpose

N

          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.

A

          A is COMPLEX array, dimension (LDA,N)
     On entry, the N-by-N matrix A.

LDA

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

AF

          AF is COMPLEX array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by CGETRF.

LDAF

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

IPIV

          IPIV is INTEGER array, dimension (N)
     The pivot indices from the factorization A = P*L*U
     as computed by CGETRF; row i of the matrix was interchanged
     with row IPIV(i).

COLEQU

          COLEQU is LOGICAL
     If .TRUE. then column equilibration was done to A before calling
     this routine. This is needed to compute the solution and error
     bounds correctly.

C

          C is REAL array, dimension (N)
     The column scale factors for A. If COLEQU = .FALSE., C
     is not accessed. If C is input, 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.

B

          B is COMPLEX array, dimension (LDB,NRHS)
     The right-hand-side matrix B.

LDB

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

Y

          Y is COMPLEX array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by CGETRS.
     On exit, the improved solution matrix Y.

LDY

          LDY is INTEGER
     The leading dimension of the array Y.  LDY >= max(1,N).

BERR_OUT

          BERR_OUT is REAL array, dimension (NRHS)
     On exit, BERR_OUT(j) contains the componentwise relative backward
     error for right-hand-side j from the formula
         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
     where abs(Z) is the componentwise absolute value of the matrix
     or vector Z. This is computed by CLA_LIN_BERR.

N_NORMS

          N_NORMS is INTEGER
     Determines which error bounds to return (see ERRS_N
     and ERRS_C).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.

ERRS_N

          ERRS_N is REAL 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 ERRS_N(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_N(:,err) contains the following
     three fields:
     err = 1 'Trust/don't trust' boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * slamch('Epsilon').

     err = 2 'Guaranteed' 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) * slamch('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) * slamch('Epsilon') to determine if the error
              estimate is 'guaranteed'. 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.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.

ERRS_C

          ERRS_C is REAL 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
     ERRS_C is not accessed.  If N_ERR_BNDS < 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERRS_C(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_C(:,err) contains the following
     three fields:
     err = 1 'Trust/don't trust' boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * slamch('Epsilon').

     err = 2 'Guaranteed' 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) * slamch('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) * slamch('Epsilon') to determine if the error
              estimate is 'guaranteed'. 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.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.

RES

          RES is COMPLEX array, dimension (N)
     Workspace to hold the intermediate residual.

AYB

          AYB is REAL array, dimension (N)
     Workspace.

DY

          DY is COMPLEX array, dimension (N)
     Workspace to hold the intermediate solution.

Y_TAIL

          Y_TAIL is COMPLEX array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.

RCOND

          RCOND is REAL
     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.

ITHRESH

          ITHRESH is INTEGER
     The maximum number of residual computations allowed for
     refinement. The default is 10. For '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
     ERRS_N and ERRS_C may no longer be trustworthy.

RTHRESH

          RTHRESH is REAL
     Determines when to stop refinement if the error estimate stops
     decreasing. Refinement will stop when the next solution no longer
     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
     default value is 0.5. For 'aggressive' set to 0.9 to permit
     convergence on extremely ill-conditioned matrices. See LAWN 165
     for more details.

DZ_UB

          DZ_UB is REAL
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we define as the relative
     change in each component being less than DZ_UB. The default value
     is 0.25, requiring the first bit to be stable. See LAWN 165 for
     more details.

IGNORE_CWISE

          IGNORE_CWISE is LOGICAL
     If .TRUE. then ignore componentwise convergence. Default value
     is .FALSE..

INFO

          INFO is INTEGER
       = 0:  Successful exit.
       < 0:  if INFO = -i, the ith argument to CGETRS had an illegal
             value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 391 of file cla_gerfsx_extended.f.