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## zla_syrfsx_extended()

subroutine zla_syrfsx_extended | ( | integer | prec_type,
|  | character | uplo,
|  | integer | n,
|  | integer | nrhs,
|  | complex*16, dimension( lda, * ) | a,
|  | integer | lda,
|  | complex*16, dimension( ldaf, * ) | af,
|  | integer | ldaf,
|  | integer, dimension( * ) | ipiv,
|  | logical | colequ,
|  | double precision, dimension( * ) | c,
|  | complex*16, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | complex*16, dimension( ldy, * ) | y,
|  | integer | ldy,
|  | double precision, dimension( * ) | berr_out,
|  | integer | n_norms,
|  | double precision, dimension( nrhs, * ) | err_bnds_norm,
|  | double precision, dimension( nrhs, * ) | err_bnds_comp,
|  | complex*16, dimension( * ) | res,
|  | double precision, dimension( * ) | ayb,
|  | complex*16, dimension( * ) | dy,
|  | complex*16, dimension( * ) | y_tail,
|  | double precision | rcond,
|  | integer | ithresh,
|  | double precision | rthresh,
|  | double precision | dz_ub,
|  | logical | ignore_cwise,
|  | integer | info )

ZLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Download ZLA_SYRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> ZLA_SYRFSX_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 ZSYRFSX to perform iterative refinement.
      !> 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. Note that this
      !> subroutine is only responsible for setting the second fields of
      !> ERR_BNDS_NORM and ERR_BNDS_COMP.
      !>

Parameters

  [in]

  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
      !>

  [in]

  UPLO

      !>          UPLO is CHARACTER*1
      !>       = 'U':  Upper triangle of A is stored;
      !>       = 'L':  Lower triangle of A is stored.
      !>

  [in]

  N

      !>          N is INTEGER
      !>     The number of linear equations, i.e., 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
      !>     matrix B.
      !>

  [in]

  A

      !>          A is COMPLEX*16 array, dimension (LDA,N)
      !>     On entry, the 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 COMPLEX*16 array, dimension (LDAF,N)
      !>     The block diagonal matrix D and the multipliers used to
      !>     obtain the factor U or L as computed by ZSYTRF.
      !>

  [in]

  LDAF

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

  [in]

  IPIV

      !>          IPIV is INTEGER array, dimension (N)
      !>     Details of the interchanges and the block structure of D
      !>     as determined by ZSYTRF.
      !>

  [in]

  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.
      !>

  [in]

  C

      !>          C is DOUBLE PRECISION 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.
      !>

  [in]

  B

      !>          B is COMPLEX*16 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]

  Y

      !>          Y is COMPLEX*16 array, dimension (LDY,NRHS)
      !>     On entry, the solution matrix X, as computed by ZSYTRS.
      !>     On exit, the improved solution matrix Y.
      !>

  [in]

  LDY

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

  [out]

  BERR_OUT

      !>          BERR_OUT is DOUBLE PRECISION 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 ZLA_LIN_BERR.
      !>

  [in]

  N_NORMS

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

  [in,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) * slamch('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) * 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 . 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.
      !>

  [in,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) * slamch('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) * 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 . 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.
      !>

  [in]

  RES

      !>          RES is COMPLEX*16 array, dimension (N)
      !>     Workspace to hold the intermediate residual.
      !>

  [in]

  AYB

      !>          AYB is DOUBLE PRECISION array, dimension (N)
      !>     Workspace.
      !>

  [in]

  DY

      !>          DY is COMPLEX*16 array, dimension (N)
      !>     Workspace to hold the intermediate solution.
      !>

  [in]

  Y_TAIL

      !>          Y_TAIL is COMPLEX*16 array, dimension (N)
      !>     Workspace to hold the trailing bits of the intermediate solution.
      !>

  [in]

  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.
      !>

  [in]

  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
      !>     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
      !>

  [in]

  RTHRESH

      !>          RTHRESH is DOUBLE PRECISION
      !>     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.
      !>

  [in]

  DZ_UB

      !>          DZ_UB is DOUBLE PRECISION
      !>     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.
      !>

  [in]

  IGNORE_CWISE

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

  [out]

  INFO

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

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