LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ cla_syrfsx_extended()

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

CLA_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 CLA_SYRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> CLA_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 CSYRFSX 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 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 array, dimension (LDAF,N)
!>     The block diagonal matrix D and the multipliers used to
!>     obtain the factor U or L as computed by CSYTRF.
!>
[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 CSYTRF.
!>
[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 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.
!>
[in]B
!>          B is COMPLEX 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 array, dimension (LDY,NRHS)
!>     On entry, the solution matrix X, as computed by CSYTRS.
!>     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 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.
!>
[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 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 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 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
!>     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 array, dimension (N)
!>     Workspace to hold the intermediate residual.
!>
[in]AYB
!>          AYB is REAL array, dimension (N)
!>     Workspace.
!>
[in]DY
!>          DY is COMPLEX array, dimension (N)
!>     Workspace to hold the intermediate solution.
!>
[in]Y_TAIL
!>          Y_TAIL is COMPLEX array, dimension (N)
!>     Workspace to hold the trailing bits of the intermediate solution.
!>
[in]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.
!>
[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 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.
!>
[in]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.
!>
[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 CLA_SYRFSX_EXTENDED had an illegal
!>             value
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.