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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine sla_porfsx_extended | ( | integer | prec_type, |
| character | uplo, | ||
| integer | n, | ||
| integer | nrhs, | ||
| real, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| real, dimension( ldaf, * ) | af, | ||
| integer | ldaf, | ||
| logical | colequ, | ||
| real, dimension( * ) | c, | ||
| real, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| real, dimension( ldy, * ) | y, | ||
| integer | ldy, | ||
| real, dimension( * ) | berr_out, | ||
| integer | n_norms, | ||
| real, dimension( nrhs, * ) | err_bnds_norm, | ||
| real, dimension( nrhs, * ) | err_bnds_comp, | ||
| real, dimension( * ) | res, | ||
| real, dimension(*) | ayb, | ||
| real, dimension( * ) | dy, | ||
| real, dimension( * ) | y_tail, | ||
| real | rcond, | ||
| integer | ithresh, | ||
| real | rthresh, | ||
| real | dz_ub, | ||
| logical | ignore_cwise, | ||
| integer | info ) |
SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
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!> !> SLA_PORFSX_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 SPORFSX 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. !>
| [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 REAL 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 REAL array, dimension (LDAF,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**T*U or A = L*L**T, as computed by SPOTRF. !> |
| [in] | LDAF | !> LDAF is INTEGER !> The leading dimension of the array AF. LDAF >= max(1,N). !> |
| [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 REAL 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 REAL array, dimension (LDY,NRHS) !> On entry, the solution matrix X, as computed by SPOTRS. !> 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 SLA_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 REAL array, dimension (N) !> Workspace to hold the intermediate residual. !> |
| [in] | AYB | !> AYB is REAL array, dimension (N) !> Workspace. This can be the same workspace passed for Y_TAIL. !> |
| [in] | DY | !> DY is REAL array, dimension (N) !> Workspace to hold the intermediate solution. !> |
| [in] | Y_TAIL | !> Y_TAIL is REAL 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 SPOTRS had an illegal !> value !> |
Definition at line 378 of file sla_porfsx_extended.f.
1.11.0