ZLA_GERFSX_EXTENDED(3)

📦
version 3.12.1 · 7 versions
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💅 NAME

This is the ZLA_GERFSX_EXTENDED(3) page from the Library Functions Manual — header vibe: SRC/zla_ge..._extended.f(3). The source file is SRC/zla_gerfsx_extended.f.

✨ SYNOPSIS

🧚 Functions/Subroutines

subroutine zla_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)

ZLA_GERFSX_EXTENDED

🦋 Function/Subroutine Documentation

subroutine zla_gerfsx_extended (integer prec_type, integer trans_type,
    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, * ) errs_n, double precision,
    dimension( nrhs, * ) errs_c, 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_GERFSX_EXTENDED

🌸 Purpose:

Okay bestie, ZLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations by doing extra-precise iterative refinement, and it provides error bounds plus backward error estimates for the solution. ZGERFSX calls this subroutine to perform iterative refinement. Besides the normwise error bound, the code provides the maximum componentwise error bound if possible. Peep the comments for ERRS_N and ERRS_C for the full tea on those error bounds. Important note: this subroutine is only responsible for setting the second fields of ERRS_N and ERRS_C.

🎀 Parameters

  • PREC_TYPE — PREC_TYPE is INTEGER. It 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. It 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*16 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*16 array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by ZGETRF.
  • 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 ZGETRF; 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 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.
  • B — B is COMPLEX*16 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*16 array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by ZGETRS. 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 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.

  • 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 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:
        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 — 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.

  • ERRS_C

ERRS_C — the componentwise error-bound tea ☕💅

ERRS_C is a 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:

                   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 — 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.
Bestie, the source sentence really does trail off at “the error estimate is .” — I’m not fixing it, I’m preserving the exact chaos 😭💀

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

RES — residual workspace 🫧

RES is a COMPLEX*16 array, dimension (N).

Workspace to hold the intermediate residual.

AYB — workspace, that’s it 🌸

AYB is a DOUBLE PRECISION array, dimension (N).

Workspace.

DY — intermediate solution holder ✨

DY is a COMPLEX*16 array, dimension (N).

Workspace to hold the intermediate solution.

Y_TAIL — trailing-bits storage 🎀

Y_TAIL is a COMPLEX*16 array, dimension (N).

Workspace to hold the trailing bits of the intermediate solution.

RCOND — reciprocal scaled condition number 🔥

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.

ITHRESH — refinement-attempt limit 👑

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 — when refinement says “enough” 💅

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.

DZ_UB — componentwise convergence starter pistol 🎯

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.

IGNORE_CWISE — the “skip componentwise convergence” switch 🙈

IGNORE_CWISE is LOGICAL.

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

INFO — exit-status bestie 🚦

INFO is INTEGER.

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

💖 Author 💖

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

Definition at line 390 of file zla_gerfsx_extended.f.

🎀 Author (the generated-by-Doxygen remix) 🎀

Generated automatically by Doxygen for LAPACK from the source code.

Footer sign-off, preserved with love: LAPACK Version 3.12.0, source SRC/zla_ge..._extended.f(3), and the footer’s date/version stamp energy — all kept exactly where it belongs, bestie 💖✨
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