SLA_GERFSX_EXTENDED(3)

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✨ NAME ✨ SRC/sla_ge..._extended.f(3) — Library Functions Manual energy, bestie

Okay so this is the Library Functions Manual page for SRC/sla_ge..._extended.f(3), and the NAME section is literally:

SRC/sla_gerfsx_extended.f

That’s the whole NAME vibe. No cap. 🫶

💅 SYNOPSIS 💅

🎀 Functions/Subroutines 🎀

Here’s the subroutine signature, copied exactly because we do not play with code:

   subroutine sla_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)

And what does it do? SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. Iconic. 💅

🌈 Function/Subroutine Documentation 🌈

Here’s the full typed-out signature again, character for character:

   subroutine sla_gerfsx_extended (integer prec_type, integer trans_type,
       integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real,
       dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
       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, * ) errs_n, real,
       dimension( nrhs, * ) errs_c, 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)

And the one-liner is: SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

🧚 Purpose 🧚

Okay so the Purpose section is giving “I handle the extra-precise refinement part”:

SLA_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 SGERFSX to perform iterative refinement.

In addition to the normwise error bound, the code provides a maximum componentwise error bound if possible. See the 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. It specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER, and P can be:

  • '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 can be:

  • '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 REAL 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 REAL array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by SGETRF.

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 SGETRF; 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 REAL 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 REAL array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by SGETRS. 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 SLA_LIN_BERR.

N_NORMS

N_NORMS is INTEGER. It 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 — 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

Part 1 ends right here at the ERRS_C heading — part 2 picks up the rest. 🌈

okay bestie, we're back in the argument list and ERRS_C is literally the main character here — she's got layers:

  • ERRS_C — 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 — 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.

  • RES — REAL array, dimension (N). Workspace to hold the intermediate residual.
  • AYB — REAL array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL.
  • DY — REAL array, dimension (N). Workspace to hold the intermediate solution.
  • Y_TAIL — REAL array, dimension (N). Workspace to hold the trailing bits of the intermediate solution.
  • RCOND — 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 — 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 — 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 — 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 — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE..
  • INFO — INTEGER.
    • = 0: Successful exit.
    • < 0: if INFO = -i, the ith argument to SGETRS had an illegal value.

👑 Author 👑

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

Definition at line 391 of file sla_gerfsx_extended.f.

🤖 Author (the auto-generated one) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

and that's a wrap on this absolute unit of a manpage, bestie! 🎀 shoutout to LAPACK Version 3.12.0 and the file SRC/sla_ge..._extended.f(3) for bringing us this iconic content. mwah 💅✨

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