CLA_GBRFSX_EXTENDED(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

okay so welcome to SRC/cla_gb..._extended.f(3) — that's the page title — and it lives in the Library Functions Manual (yes, the source header repeats the manual name, and we are folding that in near the top instead of copy-pasting the whole line, because duplicate header energy is not the vibe 💀). This is Part 1, bestie, and it is long, so hydrate before we start 🫶.

💖 NAME 💖

SRC/cla_gbrfsx_extended.f — that's the file this page is documenting, periodt.

✨ SYNOPSIS ✨

Functions/Subroutines

the whole squad of arguments, laid out exactly:

subroutine cla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs,
    ab, ldab, afb, ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out,
    n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond,
    ithresh, rthresh, dz_ub, ignore_cwise, info)

CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. iconic, honestly 🎀.

🛠️ Function/Subroutine Documentation 🛠️

subroutine cla_gbrfsx_extended

here's the full typed-out signature, copied exactly, no edits, no "I fixed a typo" behavior:

subroutine cla_gbrfsx_extended (integer prec_type, integer trans_type,
    integer n, integer kl, integer ku, integer nrhs, complex, dimension(
    ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer
    ldafb, 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_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. that's the whole moment, no cap ✨.

🎯 Purpose:

okay so the purpose block is giving main-character solver energy:

  • CLA_GBRFSX_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 CGBRFSX 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.

so yes, it's doing the extra-precise glow-up and handing you the receipts 💅.

🧾 Parameters

  • PREC_TYPE — officially: 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
    • P = 'D': Double
    • P = 'I': Indigenous
    • P = 'X' or 'E': Extra
  • TRANS_TYPE — officially: TRANS_TYPE is INTEGER. Specifies the transposition operation on A. The value is defined by ILATRANS(T) where T is a CHARACTER and:
    • T = 'N': No transpose
    • T = 'T': Transpose
    • T = 'C': Conjugate transpose
  • N — officially: N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • KL — officially: KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — officially: KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • NRHS — officially: NRHS is INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.
  • AB — officially: AB is COMPLEX array, dimension (LDAB,N). On entry, the N-by-N matrix AB.
  • LDAB — officially: LDAB is INTEGER. The leading dimension of the array AB. LDAB >= max(1,N).
  • AFB — officially: AFB is COMPLEX array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by CGBTRF.
  • LDAFB — officially: LDAFB is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — officially: IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by CGBTRF; row i of the matrix was interchanged with row IPIV(i).
  • COLEQU — officially: 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 — officially: 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. this is the "please don't mess with the scaling unless you know what you're doing" warning, bestie 🫶.
  • B — officially: B is COMPLEX array, dimension (LDB,NRHS). The right-hand-side matrix B.
  • LDB — officially: LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • Y — officially: Y is COMPLEX array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by CGBTRS. On exit, the improved solution matrix Y.
  • LDY — officially: LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
  • BERR_OUT — officially: 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.

  • N_NORMS — officially: 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.
  • ERR_BNDS_NORM — officially: 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:
        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 . (yes, the source literally leaves that blank — I am not inventing a word, that's a technical-detail no-no 💀) 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.

🎀 ERR_BNDS_COMP 🎀

Okay so ERR_BNDS_COMP is a 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:

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. 🦋

🌸 RES 🌸

RES is COMPLEX array, dimension (N). Workspace to hold the intermediate residual. Literally just holding the intermediate residual, no cap.

🎀 AYB 🎀

AYB is REAL array, dimension (N). Workspace. That's it, bestie — workspace vibes.

✨ DY ✨

DY is COMPLEX array, dimension (N). Workspace to hold the intermediate solution. Main character intermediate solution energy.

💜 Y_TAIL 💜

Y_TAIL is COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. The trailing bits are lowkey iconic.

🫶 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. So yeah — tiny RCOND does not automatically mean the error is huge, okay? 🙈

🔥 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. ⚠️

🌈 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.

⭐ 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.

🍓 IGNORE_CWISE 🍓

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

🧚 INFO 🧚

INFO is INTEGER.

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

👑 Author 👑

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

Definition at line 403 of file cla_gbrfsx_extended.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

💖 That's the whole page, bestie! This one's from LAPACK Version 3.12.0, file SRC/cla_gb..._extended.f(3) — mwah, slay, bye! 💅✨

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