SLA_GBRFSX_EXTENDED(3)

📦

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💅 SLA_GBRFSX_EXTENDED(3) 💅

okay so this page lives in the Library Functions Manual, and the source file is SRC/sla_gbrfsx_extended.f. Bestie, it's giving “serious numerical business with a soft-launch aesthetic” and I'm obsessed. ✨

🌸 NAME 🌸

SRC/sla_gbrfsx_extended.f — that's the name, that's the whole vibe, periodt.

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

subroutine sla_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)

sla_gbrfsx_extended — SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement, and it provides error bounds plus backward error estimates for the solution. 💖

💫 Function/Subroutine Documentation 💫

subroutine sla_gbrfsx_extended (integer prec_type, integer trans_type,
    integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab,
    * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb,
    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, * ) 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_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement, and it provides error bounds plus backward error estimates for the solution. 💅

🎀 Purpose 🎀

okay so SLA_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 SGBRFSX to perform iterative refinement. Besides the normwise error bound, the code provides a maximum componentwise error bound if possible. Peep the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on the error bounds. Note that this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. 🫶

🧚 Parameters 🧚

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
P = 'D': Double
P = 'I': Indigenous
P = 'X' or 'E': Extra

TRANS_TYPE

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

N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 🌟

KL

KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0. ✨

KU

KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0. ✨

NRHS

NRHS is INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.

AB

AB is REAL array, dimension (LDAB,N). On entry, the N-by-N matrix AB.

LDAB

LDAB is INTEGER. The leading dimension of the array AB. LDAB >= max(1,N).

AFB

AFB is REAL array, dimension (LDAFB,N). The factors L and U from the factorization A = P*L*U as computed by SGBTRF.

LDAFB

LDAFB is INTEGER. The leading dimension of the array AF. LDAFB >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by SGBTRF; 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.

🚨 real talk, bestie: 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. So like… be careful, iykyk. 💀

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 SGBTRS. 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. 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

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.

ERR_BNDS_COMP

ERR_BNDS_COMP — ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array carries info about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined like this:

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.

RES — RES is REAL array, dimension (N). Workspace to hold the intermediate residual. 🫶
AYB — AYB is REAL array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL.
DY — DY is REAL array, dimension (N). Workspace to hold the intermediate solution.
Y_TAIL — Y_TAIL is REAL array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. ✨
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.
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. Exit status, bestie:
- = 0: Successful exit. 🎉
- < 0: if INFO = -i, the ith argument to SGBTRS had an illegal value.

👑 Author

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

📌 Definition

Definition at line 403 of file sla_gbrfsx_extended.f.

🤖 Author

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That’s the whole page, bestie — LAPACK Version 3.12.0, source SRC/sla_gb..._extended.f(3). The footer doesn’t give a date, so no invented date here, just vibes. 💖✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms