la_gbrfsx_extended(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

, in arch rolling

🌸 NAME 🌸

okay so this is la_gbrfsx_extended(3) from the Library Functions Manual, and the name bit is literally serving:

la_gbrfsx_extended — la_gbrfsx_extended: step in gbrfsx.

✨ SYNOPSIS ✨

bestie, under Functions we have four precision-flavored subroutines. Same energy, different letter:

🧸 Functions 🧸

cla_gbrfsx_extended — CLA_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 and backward error estimates for the solution.

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)

dla_gbrfsx_extended — DLA_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 and backward error estimates for the solution.

subroutine dla_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 and backward error estimates for the solution.

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)

zla_gbrfsx_extended — ZLA_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 and backward error estimates for the solution.

subroutine zla_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)

💫 Detailed Description 💫

okay so here's the map, no cap: we got Function Documentation coming up, and it's giving subroutine details.

🫶 Function Documentation 🫶

🧚 subroutine cla_gbrfsx_extended 🧚

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)

okay so CLA_GBRFSX_EXTENDED is the main character here: it improves the computed solution to a system of linear equations for general banded matrices by doing extra-precise iterative refinement, and it provides error bounds and backward error estimates for the solution.

💖 Purpose 💖

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 the normwise error bound, the code provides the maximum componentwise error bound if possible. Peep the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those 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. 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
- P = 'D' → Double
- P = 'I' → Indigenous
- P = 'X' or P = '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 = '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 COMPLEX array, dimension (LDAB,N). On entry, it's the N-by-N matrix AB.
LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= max(1,N).
AFB — AFB is COMPLEX array, dimension (LDAF,N). It holds the factors L and U from the factorization A = P*L*U as computed by CGBTRF.
LDAFB — LDAFB 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 CGBTRF; 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.

Caveat, 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.

B — B is COMPLEX 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 array, dimension (LDY,NRHS). On entry, it's the solution matrix X, as computed by CGBTRS. On exit, it's 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 CLA_LIN_BERR.

N_NORMS — N_NORMS is INTEGER. It 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:

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 — ok so this one is a REAL array, dimension (NRHS, N_ERR_BNDS), and bestie, for every single right-hand side it's carrying around all the info about various error bounds and condition numbers matching up with 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 (the one the componentwise relative error depends on, duh) and by the type of error information, as described down below. Right now there are up to three pieces of info returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. And 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.

And the second index in ERR_BNDS_COMP(:,err) contains these 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 . (yep the source just trails off there, not me 😭) 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 a COMPLEX array, dimension (N). Workspace to hold the intermediate residual.

AYB — AYB is a REAL array, dimension (N). Workspace. That's it. It's a workspace. It's giving "I'm just here for the vibes" ✨

DY — DY is a COMPLEX array, dimension (N). Workspace to hold the intermediate solution.

Y_TAIL — Y_TAIL is a COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. (the crumbs 💅)

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 don't panic automatically, k?

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.

🌈 `dla_gbrfsx_extended` 🌈

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

DLA_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.

Purpose

DLA_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 DGBRFSX 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.

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
= 'D': Double
= 'I': Indigenous
= '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': Transpose
= '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 DOUBLE PRECISION array, dimension (LDAB,N). On entry, the N-by-N matrix AB.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDBA >= max(1,N). (yes, LDBA, that's literally what it says, I'm not fixing it 💀)

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

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 DGBTRF; 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 DOUBLE PRECISION array, dimension (LDB,NRHS). The right-hand-side matrix B.

LDB

(annnd the page cuts off right there mid-parameter — the docs keep going, bestie, hang tight 🙈)

`Y`

Y is DOUBLE PRECISION array, dimension (LDY,NRHS).

Okay so on entry this is the solution matrix X, the one you already got from DGBTRS 💅. On exit it's the improved solution matrix Y. Glow-up arc, honestly.

`LDY`

LDY is INTEGER — the leading dimension of the array Y. Constraint: LDY >= max(1,N).

Respect the max(1,N), bestie. It's there for a reason — zero-sized cases are real. 😭

`BERR_OUT`

BERR_OUT is DOUBLE PRECISION array, dimension (NRHS).

On exit, BERR_OUT(j) holds the componentwise relative backward error for right-hand-side j, calculated from this 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 one is computed by DLA_LIN_BERR. 🧮

`N_NORMS`

N_NORMS is INTEGER. It decides which error bounds come back (peep ERR_BNDS_NORM and ERR_BNDS_COMP):

If N_NORMS >= 1 → return normwise error bounds.
If N_NORMS >= 2 → return componentwise error bounds too.

`ERR_BNDS_NORM`

ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS).

For each right-hand side, this array is absolutely packed with info about various error bounds and condition numbers for the normwise relative error, which is defined like this:

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, described down 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 (and yeah the original sentence just cuts off there, no cap 💀). 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. Don't ask it for the others, it's not that girl.

See Lapack Working Note 165 for further details and extra cautions. 📚

`ERR_BNDS_COMP`

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS).

For each right-hand side, this array carries info about various error bounds and condition numbers for the componentwise relative error, defined as:

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 (the componentwise relative error depends on it) and the type of error information described below. There currently are up to three pieces of information returned for each right-hand side.

Heads up bestie: if componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. And 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 (same cut-off situation, we love a cliffhanger ✨). 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. Again. It's a one-field wonder.

See Lapack Working Note 165 for further details and extra cautions. 🎀

`RES`

RES is DOUBLE PRECISION array, dimension (N). Workspace to hold the intermediate residual. Very important little guy.

`AYB`

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

Passing this: this can be the same workspace passed for Y_TAIL. Reuse, reduce, recycle. ♻️

`DY`

DY is DOUBLE PRECISION array, dimension (N). Workspace to hold the intermediate solution.

`Y_TAIL`

Y_TAIL is DOUBLE PRECISION array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. The crumbs, the leftovers, the bits that didn't fit — iconic behaviour honestly.

`RCOND`

RCOND is DOUBLE PRECISION.

Reciprocal scaled condition number. It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that was done).

🚨 Real talk: if this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. But — and this is the plot twist — the error may still be small even if this number is very small and the matrix appears ill-conditioned. Don't panic-delete your code just because RCOND looks tragic. 💀

`ITHRESH`

ITHRESH is INTEGER.

The maximum number of residual computations allowed for refinement. The default is 10.

For 'aggressive' set it to 100 to permit convergence using approximate factorizations or factorizations other than LU. 🔥

⚠️ But like, 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. So go aggressive if you must, just know the receipts get shaky.

`RTHRESH`

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 it to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details. 🦋

`DZ_UB`

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, and we define stable 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.. She's giving "turn off the componentwise notifications" energy.

`INFO`

INFO is INTEGER.

= 0: Successful exit. 🎉
< 0: if INFO = -i, the ith argument to DGBTRS had an illegal value. Yeah, someone passed something cursed to DGBTRS — go check argument number i. 🔍

🎀 Author 🎀

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 404 of file dla_gbrfsx_extended.f. 📌

🌸 subroutine sla_gbrfsx_extended 🌸

Single-precision sibling has entered the chat. Same energy, different letters:

   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 provides error bounds and backward error
       estimates for the solution.

So SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by doing extra-precise iterative refinement, and it provides error bounds and backward error estimates for the solution. Same deal as the double-precision one, just in REAL. 🧚

Purpose:

For real, SLA_GBRFSX_EXTENDED is the glow-up routine — it takes a solution to a system of linear equations and makes it better by doing extra-precise iterative refinement, and it also hands you error bounds and backward error estimates for that solution. She's the one SGBRFSX calls when it needs iterative refinement done, iykyk. And bestie, it doesn't just do normwise error bounds — it gives you the maximum componentwise error bound too, when possible. 👑

The full deets on those bounds live in the comments for ERR_BNDS_NORM and ERR_BNDS_COMP, so go read those (they're right below, don't skip 💅). One thing to keep in your head though: this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. That's her whole lane. Periodt.

🌸 PARAMETERS 🌸

`PREC_TYPE` 🎚️

PREC_TYPE is INTEGER

Specifies the intermediate precision to be used in refinement, no cap. The value is defined by ILAPREC(P) where P is a CHARACTER, and P goes like:

'S' — Single
'D' — Double
'I' — Indigenous
'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 goes like:

'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.

`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. Yes bestie, plural right-hand-sides, she multitasks.

`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. Slay, precomputed.

`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). That's the whole tea.

`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 — so like, don't lie about it, the math is watching.

`C` 🧂

C is REAL array, dimension (N)

The column scale factors for A. If COLEQU = .FALSE., C is not accessed (she's not even looking at it). If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates.

Real talk: 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. So don't be messy with it 😭

`B` 🎯

B is REAL array, dimension (LDB,NRHS)

The right-hand-side matrix B. Iconic, essential, main character.

`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. So she walks in as X and leaves as Y — that's the arc, that's the character development.

`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. Give her her flowers 🌷

`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.

Basically a vibe check on how much you want back.

`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. (Yeah, that sentence just stops like that in the source — not me, not my business 💀) 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.

Heads up: this subroutine is only responsible for setting the second field above. And see Lapack Working Note 165 for further details and extra cautions — go read it, bestie, it's giving required reading 📖

`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 — she simply is not invited. 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. (Again, sentence just ends there, love that for us 🫠) 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.

Same deal as above: this subroutine is only responsible for setting the second field. See Lapack Working Note 165 for further details and extra cautions.

`RES` 🫧

RES is REAL array, dimension (N)

Workspace to hold the intermediate residual. Just a lil holding pen, no thoughts head empty.

`AYB` 🧺

AYB is REAL array, dimension (N)

Workspace. And listen — this can be the same workspace passed for Y_TAIL. Sharing is caring, we love efficiency.

`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. The leftovers, the crumbs, the bits that matter 💖

`RCOND` 👑

and the description for this one carries on in the next chunk — she's mid-entrance, hold that thought

RCOND — REAL. It's the reciprocal scaled condition number, bestie — a little estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration was done, that is). If this number is less than machine precision — and especially if it's flat-out zero — then the matrix is singular to working precision. 💀 But plot twist: the error can still be small even when this number is tiny and the matrix is giving full-on ill-conditioned. So don't panic-spiral just from this one number, mkay?
ITHRESH — INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For the 'aggressive' setting, crank it to 100 so it can permit convergence using approximate factorizations or factorizations that aren't LU. ⚠️ Heads up though: 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. No cap, that's literally what it says.
RTHRESH — REAL. This one decides when to stop refining 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 it to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details. 📚
DZ_UB — REAL. This is the vibe check for when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, and "stable" here means the relative change in each component is less than DZ_UB. The default value is 0.25, which requires the first bit to be stable. See LAWN 165 for more details. ✨
IGNORE_CWISE — LOGICAL. If .TRUE. then ignore componentwise convergence entirely. Default value is .FALSE.. Live your truth either way, but .FALSE. is the default for a reason.
INFO — INTEGER.
- = 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 at line 403 of file sla_gbrfsx_extended.f.

💜 subroutine zla_gbrfsx_extended

Okay so now we pivot to the complex*16 era, and honestly? The glow-up is real. 🌈

subroutine zla_gbrfsx_extended (integer prec_type, integer trans_type,
    integer n, integer kl, integer ku, integer nrhs, complex*16, dimension(
    ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb,
    integer ldafb, 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, * ) err_bnds_norm, double precision, dimension( nrhs, * )
    err_bnds_comp, 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_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. 💅

🦋 Purpose

!>
!> ZLA_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 ZGBRFSX 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 yeah — ZGBRFSX calls this icon when it needs the refinement done, and this routine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. Know your lane, slay your lane. 💅

🌸 Parameters

PREC_TYPE — INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER, and:
- = 'S': Single
- = 'D': Double
- = 'I': Indigenous
- = 'X' or 'E': Extra
TRANS_TYPE — INTEGER. Specifies the transposition operation on A. The value is defined by ILATRANS(T) where T is a CHARACTER, and:
- = 'N': No transpose
- = 'T': Transpose
- = 'C': Conjugate transpose
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. (Zero is legal, bestie, don't come for it. 🫶)
KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
NRHS — INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B.
AB — COMPLEX*16 array, dimension (LDAB,N). On entry, the N-by-N matrix A. 🎀
LDAB — INTEGER. The leading dimension of the array A. LDAB >= max(1,N).
AFB — COMPLEX*16 array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by ZGBTRF. The way ZGBTRF handed these to us… iconic. 🔥
LDAFB — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i). ✨
COLEQU — 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. Lowkey crucial, ngl.
C — 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. So like… don't do that.
B — COMPLEX*16 array, dimension (LDB,NRHS). The right-hand-side matrix B. 🧚
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Y — COMPLEX*16 array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by ZGBTRS. On exit, the improved solution matrix Y. Glow-up complete. ⭐
LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT — 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 — 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 a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) — bestie, for each right-hand side this one holds info about all the error bounds and condition numbers tied to the normwise relative error, and that's defined like this:

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 *i*th 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 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 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 *i*th 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 a COMPLEX*16 array, dimension (N). It's workspace to hold the intermediate residual. 🫶

`AYB`

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

`DY`

DY is a COMPLEX*16 array, dimension (N). Workspace to hold the intermediate solution.

`Y_TAIL`

Y_TAIL is a COMPLEX*16 array, dimension (N). Workspace to hold the trailing bits of the intermediate solution — yes, the little trailing-bit gremlins get their own spot. 💅

`RCOND`

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.

Okay but fr fr, don't panic just because RCOND looks tiny — the original says the error can still be small even when the matrix appears ill-conditioned. Read the note, bestie. 🩷

`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.

⚠️ That last bit is important: non-Gaussian-elimination factorization can make those guarantees stop being trustworthy. Not me warning you dramatically, but also yes. 💀

`RTHRESH`

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`

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`

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

`INFO`

INFO is INTEGER.

= 0: Successful exit. Slay. ✅
< 0: if INFO = -i, the *i*th argument to ZGBTRS had an illegal value.

Author 💖

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

Definition at line 403 of file zla_gbrfsx_extended.f.

Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

Okay that's the whole page, bestie — source: LAPACK, version 3.12.0, and this iconic little manual is la_gbrfsx_extended(3). Kisses, go trust your error bounds responsibly 💋✨

📦 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

la_gbrfsx_extended(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

🧸 Functions 🧸

💫 Detailed Description 💫

🫶 Function Documentation 🫶

🧚 subroutine cla_gbrfsx_extended 🧚

💖 Purpose 💖

🌈 Parameters 🌈

ERR_BNDS_COMP

Author 🎀

🌈 dla_gbrfsx_extended 🌈

Purpose

Parameters

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

🎀 Author 🎀

🌸 subroutine sla_gbrfsx_extended 🌸

Purpose:

🌸 PARAMETERS 🌸

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 👑

👑 Author

💜 subroutine zla_gbrfsx_extended

🦋 Purpose

🌸 Parameters

ERR_BNDS_COMP

RES

AYB

DY

Y_TAIL

RCOND

ITHRESH

RTHRESH

DZ_UB

IGNORE_CWISE

INFO

Author 💖

Author 🫶

`ERR_BNDS_COMP`

🌈 `dla_gbrfsx_extended` 🌈

`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`

`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` 👑

`ERR_BNDS_COMP`

`RES`

`AYB`

`DY`

`Y_TAIL`

`RCOND`

`ITHRESH`

`RTHRESH`

`DZ_UB`

`IGNORE_CWISE`

`INFO`