/home/abui...extended.f(3)

🎀 /home/abui...extended.f(3) 🎀

okay bestie, welcome to the manual page for /home/abui...extended.f(3) from the Library Functions Manual. This is the SLA_GBRFSX_EXTENDED saga, and it is giving serious LAPACK main-character energy. 💅

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sla_gbrfsx_extended.f

✨ 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 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. She ate, fr fr. 🫶

🌟 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 provides error bounds and backward error estimates for the solution. iconic behavior, honestly. ✨

🎯 Purpose:

okay so here’s the tea: SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement, and it provides error bounds and backward error estimates for the solution. This subroutine is called by SGBRFSX to perform iterative refinement. In addition to 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 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. read that again, bestie — second fields only. 💅

🌸 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

precision check, no cap. 🎀

💜 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 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. 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.

bestie, this caveat is real: if C is input, each element of C should be a power of the radix, or the solution and error estimates might not be reliable. no skipping this one. 🚨

💜 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.

threshold girlies, assemble. 🌈

💜 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. iykyk. 📚

🌈 ERR_BNDS_COMP

(ERR_BNDS_COMP keeps going in part 2, bestie — no spoilers 💖)

🌸 ERR_BNDS_COMP — componentwise error-bound central 💅

okay so ERR_BNDS_COMP is a REAL array, dimension (NRHS, N_ERR_BNDS).

For each right-hand side, this array contains info 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. 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 . (bestie, the manpage sentence literally stops there 😭💀 — I’m not inventing the missing word). 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 REAL array, dimension (N). Workspace to hold the intermediate residual.

AYB

AYB is a REAL array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL.

DY

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

Y_TAIL

Y_TAIL is a 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.

• = 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.

🤖 Author

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole LAPACK Version 3.12.0 tea, bestie: source page /home/abui...extended.f(3). The footer’s date slot was blank in the copy I got, so I’m not making up a date — no cap, just vibes 💅✨🫶