CLA_GERFSX_EXTENDED.F(3)

💅 SRC/cla_gerfsx_extended.f(3)

okay bestie, buckle up — this one lives in the Library Functions Manual and it's a whole mood, but it's also genuinely important so pay attention 🫶

✨ NAME ✨

SRC/cla_gerfsx_extended.f — the file itself, that's the whole name, iconic and minimal 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine cla_gerfsx_extended — the one and only, and honestly the star of the show 👑

subroutine cla_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)

that's CLA_GERFSX_EXTENDED in all her glory ✨

🦋 Function/Subroutine Documentation 🦋

subroutine cla_gerfsx_extended

okay so here's the whole signature, every single argument, no cap, not one of them dropped 🔥

subroutine cla_gerfsx_extended (integer prec_type, integer trans_type,
    integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda,
    complex, dimension( ldaf, * ) af, integer ldaf, 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, * )
    errs_n, real, dimension( nrhs, * ) errs_c, 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_GERFSX_EXTENDED — say it with your chest 💜

🎀 Purpose 🎀

okay so CLA_GERFSX_EXTENDED is the glow-up routine for a solution to a system of linear equations — it improves the computed solution by doing extra-precise iterative refinement, and it also hands you error bounds and backward error estimates for that solution. A full-service queen 👑

this subroutine is called by CGERFSX to do the iterative refinement — so it's not out here freelancing, it's got a job from CGERFSX ✨

besides the normwise error bound, the code provides a maximum componentwise error bound if possible — key phrase, "if possible", don't come for her 💅

see the comments for ERRS_N and ERRS_C for the deets on the error bounds. and ngl, important: this subroutine is only responsible for setting the second fields of ERRS_N and ERRS_C. that's it. that's her lane. periodt 🌈

🌟 Parameters 🌟

PREC_TYPE

PREC_TYPE is INTEGER — it specifies the intermediate precision to be used in refinement, and the value is defined by ILAPREC(P) where P is a CHARACTER and P can be any of these besties:

• = '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. and N >= 0, respect the constraint bestie 🫶

NRHS

NRHS is INTEGER — the number of right-hand-sides, i.e. the number of columns of the matrix B. iconic, plural, busy 💅

A

A is COMPLEX array, dimension (LDA,N). on entry, it's the N-by-N matrix A. the main character, literally her name 🌸

LDA

LDA is INTEGER — the leading dimension of the array A. and LDA >= max(1,N), don't test her 😭

AF

AF is COMPLEX array, dimension (LDAF,N). these are the factors L and U from the factorization A = P*L*U, as computed by CGETRF. she's pre-factored, she's ready, she ate 🔥

LDAF

LDAF is INTEGER — the leading dimension of the array AF. LDAF >= max(1,N), same energy, no exceptions ✨

IPIV

IPIV is INTEGER array, dimension (N) — the pivot indices from the factorization A = P*L*U as computed by CGETRF; row i of the matrix was interchanged with row IPIV(i). receipts! 📋

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, it actually matters, not just vibes 🙈

C

C is REAL array, dimension (N) — the column scale factors for A. details, because details are the whole thing:

• if COLEQU = .FALSE., C is not accessed, she's off duty 💀
• if C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. why? because scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows.
• and this is the part where you listen: 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, bestie, be careful 😭

B

B is COMPLEX array, dimension (LDB,NRHS) — the right-hand-side matrix B. the whole reason we're here tbh 🎀

LDB

LDB is INTEGER — the leading dimension of the array B. LDB >= max(1,N), always ⭐

Y

Y is COMPLEX array, dimension (LDY,NRHS). on entry, the solution matrix X, as computed by CGETRS. on exit, the improved solution matrix Y. that's the glow-up, that's the whole point 💖

LDY

LDY is INTEGER — the leading dimension of the array Y. LDY >= max(1,N). yes ma'am 🫡

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, credit where it's due 👑

N_NORMS

N_NORMS is INTEGER — it determines which error bounds to return (see ERRS_N and ERRS_C). the rules:

• if N_NORMS >= 1 return normwise error bounds
• if N_NORMS >= 2 return componentwise error bounds

that's it, that's the whole decision tree 🌈

ERRS_N

ERRS_N is 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 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. read that twice bestie 🔥
• 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

ERRS_C is 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. and listen:

• 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. no cap 🫶
• 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

…and this is exactly where part 1 stops — RES picks up in part 2, don't go anywhere bestie 🙈

RES

RES is COMPLEX array, dimension (N) — Workspace to hold the intermediate residual. 🫶

AYB

AYB is REAL array, dimension (N) — Workspace. ✨

DY

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

Y_TAIL

Y_TAIL is COMPLEX 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 ERRS_N and ERRS_C 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 CGETRS had an illegal value.

Author

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

Definition at line 391 of file cla_gerfsx_extended.f.

Author

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That's the whole page, bestie — footer's giving LAPACK Version 3.12.0 and the source path SRC/cla_ge..._extended.f(3). Byeee 💜