SRC/zgerfsx.f(3)

📦

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okay bestie, this is SRC/zgerfsx.f(3), from the Library Functions Manual — and it's giving “linear algebra but make it make sense.” Buckle up, hydrate, and let's go. 💖

💖 NAME 💖

SRC/zgerfsx.f — that's the name, that's the file, periodt. 🎀

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

here's the whole squad of arguments for zgerfsx — yes, it's a lot, but she's iconic for it:

   subroutine zgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c,
       b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
       err_bnds_comp, nparams, params, work, rwork, info)
       ZGERFSX

and the routine itself is ZGERFSX, no cap. 💅

🦋 Function/Subroutine Documentation 🦋

here's the full signature in all its Fortran glory — copy it exactly, bestie, character for character:

   subroutine zgerfsx (character trans, character equed, integer n, integer
       nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16,
       dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
       double precision, dimension( * ) r, double precision, dimension( * ) c,
       complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension(
       ldx , * ) x, integer ldx, double precision rcond, double precision,
       dimension( * ) berr, integer n_err_bnds, double precision, dimension(
       nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * )
       err_bnds_comp, integer nparams, double precision, dimension( * ) params,
       complex*16, dimension( * ) work, double precision, dimension( * ) rwork,
       integer info)
       ZGERFSX

ZGERFSX is the main character here. 🌟

🌸 Purpose 🌸

she improves the computed solution to a system of linear equations, and she provides error bounds and backward error estimates for that solution. slay. ✨
besides the normwise error bound, the code also 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. 👀
the original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED, R, and C below. in that case, the solution and error bounds returned are for the original unequilibrated system. important vibe check. 💚
some optional parameters are bundled in the PARAMS array. these settings determine how refinement is performed, but often the defaults are acceptable. if the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument. no cap, that's the move. 🫶

🎀 Parameters 🎀

okay, parameter parade time. every single one is here, bestie — no skips. 💅

🌟 TRANS

TRANS is CHARACTER*1 — it tells us the form of the system of equations:

= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)

🌟 EQUED

EQUED is CHARACTER*1 — it specifies the form of equilibration that was done to A before calling this routine. she needs this to compute the solution and error bounds correctly. fr fr. 🍓

= 'N': No equilibration
= 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
= 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).

if EQUED = 'B', the right hand side B has been changed accordingly. noted. 📝

🌟 N

N is INTEGER — the order of the matrix A. N >= 0. math girlie things. 🔢

🌟 NRHS

NRHS is INTEGER — the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. periodt. 💅

🌟 A

A is COMPLEX*16 array, dimension (LDA,N) — the original N-by-N matrix A. the blueprint, honestly. 👑

🌟 LDA

LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,N). ✨

🌟 AF

AF is COMPLEX*16 array, dimension (LDAF,N) — the factors L and U from the factorization A = P*L*U as computed by ZGETRF. she's already been factored, bestie. 🧚

🌟 LDAF

LDAF is INTEGER — the leading dimension of the array AF. LDAF >= max(1,N). 🦋

🌟 IPIV

IPIV is INTEGER array, dimension (N) — the pivot indices from ZGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). pivoting said “move, I'm gay.” 💀

🌟 R

R is DOUBLE PRECISION array, dimension (N) — the row scale factors for A.

if EQUED = 'R' or 'B', A is multiplied on the left by diag(R).
if EQUED = 'N' or 'C', R is not accessed.

💀 caveat, bestie: if R is accessed, each element of R 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 not a drill. 😭

🌟 C

C is DOUBLE PRECISION array, dimension (N) — the column scale factors for A.

if EQUED = 'C' or 'B', A is multiplied on the right by diag(C).
if EQUED = 'N' or 'R', C is not accessed.

💀 same caveat, same energy: if C is accessed, 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. keep it radix-powered or keep it moving. 🔥

🌟 B

B is COMPLEX*16 array, dimension (LDB,NRHS) — the right hand side matrix B. the “what we're solving for” energy. 💖

🌟 LDB

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

🌟 X

X is COMPLEX*16 array, dimension (LDX,NRHS) — on entry, the solution matrix X, as computed by ZGETRS. on exit, the improved solution matrix X. glow-up moment. 🌈

🌟 LDX

LDX is INTEGER — the leading dimension of the array X. LDX >= max(1,N). 🦋

🌟 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. she contains multitudes. 😭

🌟 BERR

BERR is DOUBLE PRECISION array, dimension (NRHS) — componentwise relative backward error. this is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). precise queen. 👑

🌟 N_ERR_BNDS

N_ERR_BNDS is INTEGER — number of error bounds to return for each right hand side and each type (normwise or componentwise). see ERR_BNDS_NORM and ERR_BNDS_COMP below. 🔍

🌟 ERR_BNDS_NORM

ERR_BNDS_NORM is 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 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) * dlamch('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) * dlamch('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) * dlamch('Epsilon') to determine if the error estimate is . (the original page literally leaves that blank, bestie — I'm just the messenger, not the editor 💀) 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.

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

💜 ERR_BNDS_COMP 💜

`ERR_BNDS_COMP`

okay so bestie, ERR_BNDS_COMP is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side it's holding all the info about various error bounds and condition numbers that line 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 (and yes, that's the one the componentwise relative error depends on), and then by the type of error information like this. 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) * dlamch('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) * dlamch('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) * dlamch('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.

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

`NPARAMS`

NPARAMS is INTEGER. It specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

`PARAMS`

PARAMS is a DOUBLE PRECISION array, dimension NPARAMS. It specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
- Default: 1.0D+0
- = 0.0: No refinement is performed, and no error bounds are computed.
- = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
- (other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) — Maximum number of residual computations allowed for refinement.
- Default: 10
- 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.
PARAMS(LA_LINRX_CWISE_I = 3) — Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (attempt componentwise convergence)

`WORK`

WORK is a COMPLEX*16 array, dimension (2*N).

`RWORK`

RWORK is a DOUBLE PRECISION array, dimension (2*N).

`INFO`

INFO is INTEGER.

= 0 — Successful exit. The solution to every right-hand side is guaranteed. 💖
< 0 — If INFO = -i, the i-th argument had an illegal value.
> 0 and <= N — U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
= N+J — The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 410 of file zgerfsx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

💖 and that's the whole page, bestie — the footer flex is LAPACK Version 3.12.0, file SRC/zgerfsx.f(3), source/version and all. May your factorization stay nonsingular and your INFO stay 0. ✨👑

📦 where this page comes from

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💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms