SRC/cgesvx.f(3)

📦

version 3.12.1 · 4 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.9.0 · centos-stream 9
3.12.0 · centos-stream 10, fedora 43, fedora 44

okay bestie, this is SRC/cgesvx.f(3) from the Library Functions Manual — the cgesvx page, and it’s giving solver main-character energy. 💅

💖 NAME 💖

SRC/cgesvx.f — that’s the name on the label, no cap.

🌸 SYNOPSIS 🌸

🎀 Functions/Subroutines 🎀

subroutine cgesvx — CGESVX computes the solution to system of linear equations A * X = B for GE matrices

subroutine cgesvx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed,
    r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

🧚 FUNCTION/SUBROUTINE DOCUMENTATION 🧚

okay so here’s the full detailed signature, bestie — copy it exactly, it’s iconic:

subroutine cgesvx (character fact, character trans, integer n, integer nrhs,
    complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, *
    ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real,
    dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * )
    b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond,
    real, dimension( * ) ferr, real, dimension( * ) berr, complex,
    dimension( * ) work, real, dimension( * ) rwork, integer info)

CGESVX computes the solution to system of linear equations A * X = B for GE matrices

✨ Purpose ✨

CGESVX uses the LU factorization to compute the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided.

🔥 Description 🔥

okay so the following steps are performed, periodt:

If FACT = 'E', real scaling factors are computed to equilibrate the system:

TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

💅 Parameters 💅

`FACT`

FACT is CHARACTER*1. It says whether the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

= 'F': On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. A, AF, and IPIV are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.

`TRANS`

TRANS is CHARACTER*1. It specifies 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)

`N`

N is INTEGER. It’s the number of linear equations, i.e., the order of the matrix A. N >= 0.

`NRHS`

NRHS is INTEGER. It’s the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

`A`

A is COMPLEX array, dimension (LDA,N).

On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).

`LDA`

LDA is INTEGER. It’s the leading dimension of the array A. LDA >= max(1,N).

`AF`

AF is COMPLEX array, dimension (LDAF,N).

If FACT = 'F', then AF is an input argument and on entry contains the factors L and U from the factorization A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the original matrix A.

If FACT = 'E', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).

`LDAF`

LDAF is INTEGER. It’s the leading dimension of the array AF. LDAF >= max(1,N).

`IPIV`

IPIV is INTEGER array, dimension (N).

If FACT = 'F', then IPIV is an input argument and on entry contains 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).

If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = P*L*U of the original matrix A.

If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = P*L*U of the equilibrated matrix A.

`EQUED`

EQUED is CHARACTER*1. It specifies the form of equilibration that was done.

= 'N': No equilibration (always true if FACT = 'N').
= '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).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

`R`

R is REAL 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. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive.

`C`

C is REAL 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. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive.

`B`

B is COMPLEX array, dimension (LDB,NRHS).

On entry, the N-by-NRHS right hand side matrix B.

On exit,

if EQUED = 'N', B is not modified;
if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.

`LDB`

LDB is INTEGER. It’s the leading dimension of the array B. LDB >= max(1,N).

`X`

X is COMPLEX array, dimension (LDX,NRHS).

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

`LDX`

LDX is INTEGER. It’s the leading dimension of the array X. LDX >= max(1,N).

`RCOND`

RCOND is REAL.

The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.

`FERR`

FERR 💖

FERR is a REAL array, dimension (NRHS).

It's the estimated forward error bound for each solution vector X(j) — aka the j-th column of the solution matrix X, bestie. If XTRUE is the true solution corresponding to X(j), then FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. Slay, but make it cautious ✨

BERR 🌸

BERR is a REAL array, dimension (NRHS).

This one 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. It's giving "how much would I have to fudge the inputs for this to be perfect" energy 💅

WORK 🎀

WORK is a COMPLEX array, dimension (2*N). That's it. That's the whole note. Iconic minimalism 🫶

RWORK 🦋

RWORK is a REAL array, dimension (MAX(1,2*N)).

On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The source literally says The norm is used. — yes, with the double space, no cap 😭. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. Important tea, do not spill it 🔥

INFO 👑

INFO is INTEGER:

= 0: successful exit. We love a clean getaway ✨
< 0: if INFO = -i, the i-th argument had an illegal value. Rude, but documented 💀
> 0: if INFO = i, and i is
- <= N: U(i,i) 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. Big yikes 😭
- = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. Plot twist, but make it numerical 🌈

Author 🫶

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

Definition at line 347 of file cgesvx.f.

Author (the Doxygen one) ✨

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole vibe check, bestie — LAPACK Version 3.12.0, source file SRC/cgesvx.f(3). The date slot is literally not printed on this page, so we're not inventing one 🫶💅

📦 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