CGESVX(3)

💅 CGESVX(3) — Library Functions Manual 💅

Bestie, welcome to CGESVX(3) from the Library Functions Manual. Buckle up, because this one is a whole linear-algebra moment. 💅

🌸 NAME 🌸

The name line is literally this path, no cap:

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

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

subroutine CGESVX (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed,
    r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
     CGESVX computes the solution to system of linear equations A * X =
    B for GE matrices

🧚 Function/Subroutine Documentation 🧚

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 💖

Okay so 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. It’s giving “solve the system and also do a vibe check on how trustworthy the answer is.” ✨

🦋 Description 🦋

The following steps are performed:

1. 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').

2. 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.

3. 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.
4. The system of equations is solved for X using the factored form of A.
5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
6. 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 a CHARACTER*1 moment. It specifies whether or not 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 a 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. The number of linear equations, i.e., the order of the matrix A. N >= 0.

NRHS

NRHS is INTEGER. 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. 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. 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. 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. 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) — okay so this is the estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X, iykyk). If XTRUE is the true solution that goes with 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). 💅

And here's the vibe check: this estimate is exactly as reliable as the estimate for RCOND — like, they're besties, they stand or fall together — and it's almost always a slight overestimate of the true error. Slightly dramatic, never wrong in the dangerous direction. 🎀

BERR

BERR is a REAL array, dimension (NRHS) — the componentwise relative backward error of each solution vector X(j). Which literally means: the smallest relative change in any element of A or B that would make X(j) an exact solution. Not me measuring how much I'd have to fudge the inputs for you to be right 😭 — it's giving "how close to the problem were we actually solving". ✨

WORK

WORK is a COMPLEX array, dimension (2*N). That's it, that's the whole note. Scratch space, mind your own business, don't touch. 🫶

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 norm is used (yes bestie, that gap is literally in the source, the norm symbol did not survive the trip 💀).

If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. That also means the solution X, the condition estimator RCOND, and the forward error bound FERR could all be unreliable — like a group chat where everyone's quoting someone who was never there. Be suspicious. 👀

If the factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. Still useful, just partial. 🧚

INFO

INFO is INTEGER — the receipts, the status report, the whole plot:

• = 0: successful exit. We ate. 👑
• < 0: if INFO = -i, the i-th argument had an illegal value. Somebody passed something cursed. 🙈
• > 0: if INFO = i, and i is…
  • <= N: U(i,i) is exactly zero. The factorization was completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. And RCOND = 0 is returned. It's giving "the math said no" 😭.
  • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning 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 actually be more accurate than the value of RCOND would suggest. Trust issues with the condition number, honestly. 🔥

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Author

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

Definition at line 347 of file cgesvx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code. Not a human, no vibes to speak of, but she did her job. 🦋

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And that's the whole page, bestie — brought to you by LAPACK Version 3.12.0, sourced straight out of /home/abui...RC/cgesvx.f(3). Go forth and factorize responsibly. 💖✨🎀