gesvx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💅 `gesvx(3)` — Library Functions Manual ✨

okay bestie, welcome to gesvx(3) from the Library Functions Manual — this is the expert-level “factor and solve” moment for GE matrices, and we are gonna get you through it without losing a single detail. no cap. 🫶

🌸 NAME 🌸

gesvx — gesvx: factor and solve, expert

🦋 SYNOPSIS 🦋

✨ Functions ✨

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

💖 Detailed Description 💖

this is the big-picture section, and honestly it’s giving main character energy 💫

🧚 Function Documentation 🧚

✨ `subroutine cgesvx` ✨

   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. 💖

Description:

The following steps are performed:

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. 😭 but it keeps going, iconic
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 and it specifies whether the factored form of matrix A is supplied on entry, and if not, whether A should be equilibrated before it is factored. the tea:
- = 'F': on entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', A has been equilibrated with scaling factors given by R and C. A, AF, and IPIV are not modified. 🫶
- = 'N': matrix A will be copied to AF and factored. ✨
- = 'E': matrix A will be equilibrated if necessary, then copied to AF and factored. 💖
TRANS — TRANS is CHARACTER*1 and 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 matrix A. N >= 0.
NRHS — NRHS is INTEGER, the number of right hand sides, i.e. the number of columns of 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 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 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 and 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 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 array X. LDX >= max(1,N).
RCOND
RCOND — it’s REAL, bestie. 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 — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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.
BERR — REAL array, dimension (NRHS). 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).
WORK — COMPLEX array, dimension (2*N).
RWORK — 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. 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.
INFO — INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 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.
  - = 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.

🌸 Author 🌸

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

Definition at line 347 of file cgesvx.f.

🦋 subroutine dgesvx 🦋

   subroutine dgesvx (character fact, character trans, integer n, integer nrhs,
       double precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
       character equed, double precision, dimension( * ) r, double precision,
       dimension( * ) c, double precision, dimension( ldb, * ) b, integer ldb,
       double precision, dimension( ldx, * ) x, integer ldx, double precision
       rcond, double precision, dimension( * ) ferr, double precision,
       dimension( * ) berr, double precision, dimension( * ) work, integer,
       dimension( * ) iwork, integer info)

DGESVX computes the solution to system of linear equations A * X = B for GE matrices — periodt 💅.

🎯 Purpose 🎯

DGESVX uses the LU factorization to compute the solution to a real 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. Iconic behavior, honestly. ✨

🌈 Description 🌈

The following steps are performed:

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

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 — CHARACTER*1. 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 — CHARACTER*1. 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 (Transpose)
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
A — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — DOUBLE PRECISION 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 DGETRF. 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 — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — 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 DGETRF; 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 — CHARACTER*1. Specifies the form of equilibration that was done. 💅 Values: '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 — 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. 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 — 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. 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 — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,N). ✨
X — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array X. LDX >= max(1,N). 🌟
RCOND — DOUBLE PRECISION. 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 — DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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. 💜
BERR — DOUBLE PRECISION array, dimension (NRHS). 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). 🌈
WORK — DOUBLE PRECISION array, dimension (MAX(1,4*N)). On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If WORK(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 WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. 😭
IWORK — INTEGER array, dimension (N). 🍓
INFO — INTEGER. Exit status. 💀
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 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.
  - = 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.

Author

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

Definition at line 346 of file dgesvx.f.

🌟 subroutine `sgesvx` 🌟

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

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

Purpose

SGESVX uses the LU factorization to compute the solution to a real 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

The following steps are performed:

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 — CHARACTER*1. 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 — CHARACTER*1. 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 (Transpose)
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 🦋
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 💖
A —

A is up next, and she’s the whole reason we’re here 💅

A — A is REAL 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 REAL 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 SGETRF. 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 SGETRF; 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. 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 REAL 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 REAL 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 is REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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.
BERR — BERR is REAL array, dimension (NRHS). 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).
WORK — WORK is REAL array, dimension (MAX(1,4*N)). On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If WORK(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 WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.
IWORK — IWORK is INTEGER array, dimension (N).
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 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.
  - = 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.

👑 Author 👑

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

Definition at line 346 of file sgesvx.f.

   subroutine zgesvx (character fact, character trans, integer n, integer nrhs,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character
       equed, 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( * ) ferr, double precision, dimension( * ) berr,
       complex*16, dimension( * ) work, double precision, dimension( * ) rwork,
       integer info)

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

💖 Purpose 💖

ZGESVX 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: ✨

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

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. slay.
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 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. lowkey the “do you already have the factorization?” switch.

= '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. no touchy, fr fr.
= '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. 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*16 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*16 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 ZGETRF. 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.

🫶 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 ZGETRF; 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. it’s giving “which scaling glow-up happened?”

= '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 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. 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 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. 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*16 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*16 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 DOUBLE PRECISION. 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 is DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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.

🔮 BERR 🔮

BERR is DOUBLE PRECISION array, dimension (NRHS). 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).

🌈 WORK 🌈

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

🎀 RWORK 🎀

RWORK is DOUBLE PRECISION array, dimension (MAX(1,2*N)). On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. 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.

💅 `INFO` 💅

okay so INFO is an INTEGER, and it’s basically the exit-tea ☕ — here’s the vibe check:

= 0 — successful exit, slay ✨
< 0 — if INFO = -i, the i-th argument had an illegal value 🙈
> 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. 💀
- = 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. 🫶

👑 Author 👑

the icons behind this:

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

Definition at line 347 of file zgesvx.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

okay bestie, that’s the whole page 🎀 source/version realness: LAPACK, Version 3.12.0, and the footer tag is gesvx(3). mwah 💋

📦 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

gesvx(3)

💅 gesvx(3) — Library Functions Manual ✨

🌸 NAME 🌸

🦋 SYNOPSIS 🦋

✨ Functions ✨

💖 Detailed Description 💖

🧚 Function Documentation 🧚

✨ subroutine cgesvx ✨

💅 Parameters 💅

🌸 Author 🌸

🦋 subroutine dgesvx 🦋

🎯 Purpose 🎯

🌈 Description 🌈

💖 Parameters 💖

Author

🌟 subroutine sgesvx 🌟

Purpose

Description

Parameters

👑 Author 👑

💖 Purpose 💖

🦋 Description 🦋

🎀 Parameters 🎀

💅 FACT 💅

🌈 TRANS 🌈

🌸 N 🌸

🦋 NRHS 🦋

✨ A ✨

💜 LDA 💜

🎀 AF 🎀

🫶 LDAF 🫶

🔥 IPIV 🔥

👑 EQUED 👑

🍓 R 🍓

🧚 C 🧚

⭐ B ⭐

💖 LDB 💖

🌟 X 🌟

🦋 LDX 🦋

💀 RCOND 💀

💎 FERR 💎

🔮 BERR 🔮

🌈 WORK 🌈

🎀 RWORK 🎀

💅 INFO 💅

👑 Author 👑

🫶 Author 🫶

💅 `gesvx(3)` — Library Functions Manual ✨

✨ `subroutine cgesvx` ✨

🌟 subroutine `sgesvx` 🌟

💅 `INFO` 💅