?gesv

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/22/2024

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?gesv

Computes the solution to the system of linear equations with a square coefficient matrix A and multiple right-hand sides.

Syntax

call sgesv( n, nrhs, a, lda, ipiv, b, ldb, info )

call dgesv( n, nrhs, a, lda, ipiv, b, ldb, info )

call cgesv( n, nrhs, a, lda, ipiv, b, ldb, info )

call zgesv( n, nrhs, a, lda, ipiv, b, ldb, info )

call dsgesv( n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork, iter, info )

call zcgesv( n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork, rwork, iter, info )

call gesv( a, b [,ipiv] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine solves for X the system of linear equations A*X = B, where A is an n-by-n matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = P*L*U, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations A*X = B.

The dsgesv and zcgesv are mixed precision iterative refinement subroutines for exploiting fast single precision hardware. They first attempt to factorize the matrix in single precision (dsgesv) or single complex precision (zcgesv) and use this factorization within an iterative refinement procedure to produce a solution with double precision (dsgesv) / double complex precision (zcgesv) normwise backward error quality (see below). If the approach fails, the method switches to a double precision or double complex precision factorization respectively and computes the solution.

The iterative refinement is not going to be a winning strategy if the ratio single precision performance over double precision performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ilaenv in the future. At present, iterative refinement is implemented.

The iterative refinement process is stopped if

iter > itermax

or for all the right-hand sides:

rnmr < sqrt(n)*xnrm*anrm*eps*bwdmax

where

iter is the number of the current iteration in the iterativerefinement process
rnmr is the infinity-norm of the residual
xnrm is the infinity-norm of the solution
anrm is the infinity-operator-norm of the matrix A
eps is the machine epsilon returned by dlamch (‘Epsilon’).

The values itermax and bwdmax are fixed to 30 and 1.0d+00 respectively.

Input Parameters

n	INTEGER. The number of linear equations, that is, the order of the matrix `A`; `n`≥ 0.
nrhs	INTEGER. The number of right-hand sides, that is, the number of columns of the matrix `B`; `nrhs`≥ 0.
a	REAL for sgesv DOUBLE PRECISION for dgesv and dsgesv COMPLEX for cgesv DOUBLE COMPLEX for zgesv and zcgesv. The array `a(size lda by *)` contains the `n`-by-`n` coefficient matrix `A`. The second dimension of `a` must be at least `max(1, n)`, the second dimension of `b` at least `max(1,nrhs)`.
b	REAL for sgesv DOUBLE PRECISION for dgesv and dsgesv COMPLEX for cgesv DOUBLE COMPLEX for zgesv and zcgesv. The array `b(size ldb by *)` contains the `n`-by-`nrhs` matrix of right hand side matrix `B`.
lda	INTEGER. The leading dimension of the array `a`; `lda≥ max(1, n)`.
ldb	INTEGER. The leading dimension of the array `b`; `ldb≥ max(1, n)`.
ldx	INTEGER. The leading dimension of the array `x`; `ldx≥ max(1, n)`.
work	DOUBLE PRECISION for dsgesv DOUBLE COMPLEX for zcgesv. Workspace array, size at least `max(1,n*nrhs)`. This array is used to hold the residual vectors.
swork	REAL for dsgesv COMPLEX for zcgesv. Workspace array, size at least `max(1,n*(n+nrhs))`. This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
rwork	DOUBLE PRECISION. Workspace array, size at least `max(1,n)`.

Output Parameters

a	Overwritten by the factors `L` and `U` from the factorization of `A = PLU`; the unit diagonal elements of `L` are not stored. If iterative refinement has been successfully used (info= 0 and iter≥ 0), then `A` is unchanged. If double precision factorization has been used (info= 0 and iter < 0), then the array `A` contains the factors `L` and `U` from the factorization `A = PLU`; the unit diagonal elements of `L` are not stored.
b	Overwritten by the solution matrix `X` for dgesv, sgesv,zgesv,zgesv. Unchanged for dsgesv and zcgesv.
ipiv	INTEGER. Array, size at least `max(1, n)`. The pivot indices that define the permutation matrix `P`; row `i` of the matrix was interchanged with row `ipiv(i)`. Corresponds to the single precision factorization (if info= 0 and iter≥ 0) or the double precision factorization (if info= 0 and iter < 0).
x	DOUBLE PRECISION for dsgesv DOUBLE COMPLEX for zcgesv. Array, size `ldx` by `nrhs`. If info = 0, contains the `n`-by-`nrhs` solution matrix `X`.
iter	INTEGER. If `iter < 0`: iterative refinement has failed, double precision factorization has been performed If `iter = -1`: the routine fell back to full precision for implementation- or machine-specific reason If `iter = -2`: narrowing the precision induced an overflow, the routine fell back to full precision If `iter = -3`: failure of sgetrf for dsgesv, or cgetrf for zcgesv If `iter = -31`: stop the iterative refinement after the 30th iteration. If `iter > 0`: iterative refinement has been successfully used. Returns the number of iterations.
info	INTEGER. If `info=0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, `U_{i, i}` (computed in double precision for mixed precision subroutines) is exactly zero. The factorization has been completed, but the factor `U` is exactly singular, so the solution could not be computed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gesv interface are as follows:

a	Holds the matrix `A` of size (`n,n`).
b	Holds the matrix `B` of size (`n,nrhs`).
ipiv	Holds the vector of length `n`.

NOTE:

Fortran 95 Interface is so far not available for the mixed precision subroutines dsgesv/zcgesv.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gesv

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

See Also