# ?gesv

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

## Syntax

FORTRAN 77:

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, iter, info )

Fortran 95:

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

## Description

This 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, b

  REAL for sgesv

  DOUBLE PRECISION for dgesv and dsgesv

  COMPLEX for cgesv

  DOUBLE COMPLEX for zgesv and zcgesv.

  Arrays: `a(lda,*)`, `b(ldb,*)`.

  The array a contains the n-by-n coefficient matrix A.

  The array b contains the n-by-nrhs matrix of right hand side matrix B.

  The second dimension of a must be at least `max(1, n)`, the second dimension of b at least `max(1,nrhs)`.

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, DIMENSION at least `max(1,n*nrhs)`. This array is used to hold the residual vectors.

swork

  REAL for dsgesv

  COMPLEX for zcgesv.

  Workspace array, DIMENSION at least `max(1,n*nrhs)`. This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.

## Output Parameters

a

  Overwritten by the factors L and U from the factorization of `A = P*L*U`; the unit diagonal elements of L are not stored.

  If iterative refinement has been successfully used (info= 0 and iter≥ 0, see description below), then A is unchanged.

  If double precision factorization has been used (info= 0 and iter < 0, see description below), then the array A contains the factors L and U from the factorization `A = P*L*U`; 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, DIMENSION 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, DIMENSION (ldx, 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`: taking into account machine parameters, n, nrhs, it is a priori not worth working in single precision

  - If `iter = -2`: overflow of an entry when moving from double to single precision

  - If `iter = -3`: failure of sgetrf

  - If `iter = -31`: stop the iterative refinement after the 30th iteration.

  If `iter > 0`: iterative refinement has been sucessfully 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.

## Fortran 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 Fortran 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.

## See Also

ilaenv Environmental enquiry function that returns values for tuning algorithmic performance.
dlamch Determines machine parameters for floating-point arithmetic.
sgetrf Computes the LU factorization of a general m-by-n matrix.
Parent topic: Driver Routines