?gesv

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

Syntax

lapack_int LAPACKE_sgesv (int matrix_layout , lapack_int n , lapack_int nrhs , float * a , lapack_int lda , lapack_int * ipiv , float * b , lapack_int ldb );

lapack_int LAPACKE_dgesv (int matrix_layout , lapack_int n , lapack_int nrhs , double * a , lapack_int lda , lapack_int * ipiv , double * b , lapack_int ldb );

lapack_int LAPACKE_cgesv (int matrix_layout , lapack_int n , lapack_int nrhs , lapack_complex_float * a , lapack_int lda , lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );

lapack_int LAPACKE_zgesv (int matrix_layout , lapack_int n , lapack_int nrhs , lapack_complex_double * a , lapack_int lda , lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );

lapack_int LAPACKE_dsgesv (int matrix_layout, lapack_int n, lapack_int nrhs, double * a, lapack_int lda, lapack_int * ipiv, double * b, lapack_int ldb, double * x, lapack_int ldx, lapack_int * iter);

lapack_int LAPACKE_zcgesv (int matrix_layout, lapack_int n, lapack_int nrhs, lapack_complex_double * a, lapack_int lda, lapack_int * ipiv, lapack_complex_double * b, lapack_int ldb, lapack_complex_double * x, lapack_int ldx, lapack_int * iter);

Include Files

  • mkl.h

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

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

n

The number of linear equations, that is, the order of the matrix A; n 0.

nrhs

The number of right-hand sides, that is, the number of columns of the matrix B; nrhs 0.

a

The array a(size max(1, lda*n)) contains the n-by-n coefficient matrix A.

b

The array bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout contains the n-by-nrhs matrix of right hand side matrix B.

lda

The leading dimension of the array a; lda max(1, n).

ldb

The leading dimension of the array b; ldb max(1, n) for column major layout and ldbnrhs for row major layout.

ldx

The leading dimension of the array x; ldx max(1, n) for column major layout and ldxnrhs for row major layout.

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), 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 = 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

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-1]. Corresponds to the single precision factorization (if info= 0 and iter≥ 0) or the double precision factorization (if info= 0 and iter < 0).

x

Array, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n) for row major layout. If info = 0, contains the n-by-nrhs solution matrix X.

iter

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.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, parameter i had an illegal value.

If info = i, Ui, 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.

For more complete information about compiler optimizations, see our Optimization Notice.