minres — SciPy v1.15.3 Manual (original) (raw)

scipy.sparse.linalg.

scipy.sparse.linalg.minres(A, b, x0=None, *, rtol=1e-05, shift=0.0, maxiter=None, M=None, callback=None, show=False, check=False)[source]#

Use MINimum RESidual iteration to solve Ax=b

MINRES minimizes norm(Ax - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular.

If shift != 0 then the method solves (A - shift*I)x = b

Parameters:

A{sparse array, ndarray, LinearOperator}

The real symmetric N-by-N matrix of the linear system Alternatively, A can be a linear operator which can produce Ax using, e.g.,scipy.sparse.linalg.LinearOperator.

bndarray

Right hand side of the linear system. Has shape (N,) or (N,1).

Returns:

xndarray

The converged solution.

infointeger

Provides convergence information:

0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters:

x0ndarray

Starting guess for the solution.

shiftfloat

Value to apply to the system (A - shift * I)x = b. Default is 0.

rtolfloat

Tolerance to achieve. The algorithm terminates when the relative residual is below rtol.

maxiterinteger

Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

M{sparse array, ndarray, LinearOperator}

Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.

callbackfunction

User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

showbool

If True, print out a summary and metrics related to the solution during iterations. Default is False.

checkbool

If True, run additional input validation to check that A and_M_ (if specified) are symmetric. Default is False.

References

Solution of sparse indefinite systems of linear equations,

C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629.https://web.stanford.edu/group/SOL/software/minres/

This file is a translation of the following MATLAB implementation:

https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip

Examples

import numpy as np from scipy.sparse import csc_array from scipy.sparse.linalg import minres A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) A = A + A.T b = np.array([2, 4, -1], dtype=float) x, exitCode = minres(A, b) print(exitCode) # 0 indicates successful convergence 0 np.allclose(A.dot(x), b) True