`scipy.sparse.linalg._isolve.minres:minres`

source: /scipy/sparse/linalg/_isolve/minres.py:12

Signature

def minres(A, b, x0 = None, *, rtol = 1e-05, shift = 0.0, maxiter = None, M = None, callback = None, show = False, check = False)

Summary

Solve Ax = b with the MINimum RESidual method, for a real symmetric or complex hermitian matrix A.

Extended Summary

MINRES minimizes norm(Ax - b) for a real symmetric or complex hermitian 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 or complex hermitian 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.
b : ndarray: Right hand side of the linear system. Has shape (N,) or (N,1).

Returns

x : ndarray

The converged solution.

info : int

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

Other Parameters

x0 : ndarray: Starting guess for the solution.
rtol : float: Tolerance to achieve. The algorithm terminates when the relative residual is below rtol.
shift : float: Value to apply to the system (A - shift * I)x = b. Default is 0.
maxiter : int: 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.
callback : function: User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
show : bool: If True, print out a summary and metrics related to the solution during iterations. Default is False.
check : bool: If True, run additional input validation to check that A and M (if specified) are symmetric. Default is False.

Notes

This file is a translation of the MATLAB implementation ^[2].

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
np.allclose(A.dot(x), b)

✓

True