scipy.sparse.linalg.

minres#

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]#

Solve Ax = b with the MINimum RESidual method, for a symmetric A.

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.

infoint
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.

rtolfloat

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

shiftfloat

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

maxiterint

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.

Notes

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

References

[1]

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/

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
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