scipy.sparse.linalg.minres¶
-
scipy.sparse.linalg.
minres
(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, M=None, callback=None, show=False, check=False)[source]¶ Use MINimum RESidual iteration to solve Ax=b
MINRES minimizes norm(A*x - 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 matrix, dense matrix, LinearOperator}
The real symmetric N-by-N matrix of the linear system Alternatively,
A
can be a linear operator which can produceAx
using, e.g.,scipy.sparse.linalg.LinearOperator
.- b{array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
- Returns
- x{array, matrix}
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
- x0{array, matrix}
Starting guess for the solution.
- tolfloat
Tolerance to achieve. The algorithm terminates when the relative residual is below tol.
- maxiterinteger
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
- M{sparse matrix, dense matrix, 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.
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_matrix >>> from scipy.sparse.linalg import minres >>> A = csc_matrix([[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