scipy.sparse.linalg.cgs#

scipy.sparse.linalg.cgs(A, b, x0=None, *, tol=<object object>, maxiter=None, M=None, callback=None, atol=0.0, rtol=1e-05)[source]#

Use Conjugate Gradient Squared iteration to solve Ax = b.

Parameters:

A{sparse matrix, ndarray, LinearOperator}: The real-valued 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).
x0ndarray: Starting guess for the solution.
rtol, atolfloat, optional: Parameters for the convergence test. For convergence, norm(b - A @ x) <= max(rtol*norm(b), atol) should be satisfied. The default is atol=0. and rtol=1e-5.
maxiterinteger: Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M{sparse matrix, 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.
tolfloat, optional, deprecated: Deprecated since version 1.12.0: cgs keyword argument tol is deprecated in favor of rtol and will be removed in SciPy 1.14.0.

Returns:

xndarray

The converged solution.

infointeger

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

Examples

>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import cgs
>>> R = np.array([[4, 2, 0, 1],
...               [3, 0, 0, 2],
...               [0, 1, 1, 1],
...               [0, 2, 1, 0]])
>>> A = csc_matrix(R)
>>> b = np.array([-1, -0.5, -1, 2])
>>> x, exit_code = cgs(A, b)
>>> print(exit_code)  # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True