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

scipy.sparse.linalg.

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

Use Conjugate Gradient iteration to solve Ax = b.

Parameters:

A{sparse array, ndarray, LinearOperator}

The real or complex N-by-N matrix of the linear system.A must represent a hermitian, positive definite matrix. 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 array, ndarray, LinearOperator}

Preconditioner for A. M must represent a hermitian, positive definite matrix. It should approximate the inverse of A (see Notes). 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.

Returns:

xndarray

The converged solution.

infointeger

Provides convergence information:

0 : successful exit >0 : convergence to tolerance not achieved, number of iterations

Notes

The preconditioner M should be a matrix such that M @ A has a smaller condition number than A, see [2].

References

Examples

import numpy as np from scipy.sparse import csc_array from scipy.sparse.linalg import cg P = np.array([[4, 0, 1, 0], ... [0, 5, 0, 0], ... [1, 0, 3, 2], ... [0, 0, 2, 4]]) A = csc_array(P) b = np.array([-1, -0.5, -1, 2]) x, exit_code = cg(A, b, atol=1e-5) print(exit_code) # 0 indicates successful convergence 0 np.allclose(A.dot(x), b) True