`scipy.sparse.linalg._isolve.iterative:cg`

source: /scipy/sparse/linalg/_isolve/iterative.py :307

Signature

   def     cg (    A    ,    b    ,    x0    =  None   ,  * ,     rtol    =  1e-05   ,    atol    =  0.0   ,    maxiter    =  None   ,    M    =  None   ,    callback    =  None     )

Summary

Solve Ax = b with the Conjugate Gradient method, for a symmetric, positive-definite A.

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.
b : ndarray: Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : ndarray: Starting guess for the solution.
rtol, atol : float, 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.
maxiter : integer: 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.
callback : function: User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Returns

x : ndarray

The converged solution.

info : integer

Provides convergence information:: 0successful 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].

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

✓

Aliases

scipy.sparse.linalg.cg