, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None, atol=None)[source]#

Use Conjugate Gradient iteration to solve Ax = b.

A{sparse matrix, 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.


Right hand side of the linear system. Has shape (N,) or (N,1).


The converged solution.

Provides convergence information:

0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

Other Parameters:

Starting guess for the solution.

tol, atolfloat, optional

Tolerances for convergence, norm(residual) <= max(tol*norm(b), atol). The default for atol is 'legacy', which emulates a different legacy behavior.


The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.


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.


User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.


>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> 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_matrix(P)
>>> b = np.array([-1, -0.5, -1, 2])
>>> x, exit_code = cg(A, b)
>>> print(exit_code)    # 0 indicates successful convergence
>>> np.allclose(, b)