scipy.sparse.linalg.

cg#

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

[1]

“Conjugate Gradient Method, Wikipedia, https://en.wikipedia.org/wiki/Conjugate_gradient_method

[2]

“Preconditioner”, Wikipedia, https://en.wikipedia.org/wiki/Preconditioner

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