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

source: /scipy/sparse/linalg/_isolve/iterative.py:25

Signature

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

Summary

Solve Ax = b with the BIConjugate Gradient method.

Parameters

A : {sparse array, ndarray, LinearOperator}: The real or complex N-by-N matrix of the linear system. Alternatively, A can be a linear operator which can produce Ax and A^T x 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 : int: 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. 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 : int

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

Notes

The preconditioner M should be a matrix such that M @ A has a smaller condition number than A, see ^[1] .

Examples

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

✓

True

Aliases

scipy.sparse.linalg.bicg