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

scipy.sparse.linalg.

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

Use BIConjugate Gradient iteration to solve Ax = b.

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.

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. 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 <0 : parameter breakdown

Notes

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

References

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