scipy.sparse.linalg.

bicg#

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

[1]

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

[2]

“Biconjugate gradient method”, Wikipedia, https://en.wikipedia.org/wiki/Biconjugate_gradient_method

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