scipy.sparse.linalg.bicg

scipy.sparse.linalg.bicg(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None, atol=None)

Use BIConjugate Gradient iteration to solve Ax = b.

Parameters
A{sparse matrix, 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).

Returns
xndarray

The converged solution.

infointeger
Provides convergence information:

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

Other Parameters
x0ndarray

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.

Warning

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

maxiterinteger

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.

callbackfunction

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

Examples

>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import bicg
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = bicg(A, b)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True