bicgstab#
- scipy.sparse.linalg.bicgstab(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None)[source]#
Use BIConjugate Gradient STABilized 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
andA^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 isatol=0.
andrtol=1e-5
.- 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. 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)
, wherexk
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 stabilized method”, Wikipedia, https://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method
Examples
>>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import bicgstab >>> R = np.array([[4, 2, 0, 1], ... [3, 0, 0, 2], ... [0, 1, 1, 1], ... [0, 2, 1, 0]]) >>> A = csc_matrix(R) >>> b = np.array([-1, -0.5, -1, 2]) >>> x, exit_code = bicgstab(A, b, atol=1e-5) >>> print(exit_code) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True