bicg#
- scipy.sparse.linalg.bicg(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None)[source]#
- Solve - Ax = bwith 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 - Axand- A^T xusing, 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- xkis 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 @ Ahas 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