xfa*ddlZddlmZdgZ ddZy)N) make_systemtfqmrc J|j} tj| tjrt} |j | }tj|jtjr|j | }t ||||\}}} }} tjj|dk(r|j} | | dfS|jd} |td| dz}||j} n||j| z } | }| j}| }|j|j| }|}dx}x}}tj|j| }|}tj|}|}|dk(r | | dfS|||z}nt!|||z}t#|D]}|dzdk(}|rBtj|j|}|dk(r | | dfcS||z }|||zz }||zz}||dz|z |z|zz}tjj||z }tjdd|dzzz }|||zz}|dz|z}|j|} | || zz } ||| |tj|dzz|kr+|rt%d j'|dz| | dfcS|sftj|j|}||z }!||!|zz}|!|z|!dz|zz}|j|j|}||z }|j|j}|}|}|rt%d j'dz| | |fS) a Use Transpose-Free Quasi-Minimal Residual 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`` using, e.g., `scipy.sparse.linalg.LinearOperator`. b : {ndarray} Right hand side of the linear system. Has shape (N,) or (N,1). x0 : {ndarray} Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b-Ax0), atol)``. The default for `tol` is 1.0e-5. The default for `atol` is ``tol * norm(b-Ax0)``. .. warning:: The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. Default is ``min(10000, ndofs * 10)``, where ``ndofs = A.shape[0]``. M : {sparse matrix, ndarray, LinearOperator} Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used. callback : function, optional User-supplied function to call after each iteration. It is called as `callback(xk)`, where `xk` is the current solution vector. show : bool, optional Specify ``show = True`` to show the convergence, ``show = False`` is to close the output of the convergence. Default is `False`. Returns ------- x : ndarray The converged solution. info : int Provides convergence information: - 0 : successful exit - >0 : convergence to tolerance not achieved, number of iterations - <0 : illegal input or breakdown Notes ----- The Transpose-Free QMR algorithm is derived from the CGS algorithm. However, unlike CGS, the convergence curves for the TFQMR method is smoothed by computing a quasi minimization of the residual norm. The implementation supports left preconditioner, and the "residual norm" to compute in convergence criterion is actually an upper bound on the actual residual norm ``||b - Axk||``. References ---------- .. [1] R. W. Freund, A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems, SIAM J. Sci. Comput., 14(2), 470-482, 1993. .. [2] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. .. [3] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, 1995. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import tfqmr >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = tfqmr(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True gri' g?rzNNNNF)numpyr utilsr__all__rr?r>rDs" )48).p%r?