U Gv faã@s(ddlZddlmZdgZddd„ZdS) éNé)Ú make_systemÚtfqmrçñh㈵øä>Fc "Cs|j} t | tj¡r"t} | | ¡}t |jtj¡r<| | ¡}t||||ƒ\}}} }} tj |¡dkrx|  ¡} | | ƒdfS|j d} |dkr˜t d| dƒ}|dkrª|  ¡} n||  | ¡} | }|   ¡}| }|  |  | ¡¡}|}d}}}t  | ¡| ¡}|}t |¡}|}|dkr | | ƒdfS|dkr4||}nt|||ƒ}t|ƒD]}|ddk}|r t  | ¡|¡}|dkrŒ| | ƒdfS||}|||}|||8}||d|||}tj |¡|}t dd |d¡}|||9}|d|}|  |¡} | || 7} |dk r*|| ƒ|t |d ¡|krj|rZtd  |d ¡ƒ| | ƒdfS|sÂt  | ¡|¡}||}!||!|}|!||!d|}|  |  |¡¡}||7}n|  |  |¡¡}|}|}qJ|rötd  |d ¡ƒ| | ƒ|fS) aÊ 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 grNi'é ééÿÿÿÿgð?rzs ÿ