tKg`BddlZddlmZddlmZdgZd ddddddd d Zy) N)_get_atol_rtol) make_systemtfqmrgh㈵>F)rtolatolmaxiterMcallbackshowc"|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| j}|}tj |}|}|dk(r | | dfSt#d|||\}}t%|D]}|dzdk(}|rBtj|j|}|dk(r | | dfcS||z }|||zz }||zz}||dz|z |z|zz}tjj||z }tj dd |dzzz } ||| zz}| dz|z}|j|}!| ||!zz } ||| |tj |d zz|kr|rt'd |d z| | dfcS|sftj|j|}||z }"||"|zz}|"|z|"dz|zz}|j|j|}||z }|j|j}|}|}|rt'd d z| | |fS) aN 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. rtol, atol : float, optional Parameters for the convergence test. For convergence, ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. The default is ``rtol=1e-5``, the default for ``atol`` is ``0.0``. 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, atol=0.0) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True rri' rg?rz:TFQMR: Linear solve converged due to reach TOL iterations z@TFQMR: Linear solve not converged due to reach MAXIT iterations )dtypenp issubdtypeint64floatastyperlinalgnormcopyshapeminmatvecinner conjugaterealsqrtrrangeprint)#Abx0rr r r r r rx postprocessndofsruwrstarvuhatdthetaetarhorhoLastr0normtau_iterevenvtrstaralphauNextczbetas# e/home/alanp/www/video.onchill/myenv/lib/python3.12/site-packages/scipy/sparse/linalg/_isolve/tfqmr.pyrr sh GGE }}UBHH% HHUO }}QWWbhh' HHUO)!QA6Aq!Q  yy~~aB FFHA"" GGAJEeURZ( z FFH  O A A E !A DA ((5??$a ( - -CG WWS\F C {A""WfdD9GD!gax1} hhu0!4G"}#A++'ME ME UT\ E!S(1, , q!C' GGB!eQh,' ( uqy!tun HHQK S1W    QK a 4 '$$(F8-.NA& &((5??,a0C=DD1H At tQw!m+A88AHHQK(D IA88AHHUO,DAGSV   F8% & NG $$)N)numpyr iterativerutilsr__all__rrAr@rGs0% )j%Bej%rA