a h! @s`ddlZddlmZddlmZddlmZmZdgZ eddd eddddd d d d dZ dS)N)_get_atol_rtol) make_system)_NoValue_deprecate_positional_argstfqmrz1.14.0)versiongh㈵>F)tolmaxiterMcallbackatolrtolshowc$Cs|j} t| tjr"t} || }t|jtjr<|| }t||||\}}} }} tj|dkrx| } | | dfS|j d} |durt d| d}|dur| }n|| | }|}| }|}| | |}|}d}}}t ||j}|}t|}|}|dkr"| | dfStd||||\}}t|D]}|ddk}|rt ||}|dkr| | dfS||}|||} |||8}||d|||}tj||}td d |d}!|||!9}|!d|}| |}"| ||"7} |dur|| |t|d |kr^| rNtd |d | | dfS|st ||}||}#||#|}|#||#d|}| | |}||7}n| | | }| }|}q>| rtd |d | | |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. 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 ``rtol``. .. warning:: The default value for ``atol`` will be changed to ``0.0`` in SciPy 1.14.0. 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`. tol : float, optional, deprecated .. deprecated:: 1.12.0 `tfqmr` keyword argument ``tol`` is deprecated in favor of ``rtol`` and will be removed in SciPy 1.14.0. 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' rg?rz:TFQMR: Linear solve converged due to reach TOL iterations z@TFQMR: Linear solve not converged due to reach MAXIT iterations )dtypenpZ issubdtypeZint64floatZastyperZlinalgZnormcopyshapeminZmatvecinner conjugaterealsqrtrrangeprint)$AbZx0r r r r r rrrx postprocessZndofsruwZrstarvZuhatdthetaetarhoZrhoLastZr0normtau_iterZevenZvtrstaralphaZuNextczbetar2]/var/www/html/assistant/venv/lib/python3.9/site-packages/scipy/sparse/linalg/_isolve/tfqmr.pyr s_                    )N) numpyrZ iterativerutilsrZscipy._lib.deprecationrr__all__rr2r2r2r3s