o h`@sDddlZddlmZddlmZdgZd ddddddd d dZdS) N)_get_atol_rtol) make_systemtfqmrgh㈵>F)rtolatolmaxiterMcallbackshowc#Cs|j} t| tjrt} || }t|jtjr|| }t||||\}}} }} tj|dkr<| } | | dfS|j d} |durLt d| d}|durU| } 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*|r"td |d | | dfS|sVt ||}||}"||"|}|"||"d|}| | |}||7}q| | |}|}|}q|rotd |d | | |fS) aN Use Transpose-Free Quasi-Minimal Residual iteration to solve ``Ax = b``. 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 ``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 array, 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_array >>> from scipy.sparse.linalg import tfqmr >>> A = csc_array([[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 rrNi' 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)#Abx0rrr r r r rx postprocessndofsruwrstarvuhatdthetaetarhorhoLastr0normtau_iterevenvtrstaralphauNextczbetar>U/var/www/vscode/kcb/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tfqmr.pyr sT                    )N)numpyr iterativerutilsr__all__rr>r>r>r?s