a h[- @sddlZddlmZmZmZmZddlmZddlm Z ddl m Z ddl m Z mZdgZed d dd e dddd d d dddZdS)N)innerzerosinffinfo)norm)sqrt) make_system)_NoValue_deprecate_positional_argsminresz1.14.0)versiongFgh㈵>)shifttolmaxiterMcallbackshowcheckrtolcJCs.t||||\}}} }} |turHd} tj| tdd|durDt|n| } |j}|j}d}d}|jd}|durvd|}gd } |rt|d t|d |d d |dt|d|d d| dtd}d}d}d}d}d}| j }t |j }|dur| }n ||| }||}t ||}|dkr:tdn|dkrP| | dfSt|}|dkrr|} | | dfSt|}| r||}||} t ||}!t || }"t|!|"}#|!||d}$|#|$krtd||} t ||}!t || }"t|!|"}#|!||d}$|#|$krtdd}%|}&d}'d}(|})|}*|}+d},d}-d}.t |j}/d}0d}1t||d}t||d}2|} |rtttd||kr|d7}d|&}!|!|}3||3}|||3}|dkr||&|%|}t |3|}4||4|&| }| }|} || }|&}%t | |}&|&dkr&tdt|&}&|-|4d|%d|&d7}-|dkrj|&|d|krjd}|(}5|0|'|1|4}6|1|'|0|4}7|1|&}(|0 |&}'t|7|'g}8|*|8}9t|7|&g}:t|:|}:|7|:}0|&|:}1|0|*};|1|*}*d|:}<|2}=|}2|3|5|=|6|2|<}| |;|} t|.|:}.t|/|:}/|+|:}#|,|6|#}+|( |#},t|-}t| }||}$|||}>||| }?|7}@|@dkr|$}@|*})|)}|dks|dkrt}An |||}A|dkrt}Bn|8|}B|.|/}|dkr`d|A}Cd|B}D|Ddkrd}|Cdkrd}||kr$d}|d|kr6d}|>|krDd}|B| krRd}|A| kr`d}d }E|d!krrd"}E|dkrd"}E||dkrd"}E|ddkrd"}E|)d|>krd"}E|)d|?krd"}E|d#|krd"}E|dkrd"}E|rb|Erb|d$d%| dd&d%|Ad'}Fd%|Bd'}Gd%|d(d%|d(d%|7|d(}Ht|F|G|H|ddkrbt|durt|| |dkrqq|rtt|d)|d d*|d+t|d,|d-d.|d-t|d/|d-d0|d-t|d1|9d-t|| |d|dkr|}Ind}I| | |IfS)2a Use MINimum RESidual iteration to solve Ax=b MINRES minimizes norm(Ax - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular. If shift != 0 then the method solves (A - shift*I)x = b Parameters ---------- A : {sparse matrix, ndarray, LinearOperator} The real symmetric 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). Returns ------- x : ndarray The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown Other Parameters ---------------- x0 : ndarray Starting guess for the solution. shift : float Value to apply to the system ``(A - shift * I)x = b``. Default is 0. rtol : float Tolerance to achieve. The algorithm terminates when the relative residual is below ``rtol``. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse matrix, ndarray, LinearOperator} Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. show : bool If ``True``, print out a summary and metrics related to the solution during iterations. Default is ``False``. check : bool If ``True``, run additional input validation to check that `A` and `M` (if specified) are symmetric. Default is ``False``. tol : float, optional, deprecated .. deprecated:: 1.12.0 `minres` keyword argument ``tol`` is deprecated in favor of ``rtol`` and will be removed in SciPy 1.14.0. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import minres >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> A = A + A.T >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = minres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True References ---------- Solution of sparse indefinite systems of linear equations, C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/ This file is a translation of the following MATLAB implementation: https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip z'scipy.sparse.linalg.minres' keyword argument `tol` is deprecated in favor of `rtol` and will be removed in SciPy v1.14. Until then, if set, it will override `rtol`.)category stacklevelNzEnter minres. zExit minres. r) z3 beta2 = 0. If M = I, b and x are eigenvectors z/ beta1 = 0. The exact solution is x0 z3 A solution to Ax = b was found, given rtol z3 A least-squares solution was found, given rtol z3 Reasonable accuracy achieved, given eps z3 x has converged to an eigenvector z3 acond has exceeded 0.1/eps z3 The iteration limit was reached z3 A does not define a symmetric matrix z3 M does not define a symmetric matrix z3 M does not define a pos-def preconditioner zSolution of symmetric Ax = bz n = Z3gz shift = z23.14ez itnlim = z rtol = z11.2ezindefinite preconditionergUUUUUU?znon-symmetric matrixznon-symmetric preconditioner)dtypezD Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|rg? g?F(Tg{Gz?Z6g z12.5ez10.3ez8.1ez istop = z itn =Z5gz Anorm = z12.4ez Acond = z rnorm = z ynorm = z Arnorm = )r r warningswarnDeprecationWarningfloatmatvecshapeprintrrepscopyr ValueErrorrrabsmaxrminr)JAbZx0rrrrrrrrx postprocessmsgr&ZpsolvefirstlastnistopitnZAnormZAcondZrnormZynormZxtyper)r1yZbeta1Zbnormwr2stzZepsaZoldbbetaZdbarZepslnZqrnormZphibarZrhs1Zrhs2Ztnorm2ZgmaxZgmincsZsnZw2vZalfaZoldepsdeltaZgbarrootZArnormgammaphidenomZw1ZepsxZepsrZdiagZtest1Ztest2t1t2ZprntZstr1Zstr2Zstr3inforK^/var/www/html/assistant/venv/lib/python3.9/site-packages/scipy/sparse/linalg/_isolve/minres.pyr sX                                                              )N)r"numpyrrrrZ numpy.linalgrmathrutilsr Zscipy._lib.deprecationr r __all__r rKrKrKrLs