a Re~*@sPddlmZmZmZmZddlmZddlmZddl m Z dgZ d d dZ dS) )innerzerosinffinfo)norm)sqrt) make_systemminresNh㈵>Fc ICst||||\}}} }} |j} |j} d}d}|jd}|durFd|}gd}|rt|dt|d|d d |d t|d |d d |dtd}d}d}d}d}d}| j}t|j}|dur|}n ||| }| |}t||}|dkrt dn|dkr| | dfSt |}|dkr@|} | | dfSt |}| r| |}| |}t||} t||}!t | |!}"| ||d}#|"|#krt d| |}t||} t||}!t | |!}"| ||d}#|"|#krt dd}$|}%d}&d}'|}(|})|}*d}+d},d}-t|j }.d}/d}0t||d}t||d}1|}|rZtttd||krT|d7}d|%} | |}2| |2}|||2}|dkr||%|$|}t|2|}3||3|%|}|}|}| |}|%}$t||}%|%dkrt dt |%}%|,|3d|$d|%d7},|dkr8|%|d|kr8d}|'}4|/|&|0|3}5|0|&|/|3}6|0|%}'|/ |%}&t |6|&g}7|)|7}8t |6|%g}9t |9|}9|6|9}/|%|9}0|/|)}:|0|)})d|9};|1}<|}1|2|4|<|5|1|;}| |:|} t |-|9}-t|.|9}.|*|9}"|+|5|"}*|' |"}+t |,}t | }||}#|||}=|||}>|6}?|?dkr`|#}?|)}(|(}|dks||dkrt}@n |||}@|dkrt}An|7|}A|-|.}|dkr.d|@}Bd|A}C|Cdkrd}|Bdkrd}||krd}|d|krd}|=|krd}|A|kr d}|@|kr.d}d}D|dkr@d }D|dkrNd }D||dkr`d }D|ddkrrd }D|(d|=krd }D|(d|>krd }D|d!|krd }D|dkrd }D|r0|Dr0|d"d#| dd$d#|@d%}Ed#|Ad%}Fd#|d&d#|d&d#|6|d&}Gt|E|F|G|ddkr0t|durB|| |dkrZqTqZ|rtt|d'|d d(|d)t|d*|d+d,|d+t|d-|d+d.|d+t|d/|8d+t|||d|dkr|}Hnd}H| | |HfS)0a 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. tol : float Tolerance to achieve. The algorithm terminates when the relative residual is below `tol`. 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``. 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 zEnter minres. zExit minres. rN) 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 matvecshapeprintrrepscopyr ValueErrorrrabsmaxrminr)IAbZx0shiftZtolmaxiterMcallbackshowcheckx postprocessrZpsolvefirstlastnmsgistopitnZAnormZAcondZrnormZynormZxtyperZr1yZbeta1ZbnormwZr2stzZepsaZoldbbetaZdbarZepslnZqrnormZphibarZrhs1Zrhs2Ztnorm2ZgmaxZgmincsZsnZw2vZalfaZoldepsdeltaZgbarrootZArnormgammaphiZdenomZw1ZepsxZepsrZdiagZtest1Ztest2t1t2ZprntZstr1Zstr2Zstr3infor?V/var/www/sunrise/env/lib/python3.9/site-packages/scipy/sparse/linalg/_isolve/minres.pyr s|R                                                             )Nr r NNNFF) numpyrrrrZ numpy.linalgrmathrutilsr __all__r r?r?r?r@s