B `*@sddlmZmZmZmZmZddlmZddlm Z dgZ dd dZ e d krdd lm Z dd lmZdZgZddZee dededgdgeeddZede dededgdgeeddZeje_eejdZe eeddedZdS))sqrtinnerzerosinffinfo)norm) make_systemminresNh㈵>Fc H Cst||||\}}} }} |j} |j} d}d}|jd}|dkrFd|}dddd d d d d dddg }|rt|dt|d||ft|d||ftd}d}d}d}d}d}| j}t|j}||| }| |}t||}|dkrtdn|dkr | | dfSt |}| r| |}| |}t||}t||} t || }!|||d}"|!|"krltd| |}t||}t||} t || }!|||d}"|!|"krtdd}#|}$d}%d}&|}'|}(|})d}*d}+d},t|j }-d}.d}/t ||d}t ||d}0|}|r&tttdx||kr|d7}d|$}||}1| |1}|||1}|dkrz||$|#|}t|1|}2||2|$|}|}|}| |}|$}#t||}$|$dkrtdt |$}$|+|2d|#d|$d7}+|dkr|$|d|krd}|&}3|.|%|/|2}4|/|%|.|2}5|/|$}&|. |$}%t |5|%g}6|(|6}7t |5|$g}8t |8|}8|5|8}.|$|8}/|.|(}9|/|(}(d|8}:|0};|}0|1|3|;|4|0|:}| |9|} t |,|8},t|-|8}-|)|8}!|*|4|!})|& |!}*t |+}t | }||}"|||}<|||}=|5}>|>dkr0|"}>|(}'|'}|dksL|dkrRt}?n |||}?|dkrnt}@n|6|}@|,|-}|dkrd|?}Ad|@}B|Bdkrd}|Adkrd}||krd}|d |krd!}|<|krd"}|@|krd}|?|krd}d#}C|d$krd%}C|dkrd%}C||dkr0d%}C|ddkrBd%}C|'d|<krTd%}C|'d|=krfd%}C|d&|krxd%}C|dkrd%}C|r|Crd'|| d|?f}Dd(|@f}Ed)|||5|f}Ft|D|E|F|ddkrt|dk r|| |dkr*Pq*W|rvtt|d*||ft|d+||ft|d,||ft|d-|7ft|||d|dkr|}Gnd}G| | |GfS).ag Use MINimum RESidual iteration to solve Ax=b MINRES minimizes norm(A*x - 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, dense matrix, 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 : {array, matrix} Right hand side of the linear system. Has shape (N,) or (N,1). Returns ------- x : {array, matrix} 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 : {array, matrix} Starting guess for the solution. 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, dense matrix, 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. 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. rNz3 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 = %3g shift = %23.14ez"itnlim = %3g rtol = %11.2ezindefinite preconditionergUUUUUU?znon-symmetric matrixznon-symmetric preconditioner)dtypezD Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|rg? g?F(Tg{Gz?z%6g %12.5e %10.3ez %10.3ez %8.1e %8.1e %8.1ez( istop = %3g itn =%5gz' Anorm = %12.4e Acond = %12.4ez' rnorm = %12.4e ynorm = %12.4ez Arnorm = %12.4e)r matvecshapeprintrrepsr ValueErrorrabsmaxrrminr)HAbZx0shifttolmaxiterMcallbackshowcheckx postprocessrpsolvefirstlastnmsgistopitnZAnormZAcondZrnormZynormZxtyperZr1yZbeta1wZr2stzZepsaZoldbbetaZdbarZepslnZqrnormZphibarZrhs1Zrhs2Ztnorm2ZgmaxZgmincsZsnZw2vZalfaZoldepsdeltaZgbarrootZArnormgammaphiZdenomZw1ZepsxZepsrZdiagZtest1Ztest2t1t2ZprntZstr1Zstr2Zstr3infor?P/home/dcms/DCMS/lib/python3.7/site-packages/scipy/sparse/linalg/isolve/minres.pyr sJ                                                      __main__)arange)spdiagsrcCstttt|dS)N) residualsappendrrr)r'r?r?r@cbnsrF)rZcsr)formatg?g-q=)r!r"r$)Nr r NNNFF)numpyrrrrrZ numpy.linalgrutilsr __all__r __name__rBZ scipy.sparserCr,rDrFfloatrr#rr)rrr'r?r?r?r@s$   ^  $(