ó î&]\c @`sœddlmZmZmZddlmZmZmZmZm Z ddl m Z ddl m Z dgZeddeeeeed „Zed kr˜dd lmZmZddlm Z dd lmZd ZgZd„ZeededdeƒgdgeeddƒZedededdeƒgdgeeddƒZeje_ deej!dƒZ"eee"dddedeƒZ#ndS(i(tdivisiontprint_functiontabsolute_import(tsqrttinnertzerostinftfinfo(tnormi(t make_systemtminresggñh㈵øä>c H C`sît||||ƒ\}}} }} |j} |j} d}d}|jd}|d+krhd|}nddddd d d d d ddg }|rÝt|dƒt|d||fƒt|d||fƒtƒnd}d}d}d}d}d}| j}t|ƒj}t|d|ƒ} |}|}| |ƒ}t ||ƒ}|dkrmt dƒ‚n|dkr‰| | ƒdfSt |ƒ}| rx| |ƒ}| |ƒ}t ||ƒ}t ||ƒ} t || ƒ}!|||d,}"|!|"krt dƒ‚n| |ƒ}t ||ƒ}t ||ƒ} t || ƒ}!|||d-}"|!|"krxt dƒ‚qxnd}#|}$d}%d}&|}'|}(|})d}*d}+d},t|ƒj }-d}.d}/t|d|ƒ}t|d|ƒ}0|}|rtƒtƒtdƒnx#||kr?|d7}d|$}||}1| |1ƒ}|||1}|dkr‚||$|#|}nt |1|ƒ}2||2|$|}|}|}| |ƒ}|$}#t ||ƒ}$|$dkrët dƒ‚nt |$ƒ}$|+|2d|#d|$d7}+|dkrA|$|d|krAd}qAn|&}3|.|%|/|2}4|/|%|.|2}5|/|$}&|. |$}%t|5|%gƒ}6|(|6}7t|5|$gƒ}8t |8|ƒ}8|5|8}.|$|8}/|.|(}9|/|(}(d|8}:|0};|}0|1|3|;|4|0|:}| |9|} t |,|8ƒ},t|-|8ƒ}-|)|8}!|*|4|!})|& |!}*t |+ƒ}t| ƒ}||}"|||}<|||}=|5}>|>dkr½|"}>n|(}'|'}|dksá|dkrêt}?n|||}?|dkr t}@n |6|}@|,|-}|dkrÛd|?}Ad|@}B|BdkrVd}n|Adkrkd}n||kr€d}n|d|kr™d }n|<|kr®d!}n|@|krÃd}n|?|krÛd}qÛnt}C|d"kröt}Cn|dkr t}Cn||dkr$t}Cn|ddkr=t}Cn|'d|<krVt}Cn|'d|=krot}Cn|d#|krˆt}Cn|dkrt}Cn|r|Crd$|| d|?f}Dd%|@f}Ed&|||5|f}Ft|D|E|Fƒ|ddkrtƒqn|d+k r,|| ƒn|dkrPqqW|rÃtƒt|d'||fƒt|d(||fƒt|d)||fƒt|d*|7fƒt|||dƒn|dkrØ|}Gnd}G| | ƒ|GfS(.s 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 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. 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 sEnter minres. sExit minres. iis3 beta2 = 0. If M = I, b and x are eigenvectors s3 beta1 = 0. The exact solution is x = 0 s3 A solution to Ax = b was found, given rtol s3 A least-squares solution was found, given rtol s3 Reasonable accuracy achieved, given eps s3 x has converged to an eigenvector s3 acond has exceeded 0.1/eps s3 The iteration limit was reached s3 A does not define a symmetric matrix s3 M does not define a symmetric matrix s3 M does not define a pos-def preconditioner sSolution of symmetric Ax = bs#n = %3g shift = %23.14es"itnlim = %3g rtol = %11.2etdtypesindefinite preconditionergð?g@snon-symmetric matrixsnon-symmetric preconditioneriÿÿÿÿsD Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|iii igš™™™™™¹?iii(g{®Gáz„?s%6g %12.5e %10.3es %10.3es %8.1e %8.1e %8.1es( istop = %3g itn =%5gs' Anorm = %12.4e Acond = %12.4es' rnorm = %12.4e ynorm = %12.4es Arnorm = %12.4eNgUUUUUUÕ?gUUUUUUÕ?(R tmatvectshapetNonetprintR RtepsRRt ValueErrorRtabstmaxRtminRtFalsetTrue(HtAtbtx0tshiftttoltmaxitertMtcallbacktshowtchecktxt postprocessR tpsolvetfirsttlasttntmsgtistoptitntAnormtAcondtrnormtynormtxtypeRtytr1tbeta1twtr2tstttztepsatoldbtbetatdbartepslntqrnormtphibartrhs1trhs2ttnorm2tgmaxtgmintcstsntw2tvtalfatoldepstdeltatgbartroottArnormtgammatphitdenomtw1tepsxtepsrtdiagttest1ttest2tt1tt2tprnttstr1tstr2tstr3tinfo((s@lib/python2.7/site-packages/scipy/sparse/linalg/isolve/minres.pyR sˆ9$                                                                                   t__main__(tonestarange(tspdiagsi cC`stjttt|ƒƒdS(N(t residualstappendRRR(R!((s@lib/python2.7/site-packages/scipy/sparse/linalg/isolve/minres.pytcbdsR tformattcsrgð?Rgê-™—q=RRN($t __future__RRRtnumpyRRRRRt numpy.linalgRtutilsR t__all__RRR t__name__tscipyR^R_t scipy.linalgt scipy.sparseR`R&RaRctfloatRRR R#R RR!(((s@lib/python2.7/site-packages/scipy/sparse/linalg/isolve/minres.pyts&(  ÿP  48