YhddlZddlZddlZddlZddlmZmZmZddlm Z m Z m Z m Z m Z mZddlZddlZddlmZddlmZddlmZddlmZmZdd lmZdd lmZgd ZGd d eZdZ dZ!dZ"dZ#e$d%d%Z&dZ' dAdZ(e'e( dBdZ)GddZ*Gd d!Z+Gd"d#Z,d$Z-Gd%d&e+Z.Gd'd(Z/d)%e&d*<Gd+d,e.Z0Gd-d.e0Z1Gd/d0e.Z2Gd1d2e.Z3Gd3d4e.Z4Gd5d6e.Z5Gd7d8e+Z6d9Z7e7d:e0Z8e7d;e1Z9e7de3Z<e7d?e5Z=e7d@e6Z>dS)CN)asarraydotvdot)normsolveinvqrsvd LinAlgError)get_blas_funcs)copy_if_needed)getfullargspec_no_self)scalar_search_wolfe1scalar_search_armijo) signature)get_close_matches) broyden1broyden2anderson linearmixing diagbroydenexcitingmixing newton_krylov BroydenFirstKrylovJacobianInverseJacobian NoConvergenceceZdZdZdS)rz\Exception raised when nonlinear solver fails to converge within the specified `maxiter`.N)__name__ __module__ __qualname____doc__h/var/www/tools.fuzzalab.pt/emblema-extractor/venv/lib/python3.11/site-packages/scipy/optimize/_nonlin.pyrr sDr%rcNtj|SN)npabsolutemaxxs r&maxnormr.&s ;q>>    r%ct|}tj|jtjst|tjS|S)z:Return `x` as an array, of either floats or complex floatsdtype)rr) issubdtyper1inexactfloat64r,s r& _as_inexactr5*s? A ="* - -,q ++++ Hr%ctj|tj|}t|d|j}||S)z;Return ndarray `x` as same array subclass and shape as `x0`__array_wrap__)r)reshapeshapegetattrr7)r-x0wraps r& _array_liker=2s= 1bhrll##A 2')9 : :D 477Nr%ctj|stjtjSt |Sr()r)isfiniteallarrayinfr)vs r& _safe_normrD9s; ;q>>     x 77Nr%z F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution a iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : {None, 'armijo' (default), 'wolfe'}, optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual. Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution. Raises ------ NoConvergence When a solution was not found. ) params_basic params_extrac@|jr|jtz|_dSdSr()r# _doc_parts)objs r&_set_docrJws( {/kJ. //r%krylovFarmijoTc | tn| } t|||| || }tfd}}t j|tj}||}t|}t|}| | |||||dz}n d|j dzz}| durd} n| durd} | d vrtd d }d }d }d}t|D]}||||}|rnt|||z}||| }t|dkrtd| rt#||||| \}}}}n!d}||z}||}t|}|| || r | ||||dzz|dzz }||dzz|krt||}n$t|t'|||dzz}|}|rQt(j|d| |dd|ddt(j|rt1t3|d}| r,|j|||dkddd|d}t3||fSt3|S)a Find a root of a function, in a way suitable for large-scale problems. Parameters ---------- %(params_basic)s jacobian : Jacobian A Jacobian approximation: `Jacobian` object or something that `asjacobian` can transform to one. Alternatively, a string specifying which of the builtin Jacobian approximations to use: krylov, broyden1, broyden2, anderson diagbroyden, linearmixing, excitingmixing %(params_extra)s full_output : bool If true, returns a dictionary `info` containing convergence information. raise_exception : bool If True, a `NoConvergence` exception is raise if no solution is found. See Also -------- asjacobian, Jacobian Notes ----- This algorithm implements the inexact Newton method, with backtracking or full line searches. Several Jacobian approximations are available, including Krylov and Quasi-Newton methods. References ---------- .. [KIM] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations". Society for Industrial and Applied Mathematics. (1995) https://archive.siam.org/books/kelley/fr16/ N)f_tolf_rtolx_tolx_rtoliterrcttt|Sr()r5r=flatten)zFr;s r&funcznonlin_solve..funcs111[B//001199;;;r%rdTrLF)NrLwolfezInvalid line searchg?gH.?g?gMbP?)tolrz[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.?z : |F(x)| = gz; step  z0A solution was found at the specified tolerance.z:The maximum number of iterations allowed has been reached.)rr\)nitfunstatussuccessmessage)r.TerminationConditionr5rTr) full_likerBr asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater+sysstdoutwriteflushrr= iteration) rVr;jacobianrRverbosemaxiterrNrOrPrQtol_norm line_searchcallback full_outputraise_exception conditionrWr-dxFxFx_normgammaeta_max eta_tresholdetanrarZs Fx_norm_neweta_Ainfos `` r& nonlin_solver|sZ#*wwH$5+0*.X???I RB<<<<<< A a B aB2hhG(##H NN16688R&&&  QhGG16!8nGd    333./// EGL C 7^^..Q++   E#s7{##nnRSn)) ) 88q==.// /  #$7aR8C%E%E !Aq"kkABAaBr((K"%%%   HQOOO Q&!3 36>L ( (gu%%CCgs5%Q,7788C   J  MMxx||MMMaMMMM N N N J        Ar 2 233 3F " * !Q; , 3% &  1b!!4''1b!!!r%:0yE>{Gz?c dg|gt|dzgttz d fd  fd}|dkrt |dd|\}} } n)|d kr#t dd | \}} |d }|zz|dkr d}n }t|} ||| fS) Nrr\Tc| dkrdS |zz}|}t|dz}|r| d<|d<|d<|S)Nrr\)rD) rstorextrCpr~rWtmp_Fxtmp_phitmp_sr-s r&phiz _nonlin_line_search..phisl a==1:  2X DHH qMM1   E!HGAJF1Ir%crt|zdzz}||zd|z |z S)NrF)r)abs)rdsrrdiffs_norms r&derphiz#_nonlin_line_search..derphi$sG!ffvo!U *AbD&&&Q/255r%rYr)xtolaminrL)rr[)T)rrr)rWr-rr~ search_typersminrrphi1phi0rrrrrrs`` ` ` @@@@@r&rnrnse CETFBxx{mG !WWtBxx F           6666666g,S&'!*26TCCC 4  &sGAJ ,02224 y  AbDAE!H}} AY T!WW2hhG aW r%c,eZdZdZdddddefdZdZdS)rdz Termination condition for an iteration. It is terminated if - |F| < f_rtol*|F_0|, AND - |F| < f_tol AND - |dx| < x_rtol*|x|, AND - |dx| < x_tol Nc|&tjtjjdz}| tj}| tj}| tj}||_||_||_||_||_ ||_ d|_ d|_ dS)NgUUUUUU?r) r)finfor4epsrBrPrQrNrOrrRf0_normrt)selfrNrOrPrQrRrs r&__init__zTerminationCondition.__init__Ks =HRZ((,6E >VF =FE >VF       r%c|xjdz c_||}||}||}|j||_|dkrdS|jd|j|jkzSt ||jko+||jz |jko||jko ||jz |kS)Nrrr\) rtrrrRintrNrOrPrQ)rfr-r~f_normx_normdx_norms r&rlzTerminationCondition.checkcs !11))B-- < !DL Q;;1 9 23 3Fdj(;t{*dl:;4:-:#DK/69<< sQ  "$d40<<<<.__array__s-$$%I%%I%IJJJ||~~%r%NN)itemsrjsetattrhasattr)rkwnamesnamevaluers r&rzJacobian.__init__s88888:: . .KD%5   !CT!C!CDDD dBtH--- 4 # # & & & & & & & & &r%c t|Sr()rrs r&aspreconditionerzJacobian.aspreconditionerst$$$r%rctr(NotImplementedErrorrrCrZs r&rzJacobian.solve!!r%cdSr(r$rr-rVs r&rozJacobian.update r%c||_|j|jf|_|j|_|jjt jur|||dSdSr()rWrir9r1 __class__rgrrorr-rVrWs r&rgzJacobian.setupsV faf% W > 8> 1 1 KK1      2 1r%Nr) r r!r"r#rrrrorgr$r%r&rr~so##J&&& %%%""""   r%rcDeZdZdZdZedZedZdS)ra A simple wrapper that inverts the Jacobian using the `solve` method. .. legacy:: class See the newer, more consistent interfaces in :mod:`scipy.optimize`. Parameters ---------- jacobian : Jacobian The Jacobian to invert. Attributes ---------- shape Matrix dimensions (M, N) dtype Data type of the matrix. c||_|j|_|j|_t |dr |j|_t |dr|j|_dSdS)Nrgr)rurrrorrgrr)rrus r&rzInverseJacobian.__init__s_  n o 8W % % (!DJ 8X & & +#?DLLL + +r%c|jjSr()rur9rs r&r9zInverseJacobian.shape }""r%c|jjSr()rur1rs r&r1zInverseJacobian.dtyperr%N)r r!r"r#rpropertyr9r1r$r%r&rrsc(+++##X###X###r%rc tjjjt t rSt jrtt r St tj rj dkrtdtj tjjdjdkrtdt fdfddfd dfd jj StjrZjdjdkrtd t fd fd dfd dfd jj St%drt%dr|t%drlt t'dt'djt'dt'dt'djjSt+r Gfddt }|St t,rGt/t0t2t4t6t8t:t<St?d)zE Convert given object to one suitable for use as a Jacobian. r\zarray must have rank <= 2rrzarray must be squarec$t|Sr()rrCJs r&zasjacobian..sQr%cRtj|Sr()rconjTrs r&rzasjacobian..s#affhhj!*<*<r%c$t|Sr()rrCrZrs r&rzasjacobian..suQ{{r%cRtj|Sr()rrrrs r&rzasjacobian..saffhhj!0D0Dr%)rrrrr1r9zmatrix must be squarec|zSr(r$rs r&rzasjacobian..s Qr%c<j|zSr(rrrs r&rzasjacobian..s!&&((*q.r%c|Sr(r$rCrZrspsolves r&rzasjacobian..swwq!}}r%cJj|Sr(rrs r&rzasjacobian..s A0F0Fr%r9r1rrrrrorg)rrrrrorgr1r9cFeZdZdZdfd ZfdZdfd ZfdZdS) asjacobian..Jacc||_dSr(r,rs r&rozasjacobian..Jac.updates r%rc|j}t|tjrt ||St j|r ||StdNzUnknown matrix type) r- isinstancer)ndarrayrscipysparseissparserjrrCrZmrrs r&rzasjacobian..Jac.solveskAdfIIa,,< A;;&\**1--<"71a==($%:;;;r%c|j}t|tjrt ||St j|r||zStdr) r-rr)rrrrrrjrrCrrs r&rzasjacobian..Jac.matvec"sdAdfIIa,,<q!99$\**1--<q5L$%:;;;r%cH|j}t|tjr't |j|Stj |r#|j|Stdr) r-rr)rrrrrrrrjrs r&rzasjacobian..Jac.rsolve+sAdfIIa,,< Q///\**1--<"716688:q111$%:;;;r%c:|j}t|tjr't |j|Stj |r|j|zStdr) r-rr)rrrrrrrrjrs r&rzasjacobian..Jac.rmatvec4s{AdfIIa,,<qvvxxz1---\**1--<6688:>)$%:;;;r%Nr)r r!r"rorrrr)rrsr&Jacrs    < < < < < < < < < < < < < < < < < < < < < < < < < >>r%c eZdZdZdZdZdS)GenericBroydenct||||||_||_t |drK|jFt |}|r*dtt |dz|z |_dSd|_dSdSdS)Nalpha?rr[)rrglast_flast_xrr rr+)rr;f0rWnormf0s r&rgzGenericBroyden.setupNstRT***  4 ! ! !dj&8"XXF ! T"XXq!1!11F:    ! !&8&8r%ctr(rrr-rr~dfrdf_norms r&_updatezGenericBroyden._update\rr%c ||jz }||jz }|||||t|t|||_||_dSr()r rrr)rr-rrr~s r&rozGenericBroyden.update_sR _ _ Q2r488T"XX666  r%N)r r!r"rgrror$r%r&r r MsA ! ! !"""r%r ceZdZdZdZedZedZdZdZ ddZ dd Z d Z dd Z d ZdZdZddZd S) LowRankMatrixz A matrix represented as .. math:: \alpha I + \sum_{n=0}^{n=M} c_n d_n^\dagger However, if the rank of the matrix reaches the dimension of the vectors, full matrix representation will be used thereon. cZ||_g|_g|_||_||_d|_dSr()r csrrr1 collapsed)rr rr1s r&rzLowRankMatrix.__init__rs0  r%ctgd|dd|gz\}}}||z}t||D]$\}} || |} ||||j| }%|S)N)axpyscaldotcr)r zipri) rCr rrrrrwcdas r&_matveczLowRankMatrix._matveczs)*B*B*B*,RaR&A3,88dD AIBKK & &DAqQ AQ161%%AAr%c t|dkr||z Stddg|dd|gz\}}|d}|tjt||jz}t |D]6\}} t |D]!\} } ||| fxx|| | z cc<"7tjt||j} t |D]\} } || || | <| |z} t|| } ||z } t|| D]\} }|| | | j | } | S)Evaluate w = M^-1 vrrrNrr0) lenr r)identityr1 enumeratezerosrr ri)rCr rrrrc0Air#jr"qr!qcs r&_solvezLowRankMatrix._solvesq r77a<<U7N$VV$4b!fslCC d U BKBrx888 8bMM % %DAq!"  % %1!A#$$q!**$ % HSWWBH - - -bMM  DAq41::AaDD U  !QKK eGQZZ ( (EArQ16B3''AAr%c|jtj|j|St||j|j|jS)zEvaluate w = M v)rr)rrr%r rrrrCs r&rzLowRankMatrix.matvecs> > %6$.!,, ,$$Q DGTWEEEr%c|j1tj|jj|St |tj|j|j|j S)zEvaluate w = M^H v) rr)rrrrr%r rrr4s r&rzLowRankMatrix.rmatvecsW > %6$.*//11155 5$$Q (;(;TWdgNNNr%rc|jt|j|St||j|j|jS)r')rrrr2r rrrs r&rzLowRankMatrix.solves< > %++ +##Atz47DGDDDr%c|j,t|jj|St|t j|j|j|j S)zEvaluate w = M^-H v) rrrrrr2r)r rrrs r&rzLowRankMatrix.rsolvesU > %)..00!44 4##Arwtz':':DGTWMMMr%cX|j;|xj|dddf|dddfzz c_dS|j||j|t |j|jkr|dSdSr()rrrappendrr(ricollapse)rr"r#s r&r9zLowRankMatrix.appends > % NNa$i!DF)..*:*:: :NN F q q tw<OOO%& ( ( ( (   MN9=NNN%& ( ( ( ( > %> ! Z DF$*=== =)) - -DAq !AAAdF)Ad111fINN,,, ,BB r%cptj|t|_d|_d|_d|_dS)z0Collapse the low-rank matrix to a full-rank one.)rhN)r)rAr rrrr rs r&r:zLowRankMatrix.collapses1$^<<< r%c|jdS|dksJt|j|kr|jdd=|jdd=dSdS)zH Reduce the rank of the matrix by dropping all vectors. Nrrr(rrrranks r&restart_reducezLowRankMatrix.restart_reducesW > % Faxxxx tw<<$      r%c|jdS|dksJt|j|kr*|jd=|jd=t|j|k(dSdS)zK Reduce the rank of the matrix by dropping oldest vectors. NrrCrDs r& simple_reducezLowRankMatrix.simple_reducesd > % Faxxxx$'llT!!  $'llT!!!!!!r%c|jdS|}||}n|dz }|jr(t|t|jd}t dt||dz }t|j}||krdSt j|jj}t j|jj}t|d\}}t||j }t|d\} } } t|t| }t|| j }t|D]N} |dd| f|j| <|dd| f|j| <O|j|d=|j|d=dS) a Reduce the rank of the matrix by retaining some SVD components. This corresponds to the "Broyden Rank Reduction Inverse" algorithm described in [1]_. Note that the SVD decomposition can be done by solving only a problem whose size is the effective rank of this matrix, which is viable even for large problems. Parameters ---------- max_rank : int Maximum rank of this matrix after reduction. to_retain : int, optional Number of SVD components to retain when reduction is done (ie. rank > max_rank). Default is ``max_rank - 2``. References ---------- .. [1] B.A. van der Rotten, PhD thesis, "A limited memory Broyden method to solve high-dimensional systems of nonlinear equations". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003). https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf Nr\rreconomic)modeF) full_matrices)rrrmr(r+r)rArrr rrr rrkrh) rmax_rank to_retainrr0rCDRUSWHks r& svd_reducezLowRankMatrix.svd_reduces: > % F   AAAA 7 (As471:''A 3q!A#;;   LL q55 F HTW    HTW   !*%%%1 1388::  q...1b 3r77OO 2499;;  q ' 'A111Q3DGAJ111Q3DGAJJ GABBK GABBKKKr%rrr()r r!r"r#r staticmethodr%r2rrrrr9rr:rFrHrVr$r%r&rrgs\\6FFF OOO EEEE NNNN   "      ??????r%ra alpha : float, optional Initial guess for the Jacobian is ``(-1/alpha)``. reduction_method : str or tuple, optional Method used in ensuring that the rank of the Broyden matrix stays low. Can either be a string giving the name of the method, or a tuple of the form ``(method, param1, param2, ...)`` that gives the name of the method and values for additional parameters. Methods available: - ``restart``: drop all matrix columns. Has no extra parameters. - ``simple``: drop oldest matrix column. Has no extra parameters. - ``svd``: keep only the most significant SVD components. Takes an extra parameter, ``to_retain``, which determines the number of SVD components to retain when rank reduction is done. Default is ``max_rank - 2``. max_rank : int, optional Maximum rank for the Broyden matrix. Default is infinity (i.e., no rank reduction). broyden_paramscHeZdZdZd dZdZdZddZd Zdd Z d Z d Z dS)ral Find a root of a function, using Broyden's first Jacobian approximation. This method is also known as "Broyden's good method". Parameters ---------- %(params_basic)s %(broyden_params)s %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='broyden1'`` in particular. Notes ----- This algorithm implements the inverse Jacobian Quasi-Newton update .. math:: H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df) which corresponds to Broyden's first Jacobian update .. math:: J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx References ---------- .. [1] B.A. van der Rotten, PhD thesis, "A limited memory Broyden method to solve high-dimensional systems of nonlinear equations". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003). https://math.leidenuniv.nl/scripties/Rotten.pdf Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.broyden1(fun, [0, 0]) >>> sol array([0.84116396, 0.15883641]) Nrestartc~t|_d_| tj}|_t|trdn|dd|d}|dz fz|dkr fd_ dS|dkr fd_ dS|dkr fd _ dStd |d ) Nr$rrr c"jjSr()r@rV reduce_paramsrsr&rz'BroydenFirst.__init__..s#547#5}#Er%simplec"jjSr()r@rHr]sr&rz'BroydenFirst.__init__..s#847#8-#Hr%rZc"jjSr()r@rFr]sr&rz'BroydenFirst.__init__..s#947#9=#Ir%zUnknown rank reduction method '') r rr r@r)rBrMrr_reducerj)rr reduction_methodrMr^s` @r&rzBroydenFirst.__init__s%%%   vH  & , , 3MM,QRR0M/2 !A-7 u $ $EEEEEDLLL  ) )HHHHHDLLL  * *IIIIIDLLLR?ORRRSS Sr%ct||||t|j |jd|j|_dS)Nr)r rgrr r9r1r@rs r&rgzBroydenFirst.setups?T1a... TZ]DJGGr%c*t|jSr()rr@rs r&rzBroydenFirst.todenses47||r%rc|j|}tj|s@||j|j|j|j|S|Sr() r@rr)r?r@rgrr rW)rrrZrs r&rzBroydenFirst.solvesf GNN1  {1~~!!## % JJt{DK ; ; ;7>>!$$ $r%c6|j|Sr()r@rrrs r&rzBroydenFirst.matvecsw}}Qr%c6|j|Sr()r@rrrrZs r&rzBroydenFirst.rsolveswq!!!r%c6|j|Sr()r@rrjs r&rzBroydenFirst.rmatvecsw~~a   r%c||j|}||j|z }|t ||z } |j|| dSr()rcr@rrrr9 rr-rr~rrrrCr"r#s r&rzBroydenFirst._updatesg  GOOB   ## # R O q!r%)NrZNr) r r!r"r#rrgrrrrrrr$r%r&rrKs22hTTTT2HHH   """"!!!r%rceZdZdZdZdS)raK Find a root of a function, using Broyden's second Jacobian approximation. This method is also known as "Broyden's bad method". Parameters ---------- %(params_basic)s %(broyden_params)s %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='broyden2'`` in particular. Notes ----- This algorithm implements the inverse Jacobian Quasi-Newton update .. math:: H_+ = H + (dx - H df) df^\dagger / ( df^\dagger df) corresponding to Broyden's second method. References ---------- .. [1] B.A. van der Rotten, PhD thesis, "A limited memory Broyden method to solve high-dimensional systems of nonlinear equations". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003). https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.broyden2(fun, [0, 0]) >>> sol array([0.84116365, 0.15883529]) c||}||j|z }||dzz } |j|| dSNr\)rcr@rr9ros r&rzBroydenSecond._updatesU   ## #  N q!r%N)r r!r"r#rr$r%r&rrs.00dr%rc.eZdZdZd dZd dZdZd ZdS) ra Find a root of a function, using (extended) Anderson mixing. The Jacobian is formed by for a 'best' solution in the space spanned by last `M` vectors. As a result, only a MxM matrix inversions and MxN multiplications are required. [Ey]_ Parameters ---------- %(params_basic)s alpha : float, optional Initial guess for the Jacobian is (-1/alpha). M : float, optional Number of previous vectors to retain. Defaults to 5. w0 : float, optional Regularization parameter for numerical stability. Compared to unity, good values of the order of 0.01. %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='anderson'`` in particular. References ---------- .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996). Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.anderson(fun, [0, 0]) >>> sol array([0.84116588, 0.15883789]) Nrct|||_||_g|_g|_d|_||_dSr()r rr Mr~rrw0)rr rwrvs r&rzAnderson.__init__BsD%%%  r%rc|j |z}t|j}|dkr|Stj||j}t |D] }t|j||||<! t|j |}n&#t$r|jdd=|jdd=|cYSwxYwt |D]1}||||j||j|j|zzzz }2|SNrr0) r r(r~r)emptyr1rkrrrr$r ) rrrZr~rdf_frUrrs r&rzAnderson.solveKsj[] LL 66Ix)))q * *A471:q))DGG $&$''EE     III   q @ @A %(DGAJDGAJ)>>? ?BB s4B B-,B-c | |jz }t|j}|dkr|Stj||j}t |D] }t|j||||<!tj||f|j}t |D]}t |D]}t|j||j||||f<||krT|j dkrI|||fxxt|j||j||j dzz|jzzcc<t||} t |D]1} || | |j| |j| |jz zzz }2|S)Nrr0r\) r r(r~r)rzr1rkrrrwr) rrr~rr{rUbr.r/rrs r&rzAnderson.matvecbsR ] LL 66Ix)))q * *A471:q))DGG HaV17 + + +q Q QA1XX Q Qdgaj$'!*55!A#66dgllacFFFd471:twqz::47A:EdjPPFFF Qaq @ @A %(DGAJDJ)>>? ?BB r%c|jdkrdS|j||j|t |j|jkrQ|jd|jdt |j|jkQt |j}t j||f|j}t|D]Y} t| |D]F} | | kr |j dz} nd} d| zt|j| |j| z|| | f<GZ|t j |dj z }||_dS)Nrr0r\r)rvr~r9rr(popr)r+r1rkrwrtriurrr$) rr-rr~rrrrr$r.r/wds r&rzAnderson._updateysZ 6Q;; F r r$'llTV## GKKNNN GKKNNN$'llTV## LL HaV17 + + +q = =A1a[[ = =66!BBBB$TWQZ < <<!A#  = RWQ]]_ ! ! # ##r%)Nrrtr)r r!r"r#rrrrr$r%r&rrse++L..r%rcHeZdZdZd dZdZd dZdZd dZd Z d Z d Z dS)ra, Find a root of a function, using diagonal Broyden Jacobian approximation. The Jacobian approximation is derived from previous iterations, by retaining only the diagonal of Broyden matrices. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. Parameters ---------- %(params_basic)s alpha : float, optional Initial guess for the Jacobian is (-1/alpha). %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='diagbroyden'`` in particular. Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.diagbroyden(fun, [0, 0]) >>> sol array([0.84116403, 0.15883384]) NcHt|||_dSr(r rr rr s r&rzDiagBroyden.__init__!%%% r%ct||||tj|jdfd|jz |j|_dS)Nrrr0)r rgr)fullr9r r1r#rs r&rgzDiagBroyden.setupsIT1a...$*Q-)1tz>LLLr%rc| |jz Sr(r#rls r&rzDiagBroyden.solverDF{r%c| |jzSr(rrjs r&rzDiagBroyden.matvecrr%c<| |jz Sr(r#rrls r&rzDiagBroyden.rsolverDFKKMM!!r%c<| |jzSr(rrjs r&rzDiagBroyden.rmatvecrr%c6tj|j Sr()r)diagr#rs r&rzDiagBroyden.todenseswwr%cN|xj||j|zz|z|dzz zc_dSrrrrs r&rzDiagBroyden._updates. 2r >2%gqj00r%r(r r r!r"r#rrgrrrrrrr$r%r&rrs&&PMMM"""""""   11111r%rcBeZdZdZd dZd dZdZd dZdZd Z d Z dS) ra Find a root of a function, using a scalar Jacobian approximation. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. Parameters ---------- %(params_basic)s alpha : float, optional The Jacobian approximation is (-1/alpha). %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='linearmixing'`` in particular. NcHt|||_dSr(rrs r&rzLinearMixing.__init__rr%rc| |jzSr(r rls r&rzLinearMixing.solver$*}r%c| |jz Sr(rrjs r&rzLinearMixing.matvecrr%c<| tj|jzSr(r)rr rls r&rzLinearMixing.rsolver"'$*%%%%r%c<| tj|jz Sr(rrjs r&rzLinearMixing.rmatvecrr%cvtjtj|jdd|jz S)Nr)r)rrr9r rs r&rzLinearMixing.todenses*wrwtz!}bm<<===r%cdSr(r$rs r&rzLinearMixing._updaterr%r(r) r r!r"r#rrrrrrrr$r%r&rrs,&&&&&&&>>>     r%rcHeZdZdZd dZdZddZdZdd Zd Z d Z d Z dS)ra Find a root of a function, using a tuned diagonal Jacobian approximation. The Jacobian matrix is diagonal and is tuned on each iteration. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='excitingmixing'`` in particular. Parameters ---------- %(params_basic)s alpha : float, optional Initial Jacobian approximation is (-1/alpha). alphamax : float, optional The entries of the diagonal Jacobian are kept in the range ``[alpha, alphamax]``. %(params_extra)s Nr[cdt|||_||_d|_dSr()r rr alphamaxbeta)rr rs r&rzExcitingMixing.__init__$s/%%%    r%ct||||tj|jdf|j|j|_dSry)r rgr)rr9r r1rrs r&rgzExcitingMixing.setup*sET1a...GTZ],dj KKK r%rc| |jzSr(rrls r&rzExcitingMixing.solve.r$)|r%c| |jz Sr(rrjs r&rzExcitingMixing.matvec1rr%c<| |jzSr(rrrls r&rzExcitingMixing.rsolve4r$)..""""r%c<| |jz Sr(rrjs r&rzExcitingMixing.rmatvec7rr%c:tjd|jz S)Nr)r)rrrs r&rzExcitingMixing.todense:swr$)|$$$r%c||jzdk}|j|xx|jz cc<|j|j|<tj|jd|j|jdS)Nr)out)r rr r)clipr)rr-rr~rrrincrs r&rzExcitingMixing._update=sa}q  $4:%: 4%  1dm;;;;;;r%)Nr[rrr$r%r&rr s4 LLL#######%%%<<<<eZdZdZ d dZdZdZdd Zd Zd Z dS)ra Find a root of a function, using Krylov approximation for inverse Jacobian. This method is suitable for solving large-scale problems. Parameters ---------- %(params_basic)s rdiff : float, optional Relative step size to use in numerical differentiation. method : str or callable, optional Krylov method to use to approximate the Jacobian. Can be a string, or a function implementing the same interface as the iterative solvers in `scipy.sparse.linalg`. If a string, needs to be one of: ``'lgmres'``, ``'gmres'``, ``'bicgstab'``, ``'cgs'``, ``'minres'``, ``'tfqmr'``. The default is `scipy.sparse.linalg.lgmres`. inner_maxiter : int, optional Parameter to pass to the "inner" Krylov solver: maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. inner_M : LinearOperator or InverseJacobian Preconditioner for the inner Krylov iteration. Note that you can use also inverse Jacobians as (adaptive) preconditioners. For example, >>> from scipy.optimize import BroydenFirst, KrylovJacobian >>> from scipy.optimize import InverseJacobian >>> jac = BroydenFirst() >>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac)) If the preconditioner has a method named 'update', it will be called as ``update(x, f)`` after each nonlinear step, with ``x`` giving the current point, and ``f`` the current function value. outer_k : int, optional Size of the subspace kept across LGMRES nonlinear iterations. See `scipy.sparse.linalg.lgmres` for details. inner_kwargs : kwargs Keyword parameters for the "inner" Krylov solver (defined with `method`). Parameter names must start with the `inner_` prefix which will be stripped before passing on the inner method. See, e.g., `scipy.sparse.linalg.gmres` for details. %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='krylov'`` in particular. scipy.sparse.linalg.gmres scipy.sparse.linalg.lgmres Notes ----- This function implements a Newton-Krylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobian-vector products, which are conveniently approximated by a finite difference: .. math:: J v \approx (f(x + \omega*v/|v|) - f(x)) / \omega Due to the use of iterative matrix inverses, these methods can deal with large nonlinear problems. SciPy's `scipy.sparse.linalg` module offers a selection of Krylov solvers to choose from. The default here is `lgmres`, which is a variant of restarted GMRES iteration that reuses some of the information obtained in the previous Newton steps to invert Jacobians in subsequent steps. For a review on Newton-Krylov methods, see for example [1]_, and for the LGMRES sparse inverse method, see [2]_. References ---------- .. [1] C. T. Kelley, Solving Nonlinear Equations with Newton's Method, SIAM, pp.57-83, 2003. :doi:`10.1137/1.9780898718898.ch3` .. [2] D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). :doi:`10.1016/j.jcp.2003.08.010` .. [3] A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). :doi:`10.1137/S0895479803422014` Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * x[1] - 1.0, ... 0.5 * (x[1] - x[0]) ** 2] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.newton_krylov(fun, [0, 0]) >>> sol array([0.66731771, 0.66536458]) Nlgmres c ||_||_ttjjjtjjjtjjjtjjj tjjj tjjj  |||_ t||j|_|j tjjjur1||jd<d|jd<|jddn|j tjjjtjjjtjjj fvr|jddn|j tjjjur||jd<d|jd<|jd g|jd d |jd d |jdddt#|j jD}|D]\}} |dst+d||dd|vrRt-|dd|d} | r d| dd} nd} t/jd|d|d| zdt2| |j|dd<dS)N)bicgstabgmresrcgsminrestfqmr)rwrvrZrrwatolrouter_kouter_vprepend_outer_vTstore_outer_AvFcg|]}|dv| S))rargskwargsr$).0rUs r& z+KrylovJacobian.__init__..s.   222 222r%inner_zUnknown parameter )rz Did you mean 'z'?zOption 'z#' is invalid for the inner method: zO. It will be ignored.Please check inner method documentation for valid options.r<)r=category)preconditionerrrrrrrrrrrrgetmethod method_kw setdefaultgcrotmkr parametersr startswithrjrr>r? UserWarning) rrr inner_maxiterinner_Mrrvalid_inner_paramskeyrinner_param_suggestionssuggestion_msgs r&rzKrylovJacobian.__init__s"% \(1,%+<&- #'<&-,%+ c&&!! mt7JKKK ;%,-3 3 3(5DN9 %()DN9 % N % %fa 0 0 0 0 [U\08"\09"\04666 N % %fa 0 0 0 0 [EL/6 6 6(/DN9 %()DN9 % N % %i 4 4 4 N % %&7 > > > N % %&6 > > > N % %fa 0 0 0   --8    ((** , ,JC>>(++ = !;c!;!;<<<122w000*;CG?+A+A+A'+('H)@)C'H'H'HNN&(N QsQQvQQQ%% !(    &+DN3qrr7 # #7 , ,r%ct|j}t|j}|jtd|ztd|z |_dS)Nr)rr;r+rromega)rmxmfs r&_update_diff_stepz KrylovJacobian._update_diff_steps[ \\     \\    Z#a**,s1bzz9 r%cZt|}|dkrd|zS|j|z }||j||zz|jz |z }t jt j|s5t jt j|rtd|S)Nrz$Function returned non-finite results) rrrWr;rr)r@r?rj)rrCnvscrhs r&rzKrylovJacobian.matvecs !WW 77Q3J Z"_ YYtwA~ & & 0B 6vbk!nn%% E"&Q*@*@ ECDD Dr%rcd|jvr|j|j|fi|j\}}n|j|j|fd|i|j\}}|S)Nrtol)rrop)rrhsrZsolrs r&rzKrylovJacobian.solve sa T^ # ## DGSCCDNCCIC# DGSMMsMdnMMIC r%c||_||_||j2t |jdr|j||dSdSdS)Nro)r;rrrrro)rr-rs r&rozKrylovJacobian.updatess       *t*H55 1#**1a00000 + * 1 1r%ct||||||_||_tjj||_|j &tj |j j dz|_ ||j3t!|jdr |j|||dSdSdS)Nr rg)rrgr;rrrraslinearoperatorrrr)rr1rrrr)rr-rrWs r&rgzKrylovJacobian.setupstQ4(((,%66t<< : !'**.48DJ       *t*G44 6#))!Q55555 + * 6 6r%)NrrNrr) r r!r"r#rrrrrorgr$r%r&rrHsccJCE')K,K,K,K,Z::: 11166666r%rcNt|j}|\}}}}}}} tt|t | d|} dd| D} | rd| z} dd| D} | r| dz} |rt d|d} | t|| |j| z} i}| tt| |||}|j |_ t||S)a Construct a solver wrapper with given name and Jacobian approx. It inspects the keyword arguments of ``jac.__init__``, and allows to use the same arguments in the wrapper function, in addition to the keyword arguments of `nonlin_solve` Nz, c"g|] \}}|d| S=r$rrUrCs r&rz#_nonlin_wrapper..<s&888A1 q 888r%c"g|] \}}|d| Srr$rs r&rz#_nonlin_wrapper..?s&888AQ****888r%zUnexpected signature a def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw): jac = %(jac)s(%(kwkw)s **kw) return nonlin_solve(F, xin, jac, iter, verbose, maxiter, f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search, callback) )rrjackwkw)_getfullargspecrlistr r(joinrjrr roglobalsexecr#rJ)rrrrvarargsvarkwdefaults kwonlyargs kwdefaults_rkw_strkwkw_strwrappernsrWs r&_nonlin_wrapperr0sL  --I@I=D'5(J A #dCMM>??+X66 7 7F YY88888 9 9F yy8888899H#d?><<<===G$6s|"*,,,,G BIIgii" d8D;DL TNNN Kr%rrrrrrr) rKNFNNNNNNrLNFT)rLrr)?rrpr>numpyr)rrr scipy.linalgrrrr r r scipy.sparse.linalgr scipy.sparser scipy._lib._utilr rr _linesearchrrrdifflibr__all__ Exceptionrr.r5r=rDrstriprHrJrrnrdrrrfr rrrrrrrrrrrrrrrrr$r%r&r s  $$$$$$$$$$????????????????''''''++++++FFFFFFCCCCCCCC%%%%%% J J J     I         T (P a111 h/// ?DKO?C48O"O"O"O"d FJ!****Z9<9<9<9<9<9<9<9<@EEEEEEEEP$#$#$#$#$#$#$#$#NY?Y?Y?@X4IIIIIIIIX * + 0mmmmm>mmm`99999L999@UUUUU~UUUxA1A1A1A1A1.A1A1A1H+ + + + + >+ + + \8<8<8<8<8<^8<8<8<~a6a6a6a6a6Xa6a6a6P)))X ?:| 4 4 ?:} 5 5 ?:x 0 0~|<< om[99  !1>BB@@ r%