^MhddlZddlZddlZddlZddlmZmZmZddlm Z m Z m Z m Z m Z mZddlZddlZddlmZddlmZddlmZddlmZmZgd ZGd d eZd Zd ZdZdZ e!d"d"Z#dZ$ d?dZ%e$e% d@dZ&GddZ'GddZ(Gd d!Z)d"Z*Gd#d$e(Z+Gd%d&Z,d'"e#d(<Gd)d*e+Z-Gd+d,e-Z.Gd-d.e+Z/Gd/d0e+Z0Gd1d2e+Z1Gd3d4e+Z2Gd5d6e(Z3d7Z4e4d8e-Z5e4d9e.Z6e4d:e/Z7e4d;e1Z8e4de3Z;dS)AN)asarraydotvdot)normsolveinvqrsvd LinAlgError)get_blas_funcs)copy_if_needed)getfullargspec_no_self)scalar_search_wolfe1scalar_search_armijo) 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__V/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/optimize/_nonlin.pyrrsDr#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' issubdtyper/inexactfloat64r*s r$ _as_inexactr3(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'reshapeshapegetattrr5)r+x0wraps r$ _array_liker;0s= 1bhrll##A 2')9 : :D 477Nr#ctj|stjtjSt |Sr&)r'isfiniteallarrayinfr)vs r$ _safe_normrB7s; ;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$_set_docrHus( {/kJ. //r#krylovFarmijoTc | tn| } t|||| || }tfd}}t j|tj}||}t|}t|}| | |||||dz}n d|j dzz}| durd} n| durd} | d vrtd d }d }d }d}t|D]}}||||}|rnt|||z}||| }t|dkrtd| rt#||||| \}}}}n!d}||z}||}t|}|| || r | ||||dzz|dzz }||dzz|krt||}n$t|t'|||dzz}|}|rLt(jd|| ||fzt(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&)r3r;flatten)zFr9s r$funcznonlin_solve..funcs111[B//001199;;;r#rdTrJF)NrJwolfezInvalid line searchg?gH.?g?gMbP?)tolrz[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.?z%d: |F(x)| = %g; step %g z0A solution was found at the specified tolerance.z:The maximum number of iterations allowed has been reached.)rrZ)nitfunstatussuccessmessage)r,TerminationConditionr3rRr' full_liker@r asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater)sysstdoutwriteflushrr; iteration) rTr9jacobianrPverbosemaxiterrLrMrNrOtol_norm line_searchcallback full_outputraise_exception conditionrUr+dxFxFx_normgammaeta_max eta_tresholdetanr]rXs Fx_norm_neweta_Ainfos `` r$ nonlin_solverzsZ#*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  :88B<<>$$ % % % J        Ar 2 233 3F " * !Q; , 3% &  1b!!4''1b!!!r#:0yE>{Gz?c dg|gt|dzgttz d fd  fd}|dkrt |dd|\}} } n)|d kr#t dd | \}} |d }|zz|dkr d}n }t|} ||| fS) NrrZTc| dkrdS |zz}|}t|dz}|r| d<|d<|d<|S)NrrZ)rB) rstorextrAprzrUtmp_Fxtmp_phitmp_sr+s r$phiz _nonlin_line_search..phisl a==1:  2X DHH qMM1   E!HGAJF1Ir#crt|zdzz}||zd|z |z S)NrF)r)abs)rdsrrdiffs_norms r$derphiz#_nonlin_line_search..derphi#sG!ffvo!U *AbD&&&Q/255r#rWr)xtolaminrJ)rrY)T)rrr)rUr+r{rz search_typersminrrphi1phi0r|rrrrrs`` ` ` @@@@@r$rjrjse CETFBxx{mG !WWtBxx F           6666666g,S&'!*26TCCC 4  &sGAJ ,02224 y  AbDAE!H}} AY T!WW2hhG aW r#c,eZdZdZdddddefdZdZdS)r`z Termination condition for an iteration. It is terminated if - |F| < f_rtol*|F_0|, AND - |F| < f_tol AND - |dx| < x_rtol*|x|, AND - |dx| < x_tol Nc|&tjtjjdz}| tj}| tj}| tj}||_||_||_||_||_ ||_ d|_ d|_ dS)NgUUUUUU?r) r'finfor2epsr@rNrOrLrMrrPf0_normrp)selfrLrMrNrOrPrs r$__init__zTerminationCondition.__init__Js =HRZ((,6E >VF =FE >VF       r#c|xjdz c_||}||}||}|j||_|dkrdS|jd|j|jkzSt ||jko+||jz |jko||jko ||jz |kS)NrrrZ) rprrrPintrLrMrNrO)rfr+rzf_normx_normdx_norms r$rhzTerminationCondition.checkbs !11))B-- < !DL Q;;1 9 23 3Fdj(;t{*dl:;4:-:#DK/69<< .__array__s-$$%I%%I%IJJJ||~~%r#NN)itemsrfsetattrhasattr)rkwnamesnamevaluers r$rzJacobian.__init__s88888:: . .KD%5   !CT!C!CDDD dBtH--- 4 # # & & & & & & & & &r#c t|Sr&)rrs r$aspreconditionerzJacobian.aspreconditionerst$$$r#rctr&NotImplementedErrorrrArXs r$rzJacobian.solve!!r#cdSr&r"rr+rTs r$rkzJacobian.update r#c||_|j|jf|_|j|_|jjt jur|||dSdSr&)rUrer7r/ __class__rcrrkrr+rTrUs r$rczJacobian.setupsV faf% W > 8> 1 1 KK1      2 1r#Nr) rrr r!rrrrkrcr"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)Nrcr)rqrrrkrrcrr)rrqs r$rzInverseJacobian.__init__s_  n o 8W % % (!DJ 8X & & +#?DLLL + +r#c|jjSr&)rqr7rs r$r7zInverseJacobian.shape }""r#c|jjSr&)rqr/rs r$r/zInverseJacobian.dtyperr#N)rrr r!rpropertyr7r/r"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. rZzarray must have rank <= 2rrzarray must be squarec$t|Sr&)rrAJs r$zasjacobian..sQr#cRtj|Sr&)rconjTrs r$rzasjacobian..s#affhhj!*<*<r#c$t|Sr&)rrArXrs r$rzasjacobian..suQ{{r#cRtj|Sr&)rrrrs r$rzasjacobian..saffhhj!0D0Dr#)rrrrr/r7zmatrix must be squarec|zSr&r"rs r$rzasjacobian..s Qr#c<j|zSr&rrrs r$rzasjacobian..s!&&((*q.r#c|Sr&r"rArXrspsolves r$rzasjacobian..swwq!}}r#cJj|Sr&rrs r$rzasjacobian..s A0F0Fr#r7r/rrrrrkrc)rrrrrkrcr/r7cFeZdZdZdfd ZfdZdfd ZfdZdS) asjacobian..Jacc||_dSr&r*rs r$rkzasjacobian..Jac.updates r#rc|j}t|tjrt ||St j|r ||StdNzUnknown matrix type) r+ isinstancer'ndarrayrscipysparseissparserfrrArXmrrs r$rzasjacobian..Jac.solveskAdfIIa,,< A;;&\**1--<"71a==($%:;;;r#c|j}t|tjrt ||St j|r||zStdr) r+rr'rrrrrrfrrArrs r$rzasjacobian..Jac.matvec!sdAdfIIa,,<q!99$\**1--<q5L$%:;;;r#cH|j}t|tjr't |j|Stj |r#|j|Stdr) r+rr'rrrrrrrrfrs r$rzasjacobian..Jac.rsolve*sAdfIIa,,< Q///\**1--<"716688:q111$%:;;;r#c:|j}t|tjr't |j|Stj |r|j|zStdr) r+rr'rrrrrrrrfrs r$rzasjacobian..Jac.rmatvec3s{AdfIIa,,<qvvxxz1---\**1--<6688:>)$%:;;;r#Nr)rrr rkrrrr)rrsr$Jacrs    < < < < < < < < < < < < < < < < < < < < < < < < < >>r#c eZdZdZdZdZdS)GenericBroydenct||||||_||_t |drK|jFt |}|r*dtt |dz|z |_dSd|_dSdSdS)Nalpha?rrY)rrclast_flast_xrrrr))rr9f0rUnormf0s r$rczGenericBroyden.setupMstRT***  4 ! ! !dj&8"XXF ! T"XXq!1!11F:    ! !&8&8r#ctr&rrr+rrzdfrdf_norms r$_updatezGenericBroyden._update[rr#c ||jz }||jz }|||||t|t|||_||_dSr&)r r rr)rr+rrrzs r$rkzGenericBroyden.update^sR _ _ Q2r488T"XX666  r#N)rrr rcrrkr"r#r$rrLsA ! ! !"""r#rceZdZdZdZedZedZdZdZ 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&)rcsrrr/ collapsed)rrrr/s r$rzLowRankMatrix.__init__qs0  r#ctgd|dd|gz\}}}||z}t||D]$\}} || |} ||||j| }%|S)N)axpyscaldotcr)r zipre) rArrrrrrwcdas r$_matveczLowRankMatrix._matvecys)*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 vrrrNrr.) lenr r'identityr/ enumeratezerosrrre)rArrrrrc0Airjrqrqcs 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!rrrrrAs r$rzLowRankMatrix.matvecs> > %6$.!,, ,$$Q DGTWEEEr#c|j1tj|jj|St |tj|j|j|j S)zEvaluate w = M^H v) rr'rrrrr!rrrr0s r$rzLowRankMatrix.rmatvecsW > %6$.*//11155 5$$Q (;(;TWdgNNNr#rc|jt|j|St||j|j|jS)r#)rrrr.rrrrs r$rzLowRankMatrix.solves< > %++ +##Atz47DGDDDr#c|j,t|jj|St|t j|j|j|j S)zEvaluate w = M^-H v) rrrrrr.r'rrrrs r$rzLowRankMatrix.rsolvesU > %)..00!44 4##Arwtz':':DGTWMMMr#cX|j;|xj|dddf|dddfzz c_dS|j||j|t |j|jkr|dSdSr&)rrrappendrr$recollapse)rrrs r$r5zLowRankMatrix.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.)rdN)r'r?r rrrrrs r$r6zLowRankMatrix.collapses1$^<<< 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. Nrr?r@s r$ simple_reducezLowRankMatrix.simple_reducesd > % Faxxxx$'llT!!  $'llT!!!!!!r#c|jdS|}||}n|dz }|jr(t|t|jd}t dt||dz }t|j}||krdSt j|jj}t j|jj}t|d\}}t||j }t|d\} } } t|t| }t|| j }t|D]N} |dd| f|j| <|dd| f|j| <O|j|d=|j|d=dS) a Reduce the rank of the matrix by retaining some SVD components. This corresponds to the "Broyden Rank Reduction Inverse" algorithm described in [1]_. Note that the SVD decomposition can be done by solving only a problem whose size is the effective rank of this matrix, which is viable even for large problems. Parameters ---------- max_rank : int Maximum rank of this matrix after reduction. to_retain : int, optional Number of SVD components to retain when reduction is done (ie. rank > max_rank). Default is ``max_rank - 2``. References ---------- .. [1] B.A. van der Rotten, PhD thesis, "A limited memory Broyden method to solve high-dimensional systems of nonlinear equations". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003). https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf NrZrreconomic)modeF) full_matrices)rrrir$r)r'r?rrr rrr rrgrd) rmax_rank to_retainrr,rCDRUSWHks 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&)rrr r!r staticmethodr!r.rrrrr5rr6rBrDrRr"r#r$rrfs\\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<rR reduce_paramsrsr$rz'BroydenFirst.__init__..s#547#5}#Er#simplec"jjSr&)r<rDrYsr$rz'BroydenFirst.__init__..s#847#8-#Hr#rVc"jjSr&)r<rBrYsr$rz'BroydenFirst.__init__..s#947#9=#Ir#zUnknown rank reduction method '') rrrr<r'r@rIrr_reducerf)rrreduction_methodrIrZs` @r$rzBroydenFirst.__init__s%%%   vH  & , , 3MM,QRR0M/2 !A-7 u $ $EEEEEDLLL  ) )HHHHHDLLL  * *IIIIIDLLLR?ORRRSS Sr#ct||||t|j |jd|j|_dS)Nr)rrcrrr7r/r<rs r$rczBroydenFirst.setups?T1a... TZ]DJGGr#c*t|jSr&)rr<rs r$rzBroydenFirst.todenses47||r#rc|j|}tj|s@||j|j|j|j|S|Sr&) r<rr'r=r>rcr r rU)rrrXrs r$rzBroydenFirst.solvesf GNN1  {1~~!!## % JJt{DK ; ; ;7>>!$$ $r#c6|j|Sr&)r<rrrs r$rzBroydenFirst.matvecsw}}Qr#c6|j|Sr&)r<rrrrXs r$rzBroydenFirst.rsolveswq!!!r#c6|j|Sr&)r<rrfs r$rzBroydenFirst.rmatvecsw~~a   r#c||j|}||j|z }|t ||z } |j|| dSr&)r_r<rrrr5 rr+rrzrrrrArrs r$rzBroydenFirst._updatesg  GOOB   ## # R O q!r#)NrVNr) rrr r!rrcrrrrrrr"r#r$rrJs22hTTTT2HHH   """"!!!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|| dSNrZ)r_r<rr5rks r$rzBroydenSecond._updatesU   ## #  N q!r#N)rrr 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&)rrrMrzrr}w0)rrrsrrs r$rzAnderson.__init__AsD%%%  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|SNrr.) rr$rzr'emptyr/rgrrrr r ) rrrXrzrdf_frQr}rs r$rzAnderson.solveJsj[] 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)Nrr.rZ) rr$rzr'rvr/rgrrrsr) rrrzrrwrQbr*r+r}rs r$rzAnderson.matvecasR ] 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)Nrr.rZr)rrrzr5rr$popr'r'r/rgrsrtriurrr ) rr+rrzrrrrr r*r+wds r$rzAnderson._updatexsZ 6Q;; F r r$'llTV## GKKNNN GKKNNN$'llTV## LL HaV17 + + +q = =A1a[[ = =66!BBBB$TWQZ < <<!A#  = RWQ]]_ ! ! # ##r#)Nrrpr)rrr 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&rrrrrs r$rzDiagBroyden.__init__!%%% r#ct||||tj|jdfd|jz |j|_dS)Nrrr.)rrcr'fullr7rr/rrs r$rczDiagBroyden.setupsIT1a...$*Q-)1tz>LLLr#rc| |jz Sr&rrhs r$rzDiagBroyden.solverDF{r#c| |jzSr&rrfs r$rzDiagBroyden.matvecrr#c<| |jz Sr&rrrhs r$rzDiagBroyden.rsolverDFKKMM!!r#c<| |jzSr&rrfs r$rzDiagBroyden.rmatvecrr#c6tj|j Sr&)r'diagrrs r$rzDiagBroyden.todenseswwr#cN|xj||j|zz|z|dzz zc_dSrnrrs r$rzDiagBroyden._updates. 2r >2%gqj00r#r&r rrr r!rrcrrrrrrr"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&rrhs r$rzLinearMixing.solver$*}r#c| |jz Sr&rrfs r$rzLinearMixing.matvecrr#c<| tj|jzSr&r'rrrhs r$rzLinearMixing.rsolver"'$*%%%%r#c<| tj|jz Sr&rrfs r$rzLinearMixing.rmatvecrr#cvtjtj|jdd|jz S)Nr)r'rrr7rrs r$rzLinearMixing.todenses*wrwtz!}bm<<===r#cdSr&r"rs r$rzLinearMixing._updaterr#r&r) rrr 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 NrYcdt|||_||_d|_dSr&)rrralphamaxbeta)rrrs r$rzExcitingMixing.__init__#s/%%%    r#ct||||tj|jdf|j|j|_dSru)rrcr'rr7rr/rrs r$rczExcitingMixing.setup)sET1a...GTZ],dj KKK r#rc| |jzSr&rrhs r$rzExcitingMixing.solve-r$)|r#c| |jz Sr&rrfs r$rzExcitingMixing.matvec0rr#c<| |jzSr&rrrhs r$rzExcitingMixing.rsolve3r$)..""""r#c<| |jz Sr&rrfs r$rzExcitingMixing.rmatvec6rr#c:tjd|jz S)Nr)r'rrrs r$rzExcitingMixing.todense9swr$)|$$$r#c||jzdk}|j|xx|jz cc<|j|j|<tj|jd|j|jdS)Nr)out)r rrr'clipr)rr+rrzrrrincrs r$rzExcitingMixing._update<sa}q  $4:%: 4%  1dm;;;;;;r#)NrYrrr"r#r$rrs4 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 |jdd|D]>\}}|dst'd|||j|dd<?dS)N)bicgstabgmresrcgsminrestfqmr)rsrrrVrrsatolrouter_kouter_vprepend_outer_vTstore_outer_AvFinner_zUnknown parameter )preconditionerrrrrrrrrrrrgetmethod method_kw setdefaultgcrotmkr startswithrf) rrr inner_maxiterinner_Mrrkeyrs 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((** , ,JC>>(++ = !;c!;!;<<<&+DN3qrr7 # # , ,r#ct|j}t|j}|jtd|ztd|z |_dS)Nr)rr9r)r romega)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) rrrUr9r r'r>r=rf)rrAnvscrds r$rzKrylovJacobian.matvecs !WW 77Q3J Z"_ YYtwA~ & & 0B 6vbk!nn%% E"&Q*@*@ ECDD Dr#rcd|jvr|j|j|fi|j\}}n|j|j|fd|i|j\}}|S)Nrtol)rrop)rrhsrXsolrs r$rzKrylovJacobian.solvesa T^ # ## DGSCCDNCCIC# DGSMMsMdnMMIC r#c||_||_||j2t |jdr|j||dSdSdS)Nrk)r9r rrrrk)rr+rs r$rkzKrylovJacobian.updatess       *t*H55 1#**1a00000 + * 1 1r#ct||||||_||_tjj||_|j &tj |j j dz|_ ||j3t!|jdr |j|||dSdSdS)Nrrc)rrcr9r rrraslinearoperatorrrr'rr/rrrr)rr+rrUs r$rczKrylovJacobian.setupstQ4(((,%66t<< : !'**.48DJ       *t*G44 6#))!Q55555 + * 6 6r#)NrrNrr) rrr r!rrrrrkrcr"r#r$rrGsccJCE')-,-,-,-,^::: 11166666r#rcNt|j}|\}}}}}}} tt|t | d|} dd| D} | rd| z} dd| D} | r| dz} |rt d|d} | t|| |j| z} i}| tt| |||}|j |_ t||S)a Construct a solver wrapper with given name and Jacobian approx. It inspects the keyword arguments of ``jac.__init__``, and allows to use the same arguments in the wrapper function, in addition to the keyword arguments of `nonlin_solve` Nz, c"g|] \}}|d| S=r".0rQrAs r$ z#_nonlin_wrapper..s&888A1 q 888r#c"g|] \}}|d| Srr"rs r$rz#_nonlin_wrapper.. s&888AQ****888r#zUnexpected signature a def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw): jac = %(jac)s(%(kwkw)s **kw) return nonlin_solve(F, xin, jac, iter, verbose, maxiter, f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search, callback) )rrjackwkw)_getfullargspecrlistrr$joinrfrrrkglobalsexecr!rH)rr signatureargsvarargsvarkwdefaults kwonlyargs kwdefaults_kwargskw_strkwkw_strwrappernsrUs r$_nonlin_wrapperrsL  --I@I=D'5(J A #dCMM>??+X66 7 7F YY88888 9 9F yy8888899H#d?><<<===G$6s|"*,,,,G BIIgii" d8D;DL TNNN Kr#rrrrrrr) rINFNNNNNNrJNFT)rJrr)rs $$$$$$$$$$????????????????''''''++++++FFFFFFCCCCCCCC J J J     I         T (P a111 h/// ?DKO?C48P"P"P"P"f 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<~C6C6C6C6C6XC6C6C6T)))X ?:| 4 4 ?:} 5 5 ?:x 0 0~|<< om[99  !1>BB@@ r#