9=eddlZddlZddlmZmZmZmZmZm Z ddlm Z m Z m Z ddl ZddlZddlmZddlZddlmZddlmZmZgdZGd d eZd Zd Zd ZdZed d Z!dZ" d>dZ#e"e# d?dZ$GddZ%GddZ&Gdd Z'd!Z(Gd"d#e&Z)Gd$d%Z*d& e!d'<Gd(d)e)Z+Gd*d+e+Z,Gd,d-e)Z-Gd.d/e)Z.Gd0d1e)Z/Gd2d3e)Z0Gd4d5e&Z1d6Z2e2d7e+Z3e2d8e,Z4e2d9e-Z5e2d:e/Z6e2d;e.Z7e2d>    r!ct|}tj|jtjst|tjS|S)z:Return `x` as an array, of either floats or complex floatsdtype)r r% issubdtyper-inexactfloat_r(s r" _as_inexactr1"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%reshapeshapegetattrr3)r)x0wraps r" _array_liker9*s= 1bhrll##A 2')9 : :D 477Nr!ctj|stjtjSt |Sr$)r%isfiniteallarrayinfr)vs r" _safe_normr@1s; ;q>>     x 77Nr!z F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution a iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : {None, 'armijo' (default), 'wolfe'}, optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'. callback : function, optional Optional callback function. It is called on every iteration as ``callback(x, f)`` where `x` is the current solution and `f` the corresponding residual. Returns ------- sol : ndarray An array (of similar array type as `x0`) containing the final solution. Raises ------ NoConvergence When a solution was not found. ) params_basic params_extrac@|jr|jtz|_dSdSr$)__doc__ _doc_parts)objs r"_set_docrGos( {/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$t|t'|||dzz}|}|rLt(jd|| ||fzt(j|rt1t3|d}| r,|j|||dkddd|d}t3||fSt3|S)a Find a root of a function, in a way suitable for large-scale problems. Parameters ---------- %(params_basic)s jacobian : Jacobian A Jacobian approximation: `Jacobian` object or something that `asjacobian` can transform to one. Alternatively, a string specifying which of the builtin Jacobian approximations to use: krylov, broyden1, broyden2, anderson diagbroyden, linearmixing, excitingmixing %(params_extra)s full_output : bool If true, returns a dictionary `info` containing convergence information. raise_exception : bool If True, a `NoConvergence` exception is raise if no solution is found. See Also -------- asjacobian, Jacobian Notes ----- This algorithm implements the inexact Newton method, with backtracking or full line searches. Several Jacobian approximations are available, including Krylov and Quasi-Newton methods. References ---------- .. [KIM] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations". Society for Industrial and Applied Mathematics. (1995) https://archive.siam.org/books/kelley/fr16/ N)f_tolf_rtolx_tolx_rtoliterrcttt|Sr$)r1r9flatten)zFr7s r"funcznonlin_solve..funcs111[B//001199;;;r!rdTrIF)NrIwolfezInvalid 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.)rrY)nitfunstatussuccessmessage)r*TerminationConditionr1rQr% full_liker>r asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater'sysstdoutwriteflushrr9 iteration) rSr7jacobianrOverbosemaxiterrKrLrMrNtol_norm line_searchcallback full_outputraise_exception conditionrTr)dxFxFx_normgammaeta_max eta_tresholdetanr\rWs Fx_norm_neweta_Ainfos `` r" nonlin_solvertsZ#*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<<>$$ % % % J        Ar 2 233 3F " * !Q; , 3% &  1b!!4''1b!!!r!:0yE>{Gz?c dg|gt|dzgttz d fd  fd}|dkrt |dd|\}} } n)|d kr#t dd | \}} |d }|zz|dkr d}n }t|} ||| fS) NrrYTc| dkrdS |zz}|}t|dz}|r| d<|d<|d<|S)NrrY)r@) rstorextr?pryrTtmp_Fxtmp_phitmp_sr)s r"phiz _nonlin_line_search..phisl a==1:  2X DHH qMM1   E!HGAJF1Ir!crt|zdzz}||zd|z |z S)NrF)r)abs)rdsrrdiffs_norms r"derphiz#_nonlin_line_search..derphisG!ffvo!U *AbD&&&Q/255r!rVr)xtolaminrI)rrX)T)rrr)rTr)rzry search_typersminrrphi1phi0r{rrrrrs`` ` ` @@@@@r"riri se CETFBxx{mG !WWtBxx F           6666666g,S&'!*26TCCC 4  &sGAJ ,02224 y  AbDAE!H}} AY T!WW2hhG aW r!c,eZdZdZdddddefdZdZdS)r_z Termination condition for an iteration. It is terminated if - |F| < f_rtol*|F_0|, AND - |F| < f_tol AND - |dx| < x_rtol*|x|, AND - |dx| < x_tol Nc|&tjtjjdz}| tj}| tj}| tj}||_||_||_||_||_ ||_ d|_ d|_ dS)NgUUUUUU?r) r%finfor0epsr>rMrNrKrLrrOf0_normro)selfrKrLrMrNrOrs r"__init__zTerminationCondition.__init__Ds =HRY''+5E >VF =FE >VF       r!c|xjdz c_||}||}||}|j||_|dkrdS|jd|j|jkzSt ||jko+||jz |jko||jko ||jz |kS)NrrrY) rorrrOintrKrLrMrN)rfr)ryf_normx_normdx_norms r"rgzTerminationCondition.check\s !11))B-- < !DL Q;;1 9 23 3Fdj(;t{*dl:;4:-:#DK/69<< z#Jacobian.__init__..sT\\^^r!)itemsresetattrhasattr __array__)rkwnamesnamevalues` r"rzJacobian.__init__s88888:: . .KD%5   !>!EFFF dBtH--- 4 # # 43333DNNN 4 4r!c t|Sr$)rrs r"aspreconditionerzJacobian.aspreconditionerst$$$r!rctr$NotImplementedErrorrr?rWs r"rzJacobian.solve!!r!cdSr$r rr)rSs r"rjzJacobian.update r!c||_|j|jf|_|j|_|jjt jur|||dSdSr$)rTrdr5r- __class__rbrrjrr)rSrTs r"rbzJacobian.setupsV faf% W > 8> 1 1 KK1      2 1r!Nr) rrrrDrrrrjrbr r!r"rrwso##J 4 4 4%%%""""   r!rc@eZdZdZedZedZdS)rc||_|j|_|j|_t |dr |j|_t |dr|j|_dSdS)Nrbr)rprrrjrrbrr)rrps r"rzInverseJacobian.__init__s_  n o 8W % % (!DJ 8X & & +#?DLLL + +r!c|jjSr$)rpr5rs r"r5zInverseJacobian.shape }""r!c|jjSr$)rpr-rs r"r-zInverseJacobian.dtyperr!N)rrrrpropertyr5r-r r!r"rrsY+++##X###X###r!rc tjjjt t rSt jrtt r St tj rj dkrtdtj tjjdjdkrtdt fdfdfdfd jj StjrXjdjdkrtd t fd fd fdfdjj St%drt%dr|t%drlt t'dt'djt'dt'dt'djjSt+r Gfddt }|St t,rGt/t0t2t4t6t8t:t<St?d)zE Convert given object to one suitable for use as a Jacobian. rYzarray must have rank <= 2rrzarray must be squarec$t|Sr$)r r?Js r"rzasjacobian..sQr!cRtj|Sr$)r conjTrs r"rzasjacobian..s#affhhj!*<*<r!c$t|Sr$)rrs r"rzasjacobian..sa r!cRtj|Sr$)rrrrs r"rzasjacobian..sqvvxxz1)=)=r!)rrrrr-r5zmatrix must be squarec|zSr$r rs r"rzasjacobian..s 1r!c<j|zSr$rrrs r"rzasjacobian..s!&&((*q.r!c|Sr$r r?rspsolves r"rzasjacobian..s1 r!cJj|Sr$rrs r"rzasjacobian..sQ)?)?r!r5r-rrrrrjrb)rrrrrjrbr-r5cFeZdZdZdfd ZfdZdfd ZfdZdS) asjacobian..Jacc||_dSr$r(rs r"rjzasjacobian..Jac.updates r!rc|j}t|tjrt ||St j|r ||StdNzUnknown matrix type) r) isinstancer%ndarrayrscipysparse isspmatrixrerr?rWmrrs r"rzasjacobian..Jac.solveskAdfIIa,,< A;;&\,,Q//<"71a==($%:;;;r!c|j}t|tjrt ||St j|r||zStdr) r)rr%rr rrrrerr?rrs r"rzasjacobian..Jac.matvecsdAdfIIa,,<q!99$\,,Q//<Q3J$%:;;;r!cH|j}t|tjr't |j|Stj |r#|j|Stdr) r)rr%rrrrrrrrers r"rzasjacobian..Jac.rsolve sAdfIIa,,< Q///\,,Q//<"716688:q111$%:;;;r!c:|j}t|tjr't |j|Stj |r|j|zStdr) r)rr%rr rrrrrrers r"rzasjacobian..Jac.rmatvecs{AdfIIa,,<qvvxxz1---\,,Q//<6688:>)$%:;;;r!Nr)rrrrjrrrr)rrsr"Jacrs    < < < < < < < < < < < < < < < < < < < < < < < < < >>r!c eZdZdZdZdZdS)GenericBroydenct||||||_||_t |drK|jFt |}|r*dtt |dz|z |_dSd|_dSdSdS)Nalpha?rrX)rrblast_flast_xrrrr')rr7f0rTnormf0s r"rbzGenericBroyden.setup/stRT***  4 ! ! !dj&8"XXF ! T"XXq!1!11F:    ! !&8&8r!ctr$rrr)rrydfrdf_norms r"_updatezGenericBroyden._update=rr!c ||jz }||jz }|||||t|t|||_||_dSr$)rrrr)rr)rr rys r"rjzGenericBroyden.update@sR _ _ Q2r488T"XX666  r!N)rrrrbrrjr r!r"rr.sA ! ! !"""r!rceZdZdZdZedZedZdZdZ ddZ dd Z d Z d Z d Zd ZdZddZdS) LowRankMatrixz A matrix represented as .. math:: \alpha I + \sum_{n=0}^{n=M} c_n d_n^\dagger However, if the rank of the matrix reaches the dimension of the vectors, full matrix representation will be used thereon. cZ||_g|_g|_||_||_d|_dSr$)rcsrrr- collapsed)rrrr-s r"rzLowRankMatrix.__init__Ss0  r!ctgd|dd|gz\}}}||z}t||D]$\}} || |} ||||j| }%|S)N)axpyscaldotcr)r ziprd) r?rrrrrrwcdas r"_matveczLowRankMatrix._matvec[s)*B*B*B*,RaR&A3,88dD AIBKK & &DAqQ AQ161%%AAr!c t|dkr||z Stddg|dd|gz\}}|d}|tjt||jz}t |D]6\}} t |D]!\} } ||| fxx|| | z cc<"7tjt||j} t |D]\} } || || | <| |z} t|| } ||z } t|| D]\} }|| | | j | } | S)Evaluate w = M^-1 vrrrNrr,) lenr r%identityr- enumeratezerosrrrd)r?rrrrrc0Airjrqrqcs r"_solvezLowRankMatrix._solveesq r77a<<U7N$VV$4b!fslCC d U BKBrx888 8bMM % %DAq!"  % %1!A#$$q!**$ % HSWWBH - - -bMM  DAq41::AaDD U  !QKK eGQZZ ( (EArQ16B3''AAr!c|jtj|j|St||j|j|jS)zEvaluate w = M v)rr%r rrrrrrr?s r"rzLowRankMatrix.matvecs> > %6$.!,, ,$$Q DGTWEEEr!c|j1tj|jj|St |tj|j|j|j S)zEvaluate w = M^H v) rr%r rrrrrrrr-s 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!rdcollapse)rrrs r"r2zLowRankMatrix.appends > % NNa$i!DF)..*:*:: :NN F q q tw< %> ! Z DF$*=== =)) - -DAq !AAAdF)Ad111fINN,,, ,BB r!cbtj||_d|_d|_d|_dS)z0Collapse the low-rank matrix to a full-rank one.N)r%r=rrrrrs r"r3zLowRankMatrix.collapses*$ 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. Nrr8r9s r" simple_reducezLowRankMatrix.simple_reducesd > % Faxxxx$'llT!!  $'llT!!!!!!r!Nc|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 NrYrreconomic)modeF) full_matrices)rrrhr!r'r%r=rrrr rrrrfrc) 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!rr$)rrrrDr staticmethodrr+rrrrr2rr3r;r=rKr r!r"rrHs\\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)ra 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://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.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 |z) Nr rrrc"jjSr$)r5rK reduce_paramsrsr"rz'BroydenFirst.__init__..ks#547#5}#Er!simplec"jjSr$)r5r=rRsr"rz'BroydenFirst.__init__..ms#847#8-#Hr!rOc"jjSr$)r5r;rRsr"rz'BroydenFirst.__init__..os#947#9=#Ir!z"Unknown rank reduction method '%s') rrrr5r%r>rBrr_reducere)rrreduction_methodrBrSs` @r"rzBroydenFirst.__init__Zs%%%   vH  & , , 3MM,QRR0M/2 !A-7 u $ $EEEEEDLLL  ) )HHHHHDLLL  * *IIIIIDLLLA-.// /r!ct||||t|j |jd|j|_dS)Nr)rrbrrr5r-r5rs r"rbzBroydenFirst.setupts?T1a... TZ]DJGGr!c*t|jSr$)rr5rs r"rzBroydenFirst.todensexs47||r!rc|j|}tj|s@||j|j|j|j|S|Sr$) r5rr%r;r<rbrrrT)rrrWrs r"rzBroydenFirst.solve{sf GNN1  {1~~!!## % JJt{DK ; ; ;7>>!$$ $r!c6|j|Sr$)r5rrrs r"rzBroydenFirst.matvecsw}}Qr!c6|j|Sr$)r5rrrrWs r"rzBroydenFirst.rsolveswq!!!r!c6|j|Sr$)r5rr^s r"rzBroydenFirst.rmatvecsw~~a   r!c||j|}||j|z }|t ||z } |j|| dSr$)rWr5rrr r2 rr)rryr rr r?rrs r"rzBroydenFirst._updatesg  GOOB   ## # R O q!r!)NrONr) rrrrDrrbrrrrrrr r!r"rr$s33j////4HHH   """"!!!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|| dSNrY)rWr5rr2rcs r"rzBroydenSecond._updatesU   ## #  N q!r!N)rrrrDrr 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$)rrrMryr r|w0)rrrkrjs r"rzAnderson.__init__sD%%%  r!rc|j |z}t|j}|dkr|Stj||j}t |D] }t|j||||<! t|j |}n&#t$r|jdd=|jdd=|cYSwxYwt |D]1}||||j||j|j|zzzz }2|SNrr,) rr!ryr%emptyr-rfr r rrr) rrrWryrdf_frJr|rs r"rzAnderson.solve&sj[] 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,rY) rr!ryr%rnr-rfr r rkr) rrryrrorJbr'r(r|rs r"rzAnderson.matvec=sR ] LL 66Ix)))q * *A471:q))DGG HaV17 + + +q Q QA1XX Q Qdgaj$'!*55!A#66dgllacFFFd471:twqz::47A:EdjPPFFF Qaq @ @A %(DGAJDJ)>>? ?BB r!c|jdkrdS|j||j|t |j|jkrQ|jd|jdt |j|jkQt |j}t j||f|j}t|D]Y} t| |D]F} | | kr |j dz} nd} d| zt|j| |j| z|| | f<GZ|t j |dj z }||_dS)Nrr,rYr)rjryr2r r!popr%r$r-rfrkr triurrr) rr)rryr rr rrr'r(wds r"rzAnderson._updateTsZ 6Q;; F r r$'llTV## GKKNNN GKKNNN$'llTV## LL HaV17 + + +q = =A1a[[ = =66!BBBB$TWQZ < <<!A#  = RWQ]]_ ! ! # ##r!)Nrrhr)rrrrDrrrrr r!r"rrse++L..r!rcHeZdZdZd dZdZd dZdZd dZd Z d Z d Z dS)ra, Find a root of a function, using diagonal Broyden Jacobian approximation. The Jacobian approximation is derived from previous iterations, by retaining only the diagonal of Broyden matrices. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. Parameters ---------- %(params_basic)s alpha : float, optional Initial guess for the Jacobian is (-1/alpha). %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='diagbroyden'`` in particular. Examples -------- The following functions define a system of nonlinear equations >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.diagbroyden(fun, [0, 0]) >>> sol array([0.84116403, 0.15883384]) NcHt|||_dSr$rrrrrs r"rzDiagBroyden.__init__!%%% r!ct||||tj|jdfd|jz |j|_dS)Nrrr,)rrbr%fullr5rr-rrs r"rbzDiagBroyden.setupsIT1a...$*Q-)1tz>LLLr!rc| |jz Sr$rr`s r"rzDiagBroyden.solverDF{r!c| |jzSr$r~r^s r"rzDiagBroyden.matvecrr!c<| |jz Sr$rrr`s r"rzDiagBroyden.rsolverDFKKMM!!r!c<| |jzSr$rr^s r"rzDiagBroyden.rmatvecrr!c6tj|j Sr$)r%diagrrs r"rzDiagBroyden.todenseswwr!cN|xj||j|zz|z|dzz zc_dSrfr~r s r"rzDiagBroyden._updates. 2r >2%gqj00r!r$r rrrrDrrbrrrrrrr r!r"rrrs&&PMMM"""""""   11111r!rcBeZdZdZd dZd dZdZd dZdZd Z d Z dS) ra Find a root of a function, using a scalar Jacobian approximation. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. Parameters ---------- %(params_basic)s alpha : float, optional The Jacobian approximation is (-1/alpha). %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='linearmixing'`` in particular. NcHt|||_dSr$rxrys r"rzLinearMixing.__init__rzr!rc| |jzSr$rr`s r"rzLinearMixing.solver$*}r!c| |jz Sr$rr^s r"rzLinearMixing.matvecrr!c<| tj|jzSr$r%rrr`s r"rzLinearMixing.rsolver"'$*%%%%r!c<| tj|jz Sr$rr^s r"rzLinearMixing.rmatvecrr!cvtjtj|jdd|jz S)Nr)r%rr|r5rrs r"rzLinearMixing.todenses*wrwtz!}bm<<===r!cdSr$r r s r"rzLinearMixing._updaterr!r$r) rrrrDrrrrrrrr 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 NrXcdt|||_||_d|_dSr$)rrralphamaxbeta)rrrs r"rzExcitingMixing.__init__s/%%%    r!ct||||tj|jdf|j|j|_dSrm)rrbr%r|r5rr-rrs r"rbzExcitingMixing.setupsET1a...GTZ],dj KKK r!rc| |jzSr$rr`s r"rzExcitingMixing.solve r$)|r!c| |jz Sr$rr^s r"rzExcitingMixing.matvec rr!c<| |jzSr$rrr`s r"rzExcitingMixing.rsolver$)..""""r!c<| |jz Sr$rr^s r"rzExcitingMixing.rmatvecrr!c:tjd|jz S)Nr)r%rrrs r"rzExcitingMixing.todenseswr$)|$$$r!c||jzdk}|j|xx|jz cc<|j|j|<tj|jd|j|jdS)Nr)out)rrrr%clipr)rr)rryr rr incrs r"rzExcitingMixing._updatesa}q  $4:%: 4%  1dm;;;;;;r!)NrXrrr 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|z||j|dd<?dS)N)bicgstabgmresrcgsminrestfqmr)rrrjrOrrratolrouter_kouter_vprepend_outer_vTstore_outer_AvFinner_zUnknown parameter %s)preconditionerrrrrrrrrrrrgetmethod method_kw setdefaultgcrotmkr startswithre) 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>>(++ ? !7#!=>>>&+DN3qrr7 # # , ,r!ct|j}t|j}|jtd|ztd|z |_dS)Nr)rr7r'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) rrrTr7rr%r<r;re)rr?nvscr\s 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)NrW)rrop)rrhsrWsolrs r"rzKrylovJacobian.solvesa DN " "# DGSCCDNCCIC# DGSLLcLT^LLIC r!c||_||_||j2t |jdr|j||dSdSdS)Nrj)r7rrrrrj)rr)rs r"rjzKrylovJacobian.updatess       *t*H55 1#**1a00000 + * 1 1r!ct||||||_||_tjj||_|j &tj |j j dz|_ ||j3t!|jdr |j|||dSdSdS)Nrrb)rrbr7rrrraslinearoperatorrrr%rr-rrrr)rr)rrTs r"rbzKrylovJacobian.setupstQ4(((,%66t<< : !'**.48DJ       *t*G44 6#))!Q55555 + * 6 6r!)NrrNrr) rrrrDrrrrrjrbr r!r"rr#sccJCE')-,-,-,-,^::: 11166666r!rcNt|j}|\}}}}}}} tt|t | d|} dd| D} | rd| z} dd| D} | r| dz} |rt d|zd} | 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 .0rJr?s r" z#_nonlin_wrapper..s&888A1 q 888r!c"g|] \}}|d| Srr rs r"rz#_nonlin_wrapper..s&888AQ****888r!zUnexpected signature %sa 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!joinrerrrjglobalsexecrDrG)rr signatureargsvarargsvarkwdefaults kwonlyargs kwdefaults_kwargskw_strkwkw_strwrappernsrTs r"_nonlin_wrapperrsJ  --I@I=D'5(J A #dCMM>??+X66 7 7F YY88888 9 9F yy8888899H#d?@2Y>???G$6s|"*,,,,G BIIgii" d8D;DL TNNN Kr!rrrrrrr) rHNFNNNNNNrINFT)rIrr):rknumpyr% scipy.linalgrrrrrrr r r scipy.sparse.linalgr scipy.sparser rscipy._lib._utilr r _linesearchrr__all__ Exceptionrr*r1r9r@rstriprErGrrir_rrrarrrrrrrrrrrrrrrrrr r!r"rs ????????????????$$$$$$$$$$''''''FFFFFFCCCCCCCC 9 9 9     I         T (P a111 h/// ?DKO?C48P"P"P"P"f FJ!****Z9<9<9<9<9<9<9<9<@AAAAAAAAH########&Y?Y?Y?@X4AAAAAAAAH * + 0ooooo>oood99999L999@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!