ʽXc@`s\dZddlmZmZmZddlZddlZddlm Z m Z m Z ddl m Z mZmZmZmZmZddlmZmZmZddlZddlZddl mZddlZddlmZd d lmZm Z d d d ddddgZ!de"fdYZ#dZ$dZ%dZ&dZ'e(ddj)ddj)Z*dZ+dde-dddddddde-e.d Z/e+e/dd d!d"Z0d#e1fd$YZ2d%e1fd&YZ3d'e1fd(YZ4d)Z5d*e3fd+YZ6d,e1fd-YZ7d.j)e*d/d>Z?e?d e8Z@e?d e9ZAe?d e:ZBe?de<ZCe?de;ZDe?de=ZEe?de>ZFdS(?s Nonlinear solvers ----------------- .. currentmodule:: scipy.optimize This is a collection of general-purpose nonlinear multidimensional solvers. These solvers find *x* for which *F(x) = 0*. Both *x* and *F* can be multidimensional. Routines ~~~~~~~~ Large-scale nonlinear solvers: .. autosummary:: newton_krylov anderson General nonlinear solvers: .. autosummary:: broyden1 broyden2 Simple iterations: .. autosummary:: excitingmixing linearmixing diagbroyden Examples ~~~~~~~~ **Small problem** >>> def F(x): ... return np.cos(x) + x[::-1] - [1, 2, 3, 4] >>> import scipy.optimize >>> x = scipy.optimize.broyden1(F, [1,1,1,1], f_tol=1e-14) >>> x array([ 4.04674914, 3.91158389, 2.71791677, 1.61756251]) >>> np.cos(x) + x[::-1] array([ 1., 2., 3., 4.]) **Large problem** Suppose that we needed to solve the following integrodifferential equation on the square :math:`[0,1]\times[0,1]`: .. math:: \nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2 with :math:`P(x,1) = 1` and :math:`P=0` elsewhere on the boundary of the square. The solution can be found using the `newton_krylov` solver: .. plot:: import numpy as np from scipy.optimize import newton_krylov from numpy import cosh, zeros_like, mgrid, zeros # parameters nx, ny = 75, 75 hx, hy = 1./(nx-1), 1./(ny-1) P_left, P_right = 0, 0 P_top, P_bottom = 1, 0 def residual(P): d2x = zeros_like(P) d2y = zeros_like(P) d2x[1:-1] = (P[2:] - 2*P[1:-1] + P[:-2]) / hx/hx d2x[0] = (P[1] - 2*P[0] + P_left)/hx/hx d2x[-1] = (P_right - 2*P[-1] + P[-2])/hx/hx d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy d2y[:,0] = (P[:,1] - 2*P[:,0] + P_bottom)/hy/hy d2y[:,-1] = (P_top - 2*P[:,-1] + P[:,-2])/hy/hy return d2x + d2y - 10*cosh(P).mean()**2 # solve guess = zeros((nx, ny), float) sol = newton_krylov(residual, guess, method='lgmres', verbose=1) print('Residual: %g' % abs(residual(sol)).max()) # visualize import matplotlib.pyplot as plt x, y = mgrid[0:1:(nx*1j), 0:1:(ny*1j)] plt.pcolor(x, y, sol) plt.colorbar() plt.show() i(tdivisiontprint_functiontabsolute_importN(tcallabletexec_txrange(tnormtsolvetinvtqrtsvdt LinAlgError(tasarraytdottvdot(tget_blas_funcs(tgetargspec_no_selfi(tscalar_search_wolfe1tscalar_search_armijotbroyden1tbroyden2tandersont linearmixingt diagbroydentexcitingmixingt newton_krylovt NoConvergencecB`seZRS((t__name__t __module__(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`stj|jS(N(tnptabsolutetmax(tx((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pytmaxnormscC`s;t|}tj|jtjs7t|dtjS|S(s:Return `x` as an array, of either floats or complex floatstdtype(R Rt issubdtypeR"tinexacttfloat_(R ((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt _as_inexacts cC`s:tj|tj|}t|d|j}||S(s;Return ndarray `x` as same array subclass and shape as `x0`t__array_wrap__(RtreshapetshapetgetattrR'(R tx0twrap((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt _array_likescC`s/tj|js%tjtjSt|S(N(RtisfinitetalltarraytinfR(tv((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt _safe_normst params_basics F : function(x) -> f Function whose root to find; should take and return an array-like object. x0 : array_like Initial guess for the solution t params_extras 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. cC`s |jr|jt|_ndS(N(t__doc__t _doc_parts(tobj((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt_set_docs tkrylovtarmijoc `sdtd|d|d|d| d|d| }tfd}j}tj}||}t|}t|}|j|j|||dkr|dk r|d}qd |j d}n| t krd } n| t krd} n| dkrt d nd }d}d}d}xt|D]}|j|||}|rePnt|||}|j|d| }t|dkrt dn| rt||||| \}}}}n(d}||}||}t|}|j|j|| r0| ||n||d|d}||d|krlt||}n t|t|||d}|}|r@tjjd|t|||ftjjq@q@W|rtt|nd}| rSi|jd6|d6|d6|dkd6idd6dd6|d6}t||fSt|SdS( s 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) http://www.siam.org/books/kelley/ tf_toltf_rtoltx_toltx_rtoltiterRc`stt|jS(N(R&R-tflatten(tz(tFR+(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pytsiidR;twolfesInvalid line searchg?gH.?g?gMbP?ttolis[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.g?is"%d: |F(x)| = %g; step %g; tol %g tnittfuntstatustsuccesss0A solution was found at the specified tolerance.s:The maximum number of iterations allowed has been reached.tmessageN(NR;RE(tTerminationConditionR&RARR1Rt asjacobiantsetuptcopytNonetsizetTruetFalset ValueErrorRtchecktminRt_nonlin_line_searchtupdateRtsyststdouttwritetflushRR-t iteration( RCR+tjacobianR@tverbosetmaxiterR<R=R>R?ttol_normt line_searchtcallbackt full_outputtraise_exceptiont conditiontfuncR tdxtFxtFx_normtgammateta_maxt eta_tresholdtetatnRIRFtst Fx_norm_newteta_Atinfo((RCR+s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt nonlin_solves,                        g:0yE>g{Gz?c `sIdg|gt|dgtttfdfd}|dkrt|dddd|\}} } n6|d krtdd d|\}} n|dkrd }n||dkr!d}n }t|} ||| fS( Niic`sm|dkrdS|}|}t|d}|ri|d<|d<|dR?R<R=R!RR@tf0_normR](tselfR<R=R>R?R@R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt__init__s$                cC`s|jd7_|j|}|j|}|j|}|jdkrW||_n|dkrgdS|jdk rd|j|jkSt||jko||j|jko||jko||j |kS(Niii( R]RRRPR@tintR<R=R>R?(RtfR Rhtf_normtx_normtdx_norm((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRUs  N(RRR6RPR!RRU(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRLs  tJacobiancB`s>eZdZdZdZddZdZdZRS(s Common interface for Jacobians or Jacobian approximations. The optional methods come useful when implementing trust region etc. algorithms that often require evaluating transposes of the Jacobian. Methods ------- solve Returns J^-1 * v update Updates Jacobian to point `x` (where the function has residual `Fx`) matvec : optional Returns J * v rmatvec : optional Returns A^H * v rsolve : optional Returns A^-H * v matmat : optional Returns A * V, where V is a dense matrix with dimensions (N,K). todense : optional Form the dense Jacobian matrix. Necessary for dense trust region algorithms, and useful for testing. Attributes ---------- shape Matrix dimensions (M, N) dtype Data type of the matrix. func : callable, optional Function the Jacobian corresponds to c `sddddddddd g }x\|jD]N\}}||krYtd |n|dk r.t|||q.q.Wtdrfd _ndS( NRRXtmatvectrmatvectrsolvetmatmatttodenseR)R"sUnknown keyword argument %sc`s jS(N(R((R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDs(titemsRTRPtsetattrthasattrt __array__(Rtkwtnamestnametvalue((Rs4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs  cC`s t|S(N(tInverseJacobian(R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pytaspreconditionersicC`s tdS(N(tNotImplementedError(RR2RF((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`sdS(N((RR RC((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRXscC`sY||_|j|jf|_|j|_|jjtjkrU|j|||ndS(N(RgRQR)R"t __class__RNRRX(RR RCRg((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRNs   (RRR6RRRRXRN(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs $   RcB`s/eZdZedZedZRS(cC`sa||_|j|_|j|_t|dr?|j|_nt|dr]|j|_ndS(NRNR(R^RRRXRRNRR(RR^((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR&s   cC`s |jjS(N(R^R)(R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR)/scC`s |jjS(N(R^R"(R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR"3s(RRRtpropertyR)R"(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR%s c`stjjjttr"StjrGttrGStt j r j dkrwt dnt j t jjdjdkrt dntdfddfd d fd d fd djdjStjjrjdjdkrHt dntdfddfdd fdd fddjdjStdr=tdr=td r=tdtddtdd jd td dtddtddjdjStrodtffdY}|SttrtdtdtdtdtdtdtdtStd d!S("sE Convert given object to one suitable for use as a Jacobian. isarray must have rank <= 2iisarray must be squareRc`s t|S(N(R (R2(tJ(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDHsRc`stjj|S(N(R tconjtT(R2(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDIsRc`s t|S(N(R(R2(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDJsRc`stjj|S(N(RRR(R2(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDKsR"R)smatrix must be squarec`s|S(N((R2(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDPsc`sjj|S(N(RR(R2(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDQsc`s |S(N((R2(Rtspsolve(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDRsc`sjj|S(N(RR(R2(RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDSsRXRNtJacc`sYeZdZdfdZfdZdfdZfdZRS(cS`s ||_dS(N(R (RR RC((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRXasic`s]|j}t|tjr.t||Stjj|rM||StddS(NsUnknown matrix type( R t isinstanceRtndarrayRtscipytsparset isspmatrixRT(RR2RFtm(RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRds   c`sX|j}t|tjr.t||Stjj|rH||StddS(NsUnknown matrix type( R RRRR RRRRT(RR2R(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRms  c`so|j}t|tjr7t|jj|Stjj |r_|jj|St ddS(NsUnknown matrix type( R RRRRRRRRRRT(RR2RFR(RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRvs c`sj|j}t|tjr7t|jj|Stjj |rZ|jj|St ddS(NsUnknown matrix type( R RRRR RRRRRRT(RR2R(R(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs (RRRXRRRR((RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR`s     RRRRRRR:s#Cannot convert object to a JacobianN( RRtlinalgRRRtinspecttisclasst issubclassRRtndimRTt atleast_2dR R)R"RRR*RRtstrtdictt BroydenFirstt BroydenSecondtAndersont DiagBroydent LinearMixingtExcitingMixingtKrylovJacobiant TypeError(RR((RRs4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRM8sZ-    ' tGenericBroydencB`s#eZdZdZdZRS(cC`stj||||||_||_t|dr|jdkrt|}|r{dtt|d||_qd|_ndS(Ntalphag?ig?( RRNtlast_ftlast_xRRRPRR(RR+tf0Rgtnormf0((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRNs   #cC`s tdS(N(R(RR RRhtdfRtdf_norm((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt_updatescC`sX||j}||j}|j||||t|t|||_||_dS(N(RRRR(RR RRRh((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRXs   ( (RRRNRRX(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs  t LowRankMatrixcB`seZdZdZedZedZdZdZddZ ddZ d Z d Z d Z d Zd ZddZRS(s 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. cC`s:||_g|_g|_||_||_d|_dS(N(RtcsR}RoR"RPt collapsed(RRRoR"((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs      c C`stdddg|d |g\}}}||}xDt||D]3\}} || |} ||||j| }qFW|S(Ntaxpytscaltdotci(RtzipRQ( R2RRR}RRRtwtctdta((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt_matvecs cC`sst|dkr||Stddg|d |g\}}|d}|tjt|d|j}xVt|D]H\}} x9t|D]+\} } ||| fc|| | 7|jdk rt|j|Stj||j|j|jS(sEvaluate w = M^-1 vN(RRPRRRRRR}(RR2RF((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`sP|jdk r(t|jjj|Stj|tj|j|j |j S(sEvaluate w = M^-H vN( RRPRRRRRRRR}R(RR2RF((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`s|jdk rL|j|dddf|dddfj7_dS|jj||jj|t|j|jkr|jndS(N( RRPRRtappendR}RRQtcollapse(RRR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs9cC`s|jdk r|jS|jtj|jd|j}xWt|j|j D]@\}}||dddf|dddfj 7}qNW|S(NR"( RRPRRRRoR"RRR}R(RtGmRR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs ""8cC`s1tj||_d|_d|_d|_dS(s0Collapse the low-rank matrix to a full-rank one.N(RR0RRPRR}R(R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs  cC`sO|jdk rdS|dks%tt|j|krK|j2|j2ndS(sH Reduce the rank of the matrix by dropping all vectors. Ni(RRPtAssertionErrorRRR}(Rtrank((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pytrestart_reduces cC`sY|jdk rdS|dks%tx-t|j|krT|jd=|jd=q(WdS(sK Reduce the rank of the matrix by dropping oldest vectors. Ni(RRPRRRR}(RR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt simple_reduce)s  c C`s|jdk rdS|}|dk r.|}n |d}|jr`t|t|jd}ntdt||d}t|j}||krdStj|jj}tj|j j}t |dd\}}t ||jj }t |dtdt\} } } t |t| }t || jj }xZt|D]L} |dd| fj|j| <|dd| fj|j | 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). http://www.math.leidenuniv.nl/scripties/Rotten.pdf Niiitmodeteconomict full_matricest compute_uv(RRPRRVRRRR0RR}R R RR RSRRRRRO( Rtmax_rankt to_retainRwRRtCtDtRtUtStWHtk((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyt svd_reduce4s0     !#' N(RRR6Rt staticmethodRRRRRRRRRRRRPR(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs         s 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 (ie., no rank reduction). tbroyden_paramsRcB`seeZdZd dd dZdZdZddZdZddZ d Z d Z RS( sL 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 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). http://www.math.leidenuniv.nl/scripties/Rotten.pdf trestartc`stj|_d_|dkr7tj}n|_t|t rXd n|d|d}|df|dkrfd_ nX|dkrfd_ n4|dkrfd_ nt d |dS( NiiR c`sjjS(N(RR((t reduce_paramsR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDstsimplec`sjjS(N(RR((RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDsRc`sjjS(N(RR((RR(s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRDss"Unknown rank reduction method '%s'(( RRRRPRRR1RRRt_reduceRT(RRtreduction_methodR((RRs4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs&            cC`s=tj||||t|j |jd|j|_dS(Ni(RRNRRR)R"R(RR RCRg((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRNscC`s t|jS(N(RR(R((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRsicC`sV|jj|}tj|jsF|j|j|j|jn|jj|S(N( RRRR.R/RNRRRg(RRRFtr((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`s|jj|S(N(RR(RR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`s|jj|S(N(RR(RRRF((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRscC`s|jj|S(N(RR(RR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRsc C`s\|j|jj|}||jj|}|t||} |jj|| dS(N(RRRRRR( RR RRhRRRR2RR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs  N( RRR6RPRRNRRRRRR(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRs       RcB`seZdZdZRS(s 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 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). http://www.math.leidenuniv.nl/scripties/Rotten.pdf c C`sK|j|}||jj|}||d} |jj|| dS(Ni(RRRR( RR RRhRRRR2RR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyR s  (RRR6R(((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRsRcB`s>eZdZddddZddZdZdZRS( s 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 References ---------- .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996). g{Gz?icC`sGtj|||_||_g|_g|_d|_||_dS(N( RRRtMRhRRPRktw0(RRRR((s4/tmp/pip-build-7oUkmx/scipy/scipy/optimize/nonlin.pyRIs      ic C`s|j |}t|j}|dkr-|Stj|d|j}x.t|D] }t|j||||>> from scipy.optimize.nonlin import BroydenFirst, KrylovJacobian >>> from scipy.optimize.nonlin 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. inner_tol, inner_maxiter, ... Parameters to pass on to the \"inner\" Krylov solver. See `scipy.sparse.linalg.gmres` for details. outer_k : int, optional Size of the subspace kept across LGMRES nonlinear iterations. See `scipy.sparse.linalg.lgmres` for details. %(params_extra)s See Also -------- 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] D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). :doi:`10.1016/j.jcp.2003.08.010` .. [2] A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). :doi:`10.1137/S0895479803422014` tlgmresii c K`st||_||_tdtjjjdtjjjdtjjjdtjjj dtjjj j |||_ td|d|j|_ |j tjjjkr||j djsh  .         )     ,CD `],/(: (