3 6^ @sPdZddlmZmZmZddlZddlZddlm Z m Z m Z ddl m Z mZmZmZmZmZddlmZmZmZddlZddlZddl mZddlZddlmZd d lmZm Z d d d ddddgZ!Gddde"Z#ddZ$ddZ%ddZ&ddZ'e(dj)dj)dZ*dd Z+dId%d&Z,e+e,dJd)d*Z-Gd+d,d,e.Z/Gd-d.d.e.Z0Gd/d0d0e.Z1d1d2Z2Gd3d4d4e0Z3Gd5d6d6e.Z4d7j)e*d8<Gd9d:d:e3Z5Gd;d<dd>e3Z7Gd?d@d@e3Z8GdAdBdBe3Z9GdCdDdDe3Z:GdEdFdFe0Z;dGdHZe>> 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() )divisionprint_functionabsolute_importN)callableexec_xrange)normsolveinvqrsvd LinAlgError)asarraydotvdot)get_blas_funcs)getargspec_no_self)scalar_search_wolfe1scalar_search_armijobroyden1broyden2anderson linearmixing diagbroydenexcitingmixing newton_krylovc@s eZdZdS) NoConvergenceN)__name__ __module__ __qualname__r!r!6/tmp/pip-build-vw4w4j08/scipy/scipy/optimize/nonlin.pyrsrcCstj|jS)N)npZabsolutemax)xr!r!r"maxnormsr&cCs*t|}tj|jtjs&t|tjdS|S)z:Return `x` as an array, of either floats or complex floats)dtype)rr#Z issubdtyper'Zinexactfloat_)r%r!r!r" _as_inexactsr)cCs(tj|tj|}t|d|j}||S)z;Return ndarray `x` as same array subclass and shape as `x0`__array_wrap__)r#Zreshapeshapegetattrr*)r%x0wrapr!r!r" _array_likesr/cCs"tj|jstjtjSt|S)N)r#isfiniteallarrayinfr)vr!r!r" _safe_norms r5z 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. )Z params_basicZ params_extracCs|jr|jt|_dS)N)__doc__ _doc_parts)objr!r!r"_set_docsr9krylovFarmijoTc sb| dkr tn| } t|||| || d}tfdd}j}tj}||}t|}t|}|j|j |||dkr|dk r|d}nd|j d}| dkrd} n | d krd} | dkrt d d }d }d}d}xHt |D] }|j |||}|rPt|||}|j||d }t|dkr0t d| rPt||||| \}}}}nd}||}||}t|}|j|j || r| ||||d|d}||d|krt||}nt|t|||d}|}|rtjjd|| ||ftjjqW|rtt|nd}| rT|j|||dkddd|d}t||fSt|SdS)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_rtoliterrcstt|jS)N)r)r/flatten)z)Fr-r!r"sznonlin_solve..rdTr;FwolfezInvalid line searchg?gH.?g?gMbP?)tolrz[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.g?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.)rrH)ZnitZfunstatussuccessmessage)Nr;rF)r&TerminationConditionr)rAr#r3r asjacobiansetupcopysize ValueErrorrcheckminr _nonlin_line_searchupdater$sysstdoutwriteflushrr/ iteration) rCr-jacobianr@verbosemaxiterr<r=r>r?Ztol_normZ line_searchcallbackZ full_outputZraise_exception conditionfuncr%dxFxFx_normgammaZeta_maxZ eta_tresholdetanrIrGsZ Fx_norm_newZeta_Ainfor!)rCr-r" nonlin_solves-      ri:0yE>{Gz?c sdg|gt|dgttdfdd fdd}|dkrxt|dd |d \}} } n&|d krtdd |d \}} |dkrd }||dkr̈d}n}t|} ||| fS)NrrHTcsT|dkrdS|}|}t|d}|rP|d<|d<|d<|S)NrrH)r5)rgstoreZxtr4p)rar`tmp_Fxtmp_phitmp_sr%r!r"phi|s   z _nonlin_line_search..phics0t|d}||dd||S)NrF)rl)abs)rgds)rqrdiffs_normr!r"derphisz#_nonlin_line_search..derphirFg{Gz?)Zxtolaminr;)rwg?)T)rrr) r`r%rbraZ search_typertZsminrvrgZphi1Zphi0rcr!) rar`rqrtrurnrorpr%r"rTus(      rTc@s.eZdZdZdddddefddZddZdS)rLz 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 NcCsx|dkrtjtjjd}|dkr(tj}|dkr6tj}|dkrDtj}||_||_||_||_||_ ||_ d|_ d|_ dS)Ng?rgUUUUUU?) r#finfor(epsr3r>r?r<r=rr@f0_normrZ)selfr<r=r>r?r@rr!r!r"__init__s zTerminationCondition.__init__cCs|jd7_|j|}|j|}|j|}|jdkr<||_|dkrHdS|jdk rbd|j|jkSt||jko||j|jko||jko||j|kS)NrrrH) rZrr{r@intr<r=r>r?)r|fr%raZf_normZx_normdx_normr!r!r"rRs       zTerminationCondition.check)rrr r6r&r}rRr!r!r!r"rLs rLc@s:eZdZdZddZddZdddZd d Zd d Zd S)Jacobiana 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 stddddddddd g }x@|jD]4\}}||kr.)itemsrQsetattrhasattr __array__)r|kwnamesnamevaluer!)r|r"r}s    zJacobian.__init__cCst|S)N)InverseJacobian)r|r!r!r"aspreconditionerszJacobian.aspreconditionerrcCstdS)N)NotImplementedError)r|r4rGr!r!r"r szJacobian.solvecCsdS)Nr!)r|r%rCr!r!r"rUszJacobian.updatecCs:||_|j|jf|_|j|_|jjtjkr6|j||dS)N)r`rPr+r' __class__rNrrU)r|r%rCr`r!r!r"rNs zJacobian.setupN)r) rrr r6r}rr rUrNr!r!r!r"rs $  rc@s,eZdZddZeddZeddZdS)rcCs>||_|j|_|j|_t|dr(|j|_t|dr:|j|_dS)NrNr)r[r rrUrrNrr)r|r[r!r!r"r}'s  zInverseJacobian.__init__cCs|jjS)N)r[r+)r|r!r!r"r+0szInverseJacobian.shapecCs|jjS)N)r[r')r|r!r!r"r'4szInverseJacobian.dtypeN)rrr r}propertyr+r'r!r!r!r"r&s rc stjjjttrStjr2ttr2Stt j rj dkrPt dt j t jjdjdkr|t dtfddfddfd dfd djjd Stjjrjdjdkrt d tfd dfddfddfddjjd StdrztdrztdrzttdtdjtdtdtdjjdStrGfdddt}|SttrttttttttdStddS)zE Convert given object to one suitable for use as a Jacobian. rHzarray must have rank <= 2rrzarray must be squarecs t|S)N)r)r4)Jr!r"rDIszasjacobian..cstjj|S)N)rconjT)r4)rr!r"rDJscs t|S)N)r )r4)rr!r"rDKscstjj|S)N)r rr)r4)rr!r"rDLs)rrr rr'r+zmatrix must be squarecs|S)Nr!)r4)rr!r"rDQscsjj|S)N)rr)r4)rr!r"rDRscs |S)Nr!)r4)rspsolver!r"rDSscsjj|S)N)rr)r4)rrr!r"rDTsr+r'r rrrrUrN)rrr rrUrNr'r+csLeZdZddZd fdd ZfddZdfdd Zfd d Zd S)zasjacobian..JaccSs ||_dS)N)r%)r|r%rCr!r!r"rUbszasjacobian..Jac.updatercsB|j}t|tjr t||Stjj|r6||StddS)NzUnknown matrix type) r% isinstancer#ndarrayr scipysparse isspmatrixrQ)r|r4rGm)rrr!r"r es      zasjacobian..Jac.solvecs@|j}t|tjr t||Stjj|r4||StddS)NzUnknown matrix type) r%rr#rrrrrrQ)r|r4r)rr!r"rns     zasjacobian..Jac.matveccsN|j}t|tjr&t|jj|Stjj |rB|jj|St ddS)NzUnknown matrix type) r%rr#rr rrrrrrQ)r|r4rGr)rrr!r"rws    zasjacobian..Jac.rsolvecsL|j}t|tjr&t|jj|Stjj |r@|jj|St ddS)NzUnknown matrix type) r%rr#rrrrrrrrQ)r|r4r)rr!r"rs    zasjacobian..Jac.rmatvecN)r)r)rrr rUr rrrr!)rrr!r"Jacas   r)rrrrrrr:z#Cannot convert object to a JacobianN) rrlinalgrrrinspectZisclass issubclassr#rndimrQZ atleast_2drr+r'rrr,r rstrdict BroydenFirst BroydenSecondAnderson DiagBroyden LinearMixingExcitingMixingKrylovJacobian TypeError)rrr!)rrr"rM9sZ            $   ' rMc@s$eZdZddZddZddZdS)GenericBroydencCs`tj||||||_||_t|dr\|jdkr\t|}|rVdtt|d||_nd|_dS)Nalphag?rg?)rrNlast_flast_xrrrr$)r|r-f0r`Znormf0r!r!r"rNszGenericBroyden.setupcCstdS)N)r)r|r%rradfrdf_normr!r!r"_updateszGenericBroyden._updatecCs@||j}||j}|j||||t|t|||_||_dS)N)rrrr)r|r%rrrar!r!r"rUs   zGenericBroyden.updateN)rrr rNrrUr!r!r!r"rsrc@seZdZdZddZeddZeddZdd Zd d Z dd dZ dddZ ddZ ddZ ddZddZddZd ddZdS)! 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. cCs(||_g|_g|_||_||_d|_dS)N)rcsrsrfr' collapsed)r|rrfr'r!r!r"r}s zLowRankMatrix.__init__c Csbtdddg|dd|g\}}}||}x0t||D]"\}} || |} ||||j| }q8W|S)Naxpyscaldotcr)rziprP) r4rrrsrrrwcdar!r!r"_matvecs  zLowRankMatrix._matveccCst|dkr||Stddg|dd|g\}}|d}|tjt||jd}xDt|D]8\}} x.t|D]"\} } ||| f|| | 7<qpWq^Wtjt||jd} x"t|D]\} } || || | <qW| |} t|| } ||} x(t|| D]\} }|| | | j | } qW| S)zEvaluate w = M^-1 vrrrNr)r') lenrr#identityr' enumeratezerosr rrP)r4rrrsrrZc0AirjrqrZqcr!r!r"_solves"  " zLowRankMatrix._solvecCs.|jdk rtj|j|Stj||j|j|jS)zEvaluate w = M vN)rr#rrrrrrs)r|r4r!r!r"rs zLowRankMatrix.matveccCs:|jdk rtj|jjj|Stj|tj|j|j|j S)zEvaluate w = M^H vN) rr#rrrrrrrsr)r|r4r!r!r"rs zLowRankMatrix.rmatvecrcCs,|jdk rt|j|Stj||j|j|jS)zEvaluate w = M^-1 vN)rr rrrrrs)r|r4rGr!r!r"r s  zLowRankMatrix.solvecCs8|jdk rt|jjj|Stj|tj|j|j|j S)zEvaluate w = M^-H vN) rr rrrrr#rrsr)r|r4rGr!r!r"rs zLowRankMatrix.rsolvecCsp|jdk r<|j|dddf|dddfj7_dS|jj||jj|t|j|jkrl|jdS)N)rrrappendrsrrPcollapse)r|rrr!r!r"rs .  zLowRankMatrix.appendcCsp|jdk r|jS|jtj|j|jd}xBt|j|jD]0\}}||dddf|dddfj 7}q8W|S)N)r') rrr#rrfr'rrrsr)r|Gmrrr!r!r"rs  ,zLowRankMatrix.__array__cCs"tj||_d|_d|_d|_dS)z0Collapse the low-rank matrix to a full-rank one.N)r#r2rrrsr)r|r!r!r"rs zLowRankMatrix.collapsecCsD|jdk rdS|dkstt|j|kr@|jdd=|jdd=dS)zH Reduce the rank of the matrix by dropping all vectors. Nr)rAssertionErrorrrrs)r|rankr!r!r"restart_reduces    zLowRankMatrix.restart_reducecCsB|jdk rdS|dkstx"t|j|kr<|jd=|jd=qWdS)zK Reduce the rank of the matrix by dropping oldest vectors. Nr)rrrrrs)r|rr!r!r" simple_reduce*s   zLowRankMatrix.simple_reduceNc Cs<|jdk rdS|}|dk r |}n|d}|jrBt|t|jd}tdt||d}t|j}||krldStj|jj}tj|jj}t |dd\}}t ||jj }t |ddd \} } } t |t | }t || jj }xDt|D]8} |dd| fj|j| <|dd| fj|j| <qW|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 NrHrrZeconomic)modeFT)Z full_matricesZ compute_uv)rrrSrr$r#r2rrsr rrr r rrO) r|max_rankZ to_retainrmrrCDRUSZWHkr!r!r" svd_reduce5s0   zLowRankMatrix.svd_reduce)r)r)N)rrr r6r} staticmethodrrrrr rrrrrrrr!r!r!r"rs        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 (ie., no rank reduction). Zbroyden_paramsc@sVeZdZdZdddZddZdd Zdd d Zd dZdddZ ddZ ddZ 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 Nrestartcstj|_d_|dkr$tj}|_t|tr:fn|dd|d}|df|dkrvfdd_ n@|dkrfdd_ n&|dkrfd d_ n t d |dS) Nrrr cs jjS)N)rrr!) reduce_paramsr|r!r"rDsz'BroydenFirst.__init__..simplecs jjS)N)rrr!)rr|r!r"rDsrcs jjS)N)rrr!)rr|r!r"rDsz"Unknown rank reduction method '%s') rr}rrr#r3rrr_reducerQ)r|rZreduction_methodrr!)rr|r"r}s&   zBroydenFirst.__init__cCs.tj||||t|j |jd|j|_dS)Nr)rrNrrr+r'r)r|r%rCr`r!r!r"rNszBroydenFirst.setupcCs t|jS)N)r r)r|r!r!r"rszBroydenFirst.todensercCs:|jj|}tj|js.|j|j|j|j|jj|S)N) rrr#r0r1rNrrr`)r|rrGrr!r!r"r s zBroydenFirst.solvecCs |jj|S)N)rr )r|rr!r!r"rszBroydenFirst.matveccCs |jj|S)N)rr)r|rrGr!r!r"rszBroydenFirst.rsolvecCs |jj|S)N)rr)r|rr!r!r"rszBroydenFirst.rmatvecc CsD|j|jj|}||jj|}|t||} |jj|| dS)N)rrrrrr) r|r%rrarrrr4rrr!r!r"rs  zBroydenFirst._update)NrN)r)r) rrr r6r}rNrr rrrrr!r!r!r"rs%   rc@seZdZdZddZdS)ra 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 c Cs:|j|}||jj|}||d} |jj|| dS)NrH)rrrr) r|r%rrarrrr4rrr!r!r"rs  zBroydenSecond._updateN)rrr r6rr!r!r!r"rs"rc@s4eZdZdZdddZddd Zd d Zd d ZdS)rav 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). N{Gz?cCs2tj|||_||_g|_g|_d|_||_dS)N)rr}rMrarrdw0)r|rrrr!r!r"r}Zs zAnderson.__init__rc Cs|j |}t|j}|dkr"|Stj||jd}x$t|D]}t|j||||<q| |j}t|j}|dkr"|Stj||jd}x$t|D]}t|j||||<q ,zAnderson.matvecc Cs|jdkrdS|jj||jj|x,t|j|jkrR|jjd|jjdq(Wt|j}tj||f|jd}xbt |D]V} xPt | |D]B} | | kr|j d} nd} d| t |j| |j| || | f<qWq|W|tj |dj j7}||_dS)Nr)r'rHr)rrarrrpopr#rr'rrrZtriurrr) r|r%rrarrrrfrrrwdr!r!r"rs"      .zAnderson._update)Nrr)r)rrr r6r}r rrr!r!r!r"r"s  rc@sVeZdZdZdddZddZddd Zd d Zdd d ZddZ ddZ ddZ 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. NcCstj|||_dS)N)rr}r)r|rr!r!r"r}s zDiagBroyden.__init__cCs6tj||||tj|jdfd|j|jd|_dS)Nrr)r')rrNr#fullr+rr'r)r|r%rCr`r!r!r"rNszDiagBroyden.setuprcCs | |jS)N)r)r|rrGr!r!r"r szDiagBroyden.solvecCs | |jS)N)r)r|rr!r!r"rszDiagBroyden.matveccCs| |jjS)N)rr)r|rrGr!r!r"rszDiagBroyden.rsolvecCs| |jjS)N)rr)r|rr!r!r"rszDiagBroyden.rmatveccCstj|j S)N)r#diagr)r|r!r!r"rszDiagBroyden.todensecCs(|j||j|||d8_dS)NrH)r)r|r%rrarrrr!r!r"rszDiagBroyden._update)N)r)r) rrr r6r}rNr rrrrrr!r!r!r"rs   rc@sNeZdZdZdddZdddZdd Zdd d Zd d ZddZ ddZ 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. NcCstj|||_dS)N)rr}r)r|rr!r!r"r}s zLinearMixing.__init__rcCs | |jS)N)r)r|rrGr!r!r"r szLinearMixing.solvecCs | |jS)N)r)r|rr!r!r"rszLinearMixing.matveccCs| tj|jS)N)r#rr)r|rrGr!r!r"rszLinearMixing.rsolvecCs| tj|jS)N)r#rr)r|rr!r!r"rszLinearMixing.rmatveccCstjtj|jdd|jS)Nrr)r#rrr+r)r|r!r!r"r szLinearMixing.todensecCsdS)Nr!)r|r%rrarrrr!r!r"r szLinearMixing._update)N)r)r) rrr r6r}r rrrrrr!r!r!r"rs   rc@sVeZdZdZdddZddZdd d Zd d Zdd dZddZ ddZ ddZ 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 N?cCs tj|||_||_d|_dS)N)rr}ralphamaxbeta)r|rrr!r!r"r},s zExcitingMixing.__init__cCs2tj||||tj|jdf|j|jd|_dS)Nr)r')rrNr#rr+rr'r)r|r%rCr`r!r!r"rN2szExcitingMixing.setuprcCs | |jS)N)r)r|rrGr!r!r"r 6szExcitingMixing.solvecCs | |jS)N)r)r|rr!r!r"r9szExcitingMixing.matveccCs| |jjS)N)rr)r|rrGr!r!r"r<szExcitingMixing.rsolvecCs| |jjS)N)rr)r|rr!r!r"r?szExcitingMixing.rmatveccCstjd|jS)Nrr)r#rr)r|r!r!r"rBszExcitingMixing.todensecCsL||jdk}|j||j7<|j|j|<tj|jd|j|jddS)Nr)out)rrrr#Zclipr)r|r%rrarrrincrr!r!r"rEszExcitingMixing._update)Nr)r)r) rrr r6r}rNr rrrrrr!r!r!r"rs   rc@sDeZdZdZdddZdd Zd d Zdd dZddZddZ 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 : {'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres'} or function 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`. The default is `scipy.sparse.linalg.lgmres`. 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.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 -------- 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] 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` Nlgmres c KsN||_||_ttjjjtjjjtjjjtjjj tjjj dj |||_ t||jd|_ |j tjjjkr||j d<d|j d<|j jddn~|j tjjjkr|j jddn^|j tjjjkr||j d<d|j d<|j jd g|j jd d |j jd d |j jddx@|jD]4\}}|jds2td|||j |dd<qWdS)N)bicgstabgmresrcgsminres)r]rZrestrtrr]Zatolrouter_kZouter_vZprepend_outer_vTZstore_outer_AvFZinner_zUnknown parameter %s)preconditionerrtrrrrrrrrrgetmethod method_kw setdefaultZgcrotmkr startswithrQ) r|rtrZ inner_maxiterZinner_Mrrkeyrr!r!r"r}s6       zKrylovJacobian.__init__cCs<t|jj}t|jj}|jtd|td||_dS)Nr)rrr-r$rrtomega)r|ZmxZmfr!r!r"_update_diff_stepsz KrylovJacobian._update_diff_stepcCsnt|}|dkrd|S|j|}|j|j|||j|}tjtj| rjtjtj|rjtd|S)Nrz$Function returned non-finite results) rrr`r-rr#r1r0rQ)r|r4nvZscrr!r!r"rs "zKrylovJacobian.matvecrcCsHd|jkr$|j|j|f|j\}}n |j|j|fd|i|j\}}|S)NrG)rrop)r|rhsrGZsolrhr!r!r"r s  zKrylovJacobian.solvecCs<||_||_|j|jdk r8t|jdr8|jj||dS)NrU)r-rrrrrU)r|r%rr!r!r"rUs   zKrylovJacobian.updatecCs|tj||||||_||_tjjj||_|j dkrJt j |j j d|_ |j|jdk rxt|jdrx|jj|||dS)Ng?rHrNg?)rrNr-rrrrZaslinearoperatorrrtr#ryr'rzrrr)r|r%rr`r!r!r"rNs   zKrylovJacobian.setup)NrrNr)r) rrr r6r}rrr rUrNr!r!r!r"rPsJ )  rc Cst|j\}}}}tt|t| d|}djdd|D}|rNd|}djdd|D}|rn|d}d} | t|||j|d} i} | jt t | | | |} |j | _ t | | S)a Construct a solver wrapper with given name and jacobian approx. It inspects the keyword arguments of ``jac.__init__``, and allows to use the same arguments in the wrapper function, in addition to the keyword arguments of `nonlin_solve` Nz, cSsg|]\}}d||fqS)z%s=%rr!).0rr4r!r!r" sz#_nonlin_wrapper..cSsg|]\}}d||fqS)z%s=%sr!)rrr4r!r!r"r sa def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw): jac = %(jac)s(%(kwkw)s **kw) return nonlin_solve(F, xin, jac, iter, verbose, maxiter, f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search, callback) )rrjacZkwkw) _getargspecr}listrrjoinrrrUglobalsrr6r9) rrargsZvarargsZvarkwdefaultskwargsZkw_strZkwkw_strwrappernsr`r!r!r"_nonlin_wrappers$      r) r:NFNNNNNNr;NFT)r;rjrk)Dr6 __future__rrrrVZnumpyr#Zscipy._lib.sixrrrZ scipy.linalgrr r r r r rrrZscipy.sparse.linalgrZ scipy.sparserrZscipy._lib._utilrrZ linesearchrr__all__ Exceptionrr&r)r/r5rstripr7r9rirTobjectrLrrrMrrrrrrrrrrrrrrrrrr!r!r!r"jsp     )   ,@D`Z b14.?,)