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nonlin.cpython-35.opt-1.pyc
==WWY� � @ sw d Z d d l m Z m Z m Z d d l Z d d l Z d d l m Z m Z m Z d d l m Z m Z m Z m Z m Z m Z d d l m Z m Z m Z d d l Z d d l Z d d l m Z d d l Z d d l m Z d d l m Z m Z d d d d d d d g Z! Gd d � d e" � Z# d d � Z$ d d � Z% d d � Z& d d � Z' e( d d j) � d d j) � � Z* d d! � Z+ d" d d# d d d d d d d$ d d# d% d& d' � Z, e+ e, � d$ d( d) d* d+ � Z- Gd, d- � d- e. � Z/ Gd. d/ � d/ e. � Z0 Gd0 d1 � d1 e. � Z1 d2 d3 � Z2 Gd4 d5 � d5 e0 � Z3 Gd6 d7 � d7 e. � Z4 d8 j) � e* d9 <Gd: d; � d; e3 � Z5 Gd< d= � d= e5 � Z6 Gd> d? � d? e3 � Z7 Gd@ dA � dA e3 � Z8 GdB dC � dC e3 � Z9 GdD dE � dE e3 � Z: GdF dG � dG e0 � Z; dH dI � Z< e< d e5 � Z= e< d e6 � Z> e< d e7 � Z? e< d e9 � Z@ e< d e8 � ZA e< d e: � ZB e< d e; � ZC d S)Ja� 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() � )�division�print_function�absolute_importN)�callable�exec_�xrange)�norm�solve�inv�qr�svd� LinAlgError)�asarray�dot�vdot)�get_blas_funcs)�getargspec_no_self� )�scalar_search_wolfe1�scalar_search_armijo�broyden1�broyden2�anderson� linearmixing� diagbroyden�excitingmixing� newton_krylovc @ s e Z d Z d S)� NoConvergenceN)�__name__� __module__� __qualname__� r! r! � /nonlin.pyr � s r c C s t j | � j � S)N)�npZabsolute�max)�xr! r! r" �maxnorm� s r& c C s; t | � } t j | j t j � s7 t | d t j �S| S)z:Return `x` as an array, of either floats or complex floats�dtype)r r# Z issubdtyper' Zinexact�float_)r% r! r! r" � _as_inexact� s r) c C s: t j | t j | � � } t | d | j � } | | � S)z;Return ndarray `x` as same array subclass and shape as `x0`�__array_wrap__)r# Zreshape�shape�getattrr* )r% �x0Zwrapr! r! r" � _array_like� s r. c C s/ t j | � j � s% t j t j � St | � S)N)r# �isfinite�all�array�infr )�vr! r! r" � _safe_norm� s r4 Z params_basicz� 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 Z params_extraa� 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. c C s | j r | j t | _ d S)N)�__doc__� _doc_parts)�objr! r! r" �_set_doc� s r8 �krylovF�armijoTc sK t d | d | d | d | d | d | � } t � � � � � f d d � } � j � } t j } | | � } t | � } t | � } | j | j � | | � | d k r� | d k r� | d } n d | j d } | d k r� d } n | d k r� d } | d" k rt d � � d } d } d } d } x�t | � D]�} | j | | | � } | r\Pt | | | � } | j | d | � } t | � d k r�t d � � | r�t | | | | | � \ } } } } n( d } | | } | | � } t | � } | j | j � | � | r!| | | � | | d | d } | | d | k r]t | | � } n t | t | | | d � � } | } | r:t j j d | t | � | | f � t j j � q:W| r�t t | � � � � n d } | r:d | j d | d | d | d k d d d d d! i | i } t | � � | f St | � � Sd 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) http://www.siam.org/books/kelley/ �f_tol�f_rtol�x_tol�x_rtol�iterr c s t � t | � � � � j � S)N)r) r. �flatten)�z)�Fr- r! r" �<lambda> s z nonlin_solve.<locals>.<lambda>Nr �d Tr: F�wolfezInvalid line searchg�������?g ��H.��?g�������?g����MbP?�tolr z[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.g �?� z"%d: |F(x)| = %g; step %g; tol %g ZnitZfun�status�success�messagez0A solution was found at the specified tolerance.z:The maximum number of iterations allowed has been reached.)Nr: rE )�TerminationConditionr) r@ r# r2 r � asjacobian�setup�copy�size� ValueErrorr �check�minr �_nonlin_line_search�updater$ �sys�stdout�write�flushr r. � iteration) rB r- �jacobianr? �verbose�maxiterr; r< r= r> Ztol_normZ line_search�callbackZ full_outputZraise_exceptionZ condition�funcr% �dx�Fx�Fx_norm�gammaZeta_maxZ eta_tresholdZeta�nrH rF �sZ Fx_norm_newZeta_A�infor! )rB r- r" � nonlin_solve� s� , rf g:�0�yE>g{�G�z�?c sI d g � | g � t | � d g � t � � t � � � d � � � � � � f d d � � � � � f d d � } | d k r� t � | � d d d d | �\ } } } n3 | d k r� t � � d � d d | �\ } } | d k r� d } � | � � | � d k r!� d } n � � � } t | � } | � | | f S)Nr rG Tc sj | � d k r � d S� | � } � | � } t | � d } | rf | � d <| � d <| � d <| S)Nr rG )r4 )rd �storeZxtr3 �p)r_ r^ �tmp_Fx�tmp_phi�tmp_sr% r! r" �phix s z _nonlin_line_search.<locals>.phic s: t | � � d � } � | | d d �� | � | S)Nr rg F)�abs)rd �ds)rl �rdiff�s_normr! r" �derphi� s z#_nonlin_line_search.<locals>.derphirE Zxtolg{�G�z�?Zaminr: g �?)r r r ) r^ r% r` r_ Z search_typero Zsminrq rd Zphi1Zphi0ra r! ) r_ r^ rl ro rp ri rj rk r% r" rS q s( $ rS c @ s@ e Z d Z d Z d d d d d e d d � Z d d � Z d S)rK 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 C s� | d k r% t j t j � j d } | d k r: t j } | d k rO t j } | d k rd t j } | | _ | | _ | | _ | | _ | d k r� t | _ n | | _ | | _ d | _ d | _ d S)Ng �?� r gUUUUUU�?)r# �finfor( �epsr2 r= r> r; r< r& r r? �f0_normrY )�selfr; r< r= r> r? r r! r! r" �__init__� s$ z TerminationCondition.__init__c C s� | j d 7_ | j | � } | j | � } | j | � } | j d k rT | | _ | d k rd d S| j d k r� d | j | j k St | | j k o� | | j | j k o� | | j k o� | | j | k � S)Nr r rG ) rY r ru r? �intr; r< r= r> )rv �fr% r_ Zf_normZx_norm�dx_normr! r! r" rQ � s zTerminationCondition.check)r r r r5 r& rw rQ r! r! r! r" rK � s rK c @ sU e Z d Z d Z d d � Z d d � Z d d d � Z d d � Z d d � Z d 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 s� d d d d d d d d d g } xV | j � D]H \ } } | | k rV t d | � � | d k r. t � | | | � q. Wt � d � r� � f d d � � _ d S) Nr rT �matvec�rmatvec�rsolveZmatmat�todenser+ r' zUnknown keyword argument %sc s � j � S)N)r r! )rv r! r" rC s z#Jacobian.__init__.<locals>.<lambda>)�itemsrP �setattr�hasattr� __array__)rv �kw�names�name�valuer! )rv r" rw s zJacobian.__init__c C s t | � S)N)�InverseJacobian)rv r! r! r" �aspreconditioner s zJacobian.aspreconditionerr c C s t � d S)N)�NotImplementedError)rv r3 rF r! r! r" r s zJacobian.solvec C s d S)Nr! )rv r% rB r! r! r" rT s zJacobian.updatec C sV | | _ | j | j f | _ | j | _ | j j t j k rR | j | | | � d S)N)r^ rO r+ r' � __class__rM r{ rT )rv r% rB r^ r! r! r" rM s zJacobian.setupN) r r r r5 rw r� r rT rM r! r! r! r" r{ � s $ r{ c @ s@ e Z d Z d d � Z e d d � � Z e d d � � Z d S)r� c C s[ | | _ | j | _ | j | _ t | d � r< | j | _ t | d � rW | j | _ d S)NrM r~ )rZ r r| rT r� rM r~ r} )rv rZ r! r! r" rw & s zInverseJacobian.__init__c C s | j j S)N)rZ r+ )rv r! r! r" r+ / s zInverseJacobian.shapec C s | j j S)N)rZ r' )rv r! r! r" r' 3 s zInverseJacobian.dtypeN)r r r rw �propertyr+ r' r! r! r! r" r� % s r� c s� t j j j � t � t � r" � St j � � rG t � t � rG � � St � t j � r� j d k rt t d � � t j t j � � � � � j d � j d k r� t d � � t d � f d d � d � f d d � d � f d d � d � f d d � d � j d � j � St j j � � r�� j d � j d k rKt d � � t d � f d d � d � f d d � d � � f d d � d � � f d d � d � j d � j � St � d � rLt � d � rLt � d � rLt d t � d � d t � d � d � j d t � d � d t � d � d t � d � d � j d � j � St � � r~G� � f d d � d t � } | � St � t � r�t d t d t d t d t d t d t d t � � � St d! � � d" S)#zE Convert given object to one suitable for use as a Jacobian. rG zarray must have rank <= 2r r zarray must be squarer| c s t � | � S)N)r )r3 )�Jr! r" rC H s z asjacobian.<locals>.<lambda>r} c s t � j � j | � S)N)r �conj�T)r3 )r� r! r" rC I s r c s t � | � S)N)r )r3 )r� r! r" rC J s r~ c s t � j � j | � S)N)r r� r� )r3 )r� r! r" rC K s r' r+ zmatrix must be squarec s � | S)Nr! )r3 )r� r! r" rC P s c s � j � j | S)N)r� r� )r3 )r� r! r" rC Q s c s � � | � S)Nr! )r3 )r� �spsolver! r" rC R s c s � � j � j | � S)N)r� r� )r3 )r� r� r! r" rC S s rT rM c sp e Z d Z d d � Z d � � f d d � Z � f d d � Z d � � f d d � Z � f d d � Z d S) zasjacobian.<locals>.Jacc S s | | _ d S)N)r% )rv r% rB r! r! r" rT a s z asjacobian.<locals>.Jac.updater c s] � | j � } t | t j � r. t | | � St j j | � rM � | | � St d � � d S)NzUnknown matrix type) r% � isinstancer# �ndarrayr �scipy�sparse� isspmatrixrP )rv r3 rF �m)r� r� r! r" r d s z asjacobian.<locals>.Jac.solvec sX � | j � } t | t j � r. t | | � St j j | � rH | | St d � � d S)NzUnknown matrix type) r% r� r# r� r r� r� r� rP )rv r3 r� )r� r! r" r| m s z asjacobian.<locals>.Jac.matvecc so � | j � } t | t j � r7 t | j � j | � St j j | � r_ � | j � j | � St d � � d S)NzUnknown matrix type) r% r� r# r� r r� r� r� r� r� rP )rv r3 rF r� )r� r� r! r" r~ v s z asjacobian.<locals>.Jac.rsolvec sj � | j � } t | t j � r7 t | j � j | � St j j | � rZ | j � j | St d � � d S)NzUnknown matrix type) r% r� r# r� r r� r� r� r� r� rP )rv r3 r� )r� r! r" r} s z asjacobian.<locals>.Jac.rmatvecN)r r r rT r r| r~ r} r! )r� r� r! r" �Jac` s r� r r r r r r r9 z#Cannot convert object to a JacobianN) r� r� �linalgr� r� r{ �inspectZisclass� issubclassr# r� �ndimrP Z atleast_2dr r+ r' r� r� r, r r �str�dict� BroydenFirst� BroydenSecond�Anderson� DiagBroyden� LinearMixing�ExcitingMixing�KrylovJacobian� TypeError)r� r� r! )r� r� r" rL 8 sZ - ' rL c @ s4 e Z d Z d d � Z d d � Z d d � Z d S)�GenericBroydenc C s� t j | | | | � | | _ | | _ t | d � r� | j d k r� t | � } | r{ d t t | � d � | | _ n d | _ d S)N�alphag �?r g �?)r{ rM �last_f�last_xr� r� r r$ )rv r- �f0r^ Znormf0r! r! r" rM � s #zGenericBroyden.setupc C s t � d S)N)r� )rv r% ry r_ �dfrz �df_normr! r! r" �_update� s zGenericBroyden._updatec C sX | | j } | | j } | j | | | | t | � t | � � | | _ | | _ d S)N)r� r� r� r )rv r% ry r� r_ r! r! r" rT � s ( zGenericBroyden.updateN)r r r rM r� rT r! r! r! r" r� � s r� c @ s� e Z d Z d Z d d � Z e d d � � Z e d d � � Z d d � Z d d � Z d d d � Z d d d � Z d d � Z d d � Z d d � Z d d � Z d d � Z d d d � Z d 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. c C s: | | _ g | _ g | _ | | _ | | _ d | _ d S)N)r� �csrn rc r' � collapsed)rv r� rc r' r! r! r" rw � s zLowRankMatrix.__init__c C s� t d d d g | d d � | g � \ } } } | | } xD t | | � D]3 \ } } | | | � } | | | | j | � } qL W| S)N�axpy�scal�dotcr )r �ziprO ) r3 r� r� rn r� r� r� �w�c�d�ar! r! r" �_matvec� s # zLowRankMatrix._matvecc C sw t | � d k r | | St d d g | d d � | g � \ } } | d } | t j t | � d | j �} xT t | � D]F \ } } x7 t | � D]) \ } } | | | f | | | � 7<q� Wq Wt j t | � d | j �} x- t | � D] \ } } | | | � | | <q� W| | } t | | � } | | } x6 t | | � D]% \ } } | | | | j | � } qJW| S)zEvaluate w = M^-1 vr r� r� Nr r' ) �lenr r# �identityr' � enumerate�zerosr r� rO )r3 r� r� rn r� r� Zc0�A�ir� �jr� �qr� Zqcr! r! r" �_solve� s" , "% zLowRankMatrix._solvec C sA | j d k r" t j | j | � St j | | j | j | j � S)zEvaluate w = M vN)r� r# r r� r� r� r� rn )rv r3 r! r! r" r| � s zLowRankMatrix.matvecc C sS | j d k r+ t j | j j j � | � St j | t j | j � | j | j � S)zEvaluate w = M^H vN) r� r# r r� r� r� r� r� rn r� )rv r3 r! r! r" r} � s zLowRankMatrix.rmatvecr c C s> | j d k r t | j | � St j | | j | j | j � S)zEvaluate w = M^-1 vN)r� r r� r� r� r� rn )rv r3 rF r! r! r" r � s zLowRankMatrix.solvec C sP | j d k r( t | j j j � | � St j | t j | j � | j | j � S)zEvaluate w = M^-H vN) r� r r� r� r� r� r# r� rn r� )rv r3 rF r! r! r" r~ � s zLowRankMatrix.rsolvec C s� | j d k rL | j | d d � d f | d d d � f j � 7_ d S| j j | � | j j | � t | j � | j k r� | j � d S)N)r� r� r� �appendrn r� rO �collapse)rv r� r� r! r! r" r� s 9zLowRankMatrix.appendc C s� | j d k r | j S| j t j | j d | j �} xW t | j | j � D]@ \ } } | | d d � d f | d d d � f j � 7} qN W| S)Nr' ) r� r� r# r� rc r' r� r� rn r� )rv �Gmr� r� r! r! r" r� s ""8zLowRankMatrix.__array__c C s1 t j | � | _ d | _ d | _ d | _ d S)z0Collapse the low-rank matrix to a full-rank one.N)r# r1 r� r� rn r� )rv r! r! r" r� s zLowRankMatrix.collapsec C sL | j d k r d St | j � | k rH | j d d � =| j d d � =d S)zH Reduce the rank of the matrix by dropping all vectors. N)r� r� r� rn )rv �rankr! r! r" �restart_reduce s z LowRankMatrix.restart_reducec C sG | j d k r d Sx- t | j � | k rB | j d =| j d =q Wd S)zK Reduce the rank of the matrix by dropping oldest vectors. Nr )r� r� r� rn )rv r� r! r! r" � simple_reduce) s zLowRankMatrix.simple_reduceNc C s� | j d k r d S| } | d k r. | } n | d } | j r] t | t | j d � � } t d t | | d � � } t | j � } | | k r� d St j | j � j } t j | j � j } t | d d �\ } } t | | j j � � } t | d d d d �\ } } } t | t | � � } t | | j j � � } xZ t | � D]L } | d d � | f j � | j | <| d d � | f j � | j | <qMW| j | d � =| j | d � =d S) 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). http://www.math.leidenuniv.nl/scripties/Rotten.pdf NrG r r �modeZeconomicZ full_matricesFZ compute_uvT)r� r� rR r� r$ r# r1 r� rn r r r� r r r rN ) rv �max_rankZ to_retainrh r� r� �C�D�R�U�SZWH�kr! r! r" � svd_reduce4 s0 !#'zLowRankMatrix.svd_reduce)r r r r5 rw � staticmethodr� r� r| r} r r~ r� r� r� r� r� r� r! r! r! r" r� � s r� a� 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 @ s� e Z d Z d Z d d d d d � Z d d � Z d d � Z d d d � Z d d � Z d d d � Z d d � Z d d � Z d S)r� aL 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 N�restartc s t j � � | � _ d � _ | d k r4 t j } | � _ t | t � rU f � n | d d � � | d } | d f � � | d k r� � � f d d � � _ n^ | d k r� � � f d d � � _ n7 | d k r� � � f d d � � _ n t d | � � d S) Nr r r c s � j j � � S)N)r� r� r! )� reduce_paramsrv r! r" rC � s z'BroydenFirst.__init__.<locals>.<lambda>Zsimplec s � j j � � S)N)r� r� r! )r� rv r! r" rC � s r� c s � j j � � S)N)r� r� r! )r� rv r! r" rC � s z"Unknown rank reduction method '%s') r� rw r� r� r# r2 r� r� r� �_reducerP )rv r� Zreduction_methodr� r! )r� rv r" rw � s& zBroydenFirst.__init__c C s= t j | | | | � t | j | j d | j � | _ d S)Nr )r� rM r� r� r+ r' r� )rv r% rB r^ r! r! r" rM � s zBroydenFirst.setupc C s t | j � S)N)r r� )rv r! r! r" r � s zBroydenFirst.todenser c C sS | j j | � } t j | � j � sC | j | j | j | j � | j j | � S)N) r� r| r# r/ r0 rM r� r� r^ )rv ry rF �rr! r! r" r � s zBroydenFirst.solvec C s | j j | � S)N)r� r )rv ry r! r! r" r| � s zBroydenFirst.matvecc C s | j j | � S)N)r� r} )rv ry rF r! r! r" r~ � s zBroydenFirst.rsolvec C s | j j | � S)N)r� r~ )rv ry r! r! r" r} � s zBroydenFirst.rmatvecc C s\ | j � | j j | � } | | j j | � } | t | | � } | j j | | � d S)N)r� r� r} r| r r� ) rv r% ry r_ r� rz r� r3 r� r� r! r! r" r� � s zBroydenFirst._update) r r r r5 rw rM r r r| r~ r} r� r! r! r! r" r� � s r� c @ s" e Z d Z d Z d d � Z d S)r� a� 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 � | } | | j j | � } | | d } | j j | | � d S)NrG )r� r� r| r� ) rv r% ry r_ r� rz r� r3 r� r� r! r! r" r� s zBroydenSecond._updateN)r r r r5 r� r! r! r! r" r� � s r� c @ sR e Z d Z d Z d d d d d � Z d d d � Z d d � Z d d � Z d S)r� a� 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). Ng{�G�z�?� c C sG t j | � | | _ | | _ g | _ g | _ d | _ | | _ d S)N)r� rw r� �Mr_ r� rb �w0)rv r� r� r� r! r! r" rw I s zAnderson.__init__r c C s | j | } t | j � } | d k r- | St j | d | j �} x. t | � D] } t | j | | � | | <qR Wy t | j | � } Wn6 t k r� | j d d � =| j d d � =| SYn Xx? t | � D]1 } | | | | j | | j | j | 7} q� W| S)Nr r' ) r� r� r_ r# �emptyr' r r r� r r� r ) rv ry rF r_ rc �df_fr� rb r� r! r! r" r R s /zAnderson.solvec C s� | | j } t | j � } | d k r- | St j | d | j �} x. t | � D] } t | j | | � | | <qR Wt j | | f d | j �} x� t | � D]� } x� t | � D]� } t | j | | j | � | | | f <| | k r� | j d k r� | | | f t | j | | j | � | j d | j 8<q� Wq� Wt | | � } x? t | � D]1 } | | | | j | | j | | j 7} q]W| S)Nr r' rG ) r� r� r_ r# r� r' r r r� r� r ) rv ry r_ rc r� r� �br� r� rb r� r! r! r" r| i s" 'E/zAnderson.matvecc C sM | j d k r d S| j j | � | j j | � x< t | j � | j k rq | j j d � | j j d � q6 Wt | j � } t j | | f d | j �} x t | � D]q } xh t | | � D]W } | | k r� | j d } n d } d | t | j | | j | � | | | f <q� Wq� W| t j | d � j j � 7} | | _ d S)Nr r' rG r )r� r_ r� r� r� �popr# r� r' r r� r Ztriur� r� r� ) rv r% ry r_ r� rz r� rc r� r� r� Zwdr! r! r" r� � s" 7 zAnderson._update)r r r r5 rw r r| r� r! r! r! r" r� s r� c @ s e Z d Z d Z d d d � Z d d � Z d d d � Z d d � Z d d d � Z d d � Z d d � Z d d � Z d S)r� a� 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 Nc C s t j | � | | _ d S)N)r� rw r� )rv r� r! r! r" rw � s zDiagBroyden.__init__c C sF t j | | | | � t j | j d f d | j �| j | _ d S)Nr r' )r� rM r# �onesr+ r' r� r� )rv r% rB r^ r! r! r" rM � s zDiagBroyden.setupr c C s | | j S)N)r� )rv ry rF r! r! r" r � s zDiagBroyden.solvec C s | | j S)N)r� )rv ry r! r! r" r| � s zDiagBroyden.matvecc C s | | j j � S)N)r� r� )rv ry rF r! r! r" r~ � s zDiagBroyden.rsolvec C s | | j j � S)N)r� r� )rv ry r! r! r" r} � s zDiagBroyden.rmatvecc C s t j | j � S)N)r# �diagr� )rv r! r! r" r � s zDiagBroyden.todensec C s* | j | | j | | | d 8_ d S)NrG )r� )rv r% ry r_ r� rz r� r! r! r" r� � s zDiagBroyden._update) r r r r5 rw rM r r| r~ r} r r� r! r! r! r" r� � s r� c @ ss e Z d Z d Z d d d � Z d d d � Z d d � Z d d d � Z d d � Z d d � Z d d � Z d S)r� at 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 Nc C s t j | � | | _ d S)N)r� rw r� )rv r� r! r! r" rw � s zLinearMixing.__init__r c C s | | j S)N)r� )rv ry rF r! r! r" r � s zLinearMixing.solvec C s | | j S)N)r� )rv ry r! r! r" r| � s zLinearMixing.matvecc C s | t j | j � S)N)r# r� r� )rv ry rF r! r! r" r~ � s zLinearMixing.rsolvec C s | t j | j � S)N)r# r� r� )rv ry r! r! r" r} � s zLinearMixing.rmatvecc C s% t j t j | j d � | j � S)Nr )r# r� r� r+ r� )rv r! r! r" r � s zLinearMixing.todensec C s d S)Nr! )rv r% ry r_ r� rz r� r! r! r" r� � s zLinearMixing._update) r r r r5 rw r r| r~ r} r r� r! r! r! r" r� � s r� c @ s� e Z d Z d Z d d d d � Z d d � Z d d d � Z d d � Z d d d � Z d d � Z d d � Z d d � Z d S)r� aF 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. 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 Ng �?c C s, t j | � | | _ | | _ d | _ d S)N)r� rw r� �alphamax�beta)rv r� r� r! r! r" rw s zExcitingMixing.__init__c C sF t j | | | | � | j t j | j d f d | j �| _ d S)Nr r' )r� rM r� r# r� r+ r' r� )rv r% rB r^ r! r! r" rM s zExcitingMixing.setupr c C s | | j S)N)r� )rv ry rF r! r! r" r s zExcitingMixing.solvec C s | | j S)N)r� )rv ry r! r! r" r| s zExcitingMixing.matvecc C s | | j j � S)N)r� r� )rv ry rF r! r! r" r~ s zExcitingMixing.rsolvec C s | | j j � S)N)r� r� )rv ry r! r! r" r} s zExcitingMixing.rmatvecc C s t j d | j � S)Nr ���)r# r� r� )rv r! r! r" r ! s zExcitingMixing.todensec C s^ | | j d k } | j | | j 7<| j | j | <t j | j d | j d | j �d S)Nr �out)r� r� r� r# Zclipr� )rv r% ry r_ r� rz r� Zincrr! r! r" r� $ s zExcitingMixing._update) r r r r5 rw rM r r| r~ r} r r� r! r! r! r" r� � s r� c @ sp e Z d Z d Z d d d d d d d � Z d d � Z d d � Z d d d � Z d d � Z d d � Z d S)r� a; 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 -------- 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 N�lgmres� � c K st | | _ | | _ t d t j j j d t j j j d t j j j d t j j j d t j j j � j | | � | _ t d | d | j � | _ | j t j j j k r� | | j d <d | j d <nX | j t j j j k r | | j d <d | j d <| j j d g � | j j d d � xP | j � D]B \ } } | j d � sUt d | � � | | j | d d � <q*Wd S)N�bicgstab�gmresr� �cgs�minresr\ r� Zrestrtr �outer_kZouter_vZstore_outer_AvFZinner_zUnknown parameter %s� )�preconditionerro r� r� r� r� r� r� r� r� r� �get�method� method_kw� setdefaultr� � startswithrP ) rv ro r� Z inner_maxiterZinner_Mr� r� �keyr� r! r! r" rw y s, zKrylovJacobian.__init__c C sT t | j � j � } t | j � j � } | j t d | � t d | � | _ d S)Nr )rm r- r$ r� ro �omega)rv ZmxZmfr! r! r" �_update_diff_step� s z KrylovJacobian._update_diff_stepc C s� t | � } | d k r d | S| j | } | j | j | | � | j | } t j t j | � � r� t j t j | � � r� t d � � | S)Nr z$Function returned non-finite results) r r� r^ r- r� r# r0 r/ rP )rv r3 ZnvZscr� r! r! r" r| � s %1 zKrylovJacobian.matvecr c C s^ d | j k r3 | j | j | | j � \ } } n' | j | j | d | | j �\ } } | S)NrF )r� r� �op)rv ZrhsrF Zsolre r! r! r" r � s $'zKrylovJacobian.solvec C sT | | _ | | _ | j � | j d k rP t | j d � rP | j j | | � d S)NrT )r- r� r� r� r� rT )rv r% ry r! r! r" rT � s zKrylovJacobian.updatec C s� t j | | | | � | | _ | | _ t j j j | � | _ | j d k rk t j | j � j d | _ | j � | j d k r� t | j d � r� | j j | | | � d S)Ng �?rG rM g �?)r{ rM r- r� r� r� r� Zaslinearoperatorr� ro r# rs r' rt r� r� r� )rv r% ry r^ r! r! r" rM � s zKrylovJacobian.setup) r r r r5 rw r� r| r rT rM r! r! r! r" r� / s H $ r� c C s t | j � \ } } } } t t | t | � d � | � � } d j d d � | D� � } | rm d | } d j d d � | D� � } | r� | d } d } | t d | d | d | j d | � } i } | j t � � 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, c S s& g | ] \ } } d | | f � q S)z%s=%rr! )�.0r� r3 r! r! r" � <listcomp>� s z#_nonlin_wrapper.<locals>.<listcomp>c S s& g | ] \ } } d | | f � q S)z%s=%sr! )r� r� r3 r! r! r" r� � s 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) r� r� �jacZkwkw) � _getargspecrw �listr� r� �joinr� r rT �globalsr r5 r8 ) r� r� �argsZvarargsZvarkwZdefaults�kwargsZkw_strZkwkw_str�wrapper�nsr^ r! r! r" �_nonlin_wrapper� s$ & r )Dr5 Z __future__r r r rU Znumpyr# Zscipy._lib.sixr r r Z scipy.linalgr r r r r r r r r Zscipy.sparse.linalgr� Z scipy.sparser r� Zscipy._lib._utilr r� Z linesearchr r �__all__� Exceptionr r& r) r. r4 r� �stripr6 r8 rf rS �objectrK r{ r� rL r� r� r� r� r� r� r� r� r� r r r r r r r r r! r! r! r" �<module>j sh . ) � ,CD `�],�/(:� (
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Size
Permission
Action
__init__.cpython-35.opt-1.pyc
6964 bytes
0644
__init__.cpython-35.pyc
6964 bytes
0644
_basinhopping.cpython-35.opt-1.pyc
24677 bytes
0644
_basinhopping.cpython-35.pyc
24677 bytes
0644
_differentialevolution.cpython-35.opt-1.pyc
26738 bytes
0644
_differentialevolution.cpython-35.pyc
26738 bytes
0644
_hungarian.cpython-35.opt-1.pyc
7804 bytes
0644
_hungarian.cpython-35.pyc
7804 bytes
0644
_linprog.cpython-35.opt-1.pyc
31172 bytes
0644
_linprog.cpython-35.pyc
31172 bytes
0644
_minimize.cpython-35.opt-1.pyc
24043 bytes
0644
_minimize.cpython-35.pyc
24043 bytes
0644
_numdiff.cpython-35.opt-1.pyc
18899 bytes
0644
_numdiff.cpython-35.pyc
18899 bytes
0644
_root.cpython-35.opt-1.pyc
25483 bytes
0644
_root.cpython-35.pyc
25483 bytes
0644
_spectral.cpython-35.opt-1.pyc
7761 bytes
0644
_spectral.cpython-35.pyc
7761 bytes
0644
_trustregion.cpython-35.opt-1.pyc
7169 bytes
0644
_trustregion.cpython-35.pyc
7169 bytes
0644
_trustregion_dogleg.cpython-35.opt-1.pyc
3735 bytes
0644
_trustregion_dogleg.cpython-35.pyc
3735 bytes
0644
_trustregion_ncg.cpython-35.opt-1.pyc
3464 bytes
0644
_trustregion_ncg.cpython-35.pyc
3464 bytes
0644
_tstutils.cpython-35.opt-1.pyc
1995 bytes
0644
_tstutils.cpython-35.pyc
1995 bytes
0644
cobyla.cpython-35.opt-1.pyc
9239 bytes
0644
cobyla.cpython-35.pyc
9239 bytes
0644
lbfgsb.cpython-35.opt-1.pyc
15952 bytes
0644
lbfgsb.cpython-35.pyc
15952 bytes
0644
linesearch.cpython-35.opt-1.pyc
18967 bytes
0644
linesearch.cpython-35.pyc
18967 bytes
0644
minpack.cpython-35.opt-1.pyc
28403 bytes
0644
minpack.cpython-35.pyc
28403 bytes
0644
nnls.cpython-35.opt-1.pyc
1659 bytes
0644
nnls.cpython-35.pyc
1659 bytes
0644
nonlin.cpython-35.opt-1.pyc
47479 bytes
0644
nonlin.cpython-35.pyc
47545 bytes
0644
optimize.cpython-35.opt-1.pyc
83901 bytes
0644
optimize.cpython-35.pyc
83901 bytes
0644
setup.cpython-35.opt-1.pyc
2801 bytes
0644
setup.cpython-35.pyc
2801 bytes
0644
slsqp.cpython-35.opt-1.pyc
15484 bytes
0644
slsqp.cpython-35.pyc
15484 bytes
0644
tnc.cpython-35.opt-1.pyc
13998 bytes
0644
tnc.cpython-35.pyc
13998 bytes
0644
zeros.cpython-35.opt-1.pyc
19335 bytes
0644
zeros.cpython-35.pyc
19335 bytes
0644
N4ST4R_ID | Naxtarrr
