U |e @s"ddlZddlZddlmZmZmZmZmZm Z ddlm Z m Z m Z ddl ZddlZddlmZddlZddlmZddlmZmZdd d d d d ddddg ZGdddeZddZddZddZddZed d dZ!dd Z"dFd%d&Z#e"e#dGd)d*Z$Gd+d,d,Z%Gd-d.d.Z&Gd/ddZ'd0d1Z(Gd2d3d3e&Z)Gd4d5d5Z*d6 e!d7<Gd8dde)Z+Gd9d:d:e+Z,Gd;d<dd>e)Z.Gd?d@d@e)Z/GdAdBdBe)Z0GdCdde&Z1dDdEZ2e2de+Z3e2d e,Z4e2d e-Z5e2d e/Z6e2d e.Z7e2d e0Z8e2de1Z9dS)HN)normsolveinvqrsvd LinAlgError)asarraydotvdot)get_blas_funcs)getfullargspec_no_self)scalar_search_wolfe1scalar_search_armijobroyden1broyden2anderson linearmixing diagbroydenexcitingmixing newton_krylov BroydenFirstKrylovJacobianInverseJacobianc@s eZdZdS) NoConvergenceN)__name__ __module__ __qualname__rrS/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/optimize/_nonlin.pyrsrcCst|SN)npabsolutemaxxrrrmaxnormsr&cCs*t|}t|jtjs&t|tjdS|S)z:Return `x` as an array, of either floats or complex floatsdtype)rr! issubdtyper(inexactfloat_r$rrr _as_inexact"sr,cCs(t|t|}t|d|j}||S)z;Return ndarray `x` as same array subclass and shape as `x0`__array_wrap__)r!reshapeshapegetattrr-)r%x0wraprrr _array_like*sr3cCs"t|sttjSt|Sr )r!isfiniteallarrayinfrvrrr _safe_norm1s r: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. )Z params_basicZ params_extracCs|jr|jt|_dSr )__doc__ _doc_parts)objrrr_set_docosr>krylovFarmijoTc sh| dkr tn| } t|||| || d}tfdd}}t|tj}||}t|}t|}| | |||dkr|dk r|d}nd|j d}| dkrd} n | d krd} | d krt d d }d }d}d}t |D]$}||||}|rq&t|||}|j||d }t|dkr8t d| rXt||||| \}}}}nd}||}||}t|}|| || r| ||||d|d}||d|krt||}nt|t|||d}|}|rtjd|| ||ftjq|r"tt|nd}| rZ|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|Sr )r,r3flatten)zFr1rrznonlin_solve..r dTr@F)Nr@wolfezInvalid 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.)r rP)nitfunstatussuccessmessage)r&TerminationConditionr,rFr! full_liker7r asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater#sysstdoutwriteflushrr3 iteration) rIr1jacobianrEverbosemaxiterrArBrCrDtol_norm line_searchcallback full_outputraise_exception conditionfuncr%dxFxFx_normgammaZeta_maxZ eta_tresholdetanrSrNsZ Fx_norm_newZeta_AinforrHr nonlin_solvets-      ry:0yE>{Gz?c sdg|gt|dgttdfdd fdd}|dkrxt|dd |d \}} } n&|d krtdd |d \}} |dkrd }||dkr̈d}n}t|} ||| fS)NrrPTcsT|dkrdS|}|}t|d}|rP|d<|d<|d<|S)NrrP)r:)rwstoreZxtr9p)rqrptmp_Fxtmp_phitmp_sr%rrphis   z _nonlin_line_search..phics0t|d}||dd||S)Nr F)r|)abs)rwds)rrdiffs_normrrderphisz#_nonlin_line_search..derphirMr{)xtolaminr@)rrO)T)rrr) rpr%rrrqZ search_typersminrrwphi1phi0rsr) rqrprrrr~rrr%rr` s.      r`c@s.eZdZdZdddddefddZddZdS)rVz 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|dkrttjjd}|dkr(tj}|dkr6tj}|dkrDtj}||_||_||_||_||_ ||_ d|_ d|_ dS)NgUUUUUU?r) r!finfor+epsr7rCrDrArBrrEf0_normrf)selfrArBrCrDrErrrr__init__Cs zTerminationCondition.__init__cCs|jd7_||}||}||}|jdkr<||_|dkrHdS|jdk rbd|j|jkSt||jko||j|jko||jko||j|kS)Nr rrP) rfrrrEintrArBrCrD)rfr%rqZf_normx_normdx_normrrrr^[s         zTerminationCondition.check)rrrr;r&rr^rrrrrV6s   rVc@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 spddddddddd g }|D]4\}}||kr:td ||dk rt|||qtdrlfd d _dS) Nrramatvecrmatvecrsolvematmattodenser/r(zUnknown keyword argument %scsSr )rrrrrrJrKz#Jacobian.__init__..)itemsr\setattrhasattr __array__)rkwnamesnamevaluerrrrs   zJacobian.__init__cCst|Sr )rrrrraspreconditionerszJacobian.aspreconditionerrcCstdSr NotImplementedErrorrr9rNrrrrszJacobian.solvecCsdSr rrr%rIrrrraszJacobian.updatecCs:||_|j|jf|_|j|_|jjtjkr6|||dSr )rpr[r/r( __class__rYrrarr%rIrprrrrYs zJacobian.setupN)r) rrrr;rrrrarYrrrrrvs %  rc@s,eZdZddZeddZeddZdS)rcCs>||_|j|_|j|_t|dr(|j|_t|dr:|j|_dS)NrYr)rgrrrarrYrr)rrgrrrrs  zInverseJacobian.__init__cCs|jjSr )rgr/rrrrr/szInverseJacobian.shapecCs|jjSr )rgr(rrrrr(szInverseJacobian.dtypeN)rrrrpropertyr/r(rrrrrs   c stjjjttrStr2ttr2Stt j rj dkrPt dt t jdjdkr|t dtfddfddfd dfd djjd Stjrjdjdkrt d tfd dfddfddfddjjd StdrztdrztdrzttdtdjtdtdtdjjdStrGfdddt}|SttrttttttttdStddS)zE Convert given object to one suitable for use as a Jacobian. rPzarray must have rank <= 2rr zarray must be squarecs t|Sr )r r8JrrrJrKzasjacobian..cstj|Sr )r conjTr8rrrrJrKcs t|Sr )rr8rrrrJrKcstj|Sr )rrrr8rrrrJrK)rrrrr(r/zmatrix must be squarecs|Sr rr8rrrrJrKcsj|Sr rrr8rrrrJrKcs |Sr rr8rspsolverrrJrKcsj|Sr rr8rrrrJrKr/r(rrrrrarY)rrrrrarYr(r/csLeZdZddZd fdd ZfddZdfdd Zfd d Zd S)zasjacobian..JaccSs ||_dSr r$rrrrraszasjacobian..Jac.updatercsB|j}t|tjr t||Stj|r6||StddSNzUnknown matrix type) r% isinstancer!ndarrayrscipysparse isspmatrixr\rr9rNmrrrrs      zasjacobian..Jac.solvecs@|j}t|tjr t||Stj|r4||StddSr) r%rr!rr rrrr\rr9rrrrrs     zasjacobian..Jac.matveccsN|j}t|tjr&t|j|Stj |rB|j|St ddSr) r%rr!rrrrrrrr\rrrrr s    zasjacobian..Jac.rsolvecsL|j}t|tjr&t|j|Stj |r@|j|St ddSr) r%rr!rr rrrrrr\rrrrrs    zasjacobian..Jac.rmatvecN)r)r)rrrrarrrrrrrrJacs   r)rrrrrrr?z#Cannot convert object to a JacobianN) rrlinalgrrrinspectisclass issubclassr!rndimr\ atleast_2drr/r(rrr0rcallablestrdictr BroydenSecondAnderson DiagBroyden LinearMixingExcitingMixingr TypeError)rrrrrrXsj            $  ' rXc@s$eZdZddZddZddZdS)GenericBroydencCs`t||||||_||_t|dr\|jdkr\t|}|rVdtt|d||_nd|_dS)Nalpha?r rO)rrYlast_flast_xrrrr#)rr1f0rpZnormf0rrrrY.szGenericBroyden.setupcCstdSr rrr%rrqdfrdf_normrrr_update<szGenericBroyden._updatec Cs@||j}||j}|||||t|t|||_||_dSr )rrrr)rr%rrrqrrrra?s   zGenericBroyden.updateN)rrrrYrrarrrrr-src@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|_dSr )rcsrrvr( collapsed)rrrvr(rrrrRs zLowRankMatrix.__init__c Cs^tdddg|dd|g\}}}||}t||D]"\}} || |} ||||j| }q6|S)Naxpyscaldotcr )r zipr[) r9rrrrrrwcdarrr_matvecZs   zLowRankMatrix._matveccCs t|dkr||Stddg|dd|g\}}|d}|tjt||jd}t|D]4\}} t|D]"\} } ||| f|| | 7<qlq\tjt||jd} t|D]\} } || || | <q| |} t|| } ||} t|| D]\} }|| | | j | } q| S)Evaluate w = M^-1 vrrrNr r') lenr r!identityr( enumeratezerosrrr[)r9rrrrrc0Airjrqrqcrrr_solveds"   zLowRankMatrix._solvecCs.|jdk rt|j|St||j|j|jS)zEvaluate w = M vN)rr!r rrrrrrr9rrrrs zLowRankMatrix.matveccCs:|jdk rt|jj|St|t|j|j|j S)zEvaluate w = M^H vN) rr!r rrrrrrrrrrrrs zLowRankMatrix.rmatvecrcCs,|jdk rt|j|St||j|j|jS)rN)rrrrrrrrrrrrs  zLowRankMatrix.solvecCs8|jdk rt|jj|St|t|j|j|j S)zEvaluate w = M^-H vN) rrrrrrr!rrrrrrrrs zLowRankMatrix.rsolvecCsp|jdk r<|j|dddf|dddf7_dS|j||j|t|j|jkrl|dSr )rrrappendrrr[collapse)rrrrrrrs .  zLowRankMatrix.appendcCsl|jdk r|jS|jtj|j|jd}t|j|jD]0\}}||dddf|dddf 7}q6|S)Nr') rrr!rrvr(rrrr)rGmrrrrrrs  *zLowRankMatrix.__array__cCs"t||_d|_d|_d|_dS)z0Collapse the low-rank matrix to a full-rank one.N)r!r6rrrrrrrrrs zLowRankMatrix.collapsecCsD|jdk rdS|dkstt|j|kr@|jdd=|jdd=dS)zH Reduce the rank of the matrix by dropping all vectors. NrrAssertionErrorrrrrrankrrrrestart_reduces    zLowRankMatrix.restart_reducecCs>|jdk rdS|dkstt|j|kr:|jd=|jd=qdS)zK Reduce the rank of the matrix by dropping oldest vectors. Nrrrrrr simple_reduces   zLowRankMatrix.simple_reduceNc Cs6|jdk rdS|}|dk r |}n|d}|jrBt|t|jd}tdt||d}t|j}||krldSt|jj}t|jj}t |dd\}}t ||j }t |dd\} } } t |t | }t || j }t|D]8} |dd| f|j| <|dd| f|j| <q|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 NrPrr economic)modeF) full_matrices)rrr_rr#r!r6rrrr rrrr]rZ) rmax_rankZ to_retainr}rrCDRUSZWHkrrr svd_reduces0    zLowRankMatrix.svd_reduce)r)r)N)rrrr;r staticmethodrrrrrrrrrrrrrrrrrGs          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). 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 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]) Nrestartcst|_d_|dkr$tj}|_t|tr:dn|dd|d}|df|dkrvfdd_ n@|dkrfdd_ n&|d krfd d_ n t d |dS) Nrr rrcs jjSr )rrrZ reduce_paramsrrrrJjrKz'BroydenFirst.__init__..Zsimplecs jjSr )rrrrrrrJlrKrcs jjSr )rrrrrrrJnrKz"Unknown rank reduction method '%s') rrrrr!r7rrr_reducer\)rrZreduction_methodrrrrrYs(   zBroydenFirst.__init__cCs.t||||t|j |jd|j|_dS)Nr)rrYrrr/r(rrrrrrYsszBroydenFirst.setupcCs t|jSr )rrrrrrrwszBroydenFirst.todensercCs>|j|}t|s:||j|j|j|j|S|Sr ) rrr!r4r5rYrrrp)rrrNrrrrrzs   zBroydenFirst.solvecCs |j|Sr )rrrrrrrrszBroydenFirst.matveccCs |j|Sr )rrrrrNrrrrszBroydenFirst.rsolvecCs |j|Sr )rrrrrrrszBroydenFirst.rmatvecc CsD||j|}||j|}|t||} |j|| dSr )rrrrr r rr%rrqrrrr9rrrrrrs  zBroydenFirst._update)NrN)r)r) rrrr;rrYrrrrrrrrrrr#s5   c@seZdZdZddZdS)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 Cs:||}||j|}||d} |j|| dSNrP)rrrrr rrrrs  zBroydenSecond._updateN)rrrr;rrrrrrs2rc@s4eZdZdZdddZddd Zd d Zd d 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]) Nr{cCs2t|||_||_g|_g|_d|_||_dSr )rrrMrqrrtw0)rrrr rrrrs zAnderson.__init__rc Cs|j |}t|j}|dkr"|Stj||jd}t|D]}t|j||||<q:zt |j |}Wn0t k r|jdd=|jdd=|YSXt|D]*}||||j||j|j|7}q|SNrr') rrrqr!emptyr(r]r rrrr) rrrNrqrvdf_frrtrrrrr%s        (zAnderson.solvec Cs,| |j}t|j}|dkr"|Stj||jd}t|D]}t|j||||<q:tj||f|jd}t|D]x}t|D]j}t|j||j||||f<||kr||j dkr||||ft|j||j||j d|j8<q|qpt ||} t|D]*} || | |j| |j| |j7}q|S)Nrr'rP) rrrqr!rr(r]r rrr) rrrqrvrrbrrrtrrrrr<s"     :  (zAnderson.matvecc Cs|jdkrdS|j||j|t|j|jkrP|jd|jdq&t|j}tj||f|jd}t |D]R} t | |D]B} | | kr|j d} nd} d| t |j| |j| || | f<qqv|t |dj 7}||_dS)Nrr'rPr )r rqrrrpopr!rr(r]rr triurrr) rr%rrqrrrrvrrrwdrrrrSs"       *zAnderson._update)Nr{r )r)rrrr;rrrrrrrrrs F 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. 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]) NcCst|||_dSr rrrrrrrrrs zDiagBroyden.__init__cCs6t||||tj|jdfd|j|jd|_dS)Nrr r')rrYr!fullr/rr(rrrrrrYszDiagBroyden.setuprcCs | |jSr rr rrrrszDiagBroyden.solvecCs | |jSr rrrrrrszDiagBroyden.matveccCs| |jSr rrr rrrrszDiagBroyden.rsolvecCs| |jSr rrrrrrszDiagBroyden.rmatveccCst|j Sr )r!diagrrrrrrszDiagBroyden.todensecCs(|j||j|||d8_dSr rrrrrrszDiagBroyden._update)N)r)r rrrr;rrYrrrrrrrrrrrqs(   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. NcCst|||_dSr rrrrrrs zLinearMixing.__init__rcCs | |jSr rr rrrrszLinearMixing.solvecCs | |jSr rrrrrrszLinearMixing.matveccCs| t|jSr r!rrr rrrrszLinearMixing.rsolvecCs| t|jSr rrrrrrszLinearMixing.rmatveccCstt|jdd|jS)Nr)r!rrr/rrrrrrszLinearMixing.todensecCsdSr rrrrrrszLinearMixing._update)N)r)r) rrrr;rrrrrrrrrrrrs   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 NrOcCs t|||_||_d|_dSr )rrralphamaxbeta)rrr rrrrs zExcitingMixing.__init__cCs2t||||tj|jdf|j|jd|_dSr)rrYr!rr/rr(r!rrrrrYszExcitingMixing.setuprcCs | |jSr r!r rrrrszExcitingMixing.solvecCs | |jSr r"rrrrr szExcitingMixing.matveccCs| |jSr r!rr rrrrszExcitingMixing.rsolvecCs| |jSr r#rrrrrszExcitingMixing.rmatveccCstd|jS)Nr)r!rr!rrrrrszExcitingMixing.todensecCsL||jdk}|j||j7<|j|j|<tj|jd|j|jddS)Nr)out)rr!rr!clipr )rr%rrqrrrincrrrrrszExcitingMixing._update)NrO)r)rrrrrrrs   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 : 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 Ksd||_||_ttjjjtjjjtjjjtjjj tjjj tjjj d |||_ t||jd|_|j tjjjkr||jd<d|jd<|jddn|j tjjjtjjjtjjj fkr|jddn^|j tjjjkr"||jd<d|jd<|jd g|jd d |jd d |jdd|D]4\}}|dsJtd|||j|dd<q*dS)N)bicgstabgmresr'cgsminrestfqmr)rir rr riatolrouter_kouter_vprepend_outer_vTstore_outer_AvFZinner_zUnknown parameter %s)preconditionerrrrrrr*r+r'r,r-r.getmethod method_kw setdefaultgcrotmkr startswithr\) rrr7Z inner_maxiterZinner_Mr0rkeyrrrrrsD        zKrylovJacobian.__init__cCs<t|j}t|j}|jtd|td||_dS)Nr )rr1r#rromega)rmxZmfrrr_update_diff_stepsz KrylovJacobian._update_diff_stepcCslt|}|dkrd|S|j|}||j|||j|}tt|shtt|rhtd|S)Nrz$Function returned non-finite results) rr=rpr1rr!r5r4r\)rr9nvZscrrrrrs  zKrylovJacobian.matvecrcCsHd|jkr$|j|j|f|j\}}n |j|j|fd|i|j\}}|S)NrN)r8r7op)rrhsrNsolrxrrrrs  zKrylovJacobian.solvecCs<||_||_||jdk r8t|jdr8|j||dS)Nra)r1rr?r5rra)rr%rrrrras   zKrylovJacobian.updatecCs|t||||||_||_tjj||_|j dkrJt |j j d|_ ||jdk rxt|jdrx|j|||dS)NrrY)rrYr1rrrraslinearoperatorrArr!rr(rr?r5r)rr%rrprrrrYs   zKrylovJacobian.setup)Nr'r(Nr))r) rrrr;rr?rrrarYrrrrr"se /  cCst|j}|\}}}}}}} tt|t| d|} ddd| D} | rXd| } ddd| D} | rx| d} |rtd|d} | t|| |j| d} i}| 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, cSsg|]\}}d||fqS)z%s=%rr.0rr9rrr sz#_nonlin_wrapper..cSsg|]\}}d||fqS)z%s=%srrErrrrGszUnexpected 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) )rrjacZkwkw)_getfullargspecrlistrrjoinr\rrraglobalsexecr;r>)rrH signatureargsvarargsvarkwdefaults kwonlyargs kwdefaults_kwargsZkw_strZkwkw_strwrappernsrprrr_nonlin_wrappers,     rY) r?NFNNNNNNr@NFT)r@rzr{):rbnumpyr! scipy.linalgrrrrrrrr r scipy.sparse.linalgr scipy.sparser rscipy._lib._utilr rI _linesearchrr__all__ Exceptionrr&r,r3r:rstripr<r>ryr`rVrrrXrrrrrrrrrrYrrrrrrrrrrrs     4  -@D`E r@D.?K,