B &]\h @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(d)d)dZ*dd Z+dId%d&Z,e+e,dJd)d*Z-Gd+d,d,e.Z/Gd-d.d.e.Z0Gd/d0d0e.Z1d1d2Z2Gd3d4d4e0Z3Gd5d6d6e.Z4d7)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!4lib/python3.7/site-packages/scipy/optimize/nonlin.pyrsrcCst|S)N)npZabsolutemax)xr!r!r"maxnormsr&cCs*t|}t|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(t|t|}t|d|j}||S)z;Return ndarray `x` as same array subclass and shape as `x0`__array_wrap__)r#Zreshapeshapegetattrr*)r%x0Zwrapr!r!r" _array_likesr.cCs"t|sttjSt|S)N)r#isfiniteallarrayinfr)vr!r!r" _safe_norms r4z 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_docsr8krylovFarmijoTc s`| dkr tn| } t|||| || d}tfdd}}tj}||}t|}t|}|| |||dkr|dk r|d}nd|j d}| dkrd} n | d krd} | d krt d d }d }d}d}xFt |D]}| |||}|rPt|||}|j||d }t|dkr.t d| rNt||||| \}}}}nd}||}||}t|}|| || r| ||||d|d}||d|krt||}nt|t|||d}|}|rtjd|| ||ftjqW|rtt|nd}| rR|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|S)N)r)r.flatten)z)Fr-r!r"sznonlin_solve..rdTr:F)Nr:wolfezInvalid 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.)rrG)ZnitZfunstatusZsuccessmessage)r&TerminationConditionr)r@r#r2r asjacobiansetupcopysize ValueErrorrcheckminr _nonlin_line_searchupdater$sysstdoutwriteflushrr. iteration) rBr-jacobianr?verbosemaxiterr;r<r=r>Ztol_normZ line_searchcallbackZ full_outputZraise_exceptionZ conditionfuncr%dxFxFx_normgammaZeta_maxZ eta_tresholdZetanrHrFsZ Fx_norm_newZeta_Ainfor!)rBr-r" nonlin_solves-      re:0yE>{Gz?c sdg|gt|dgttdfdd fdd}|dkrxt|dd |d \}} } n&|d krtdd |d \}} |dkrd }||dkr̈d}n}t|} ||| fS)NrrGTcsT|dkrdS|}|}t|d}|rP|d<|d<|d<|S)NrrG)r4)rcstoreZxtr3p)r^r]tmp_Fxtmp_phitmp_sr%r!r"phi|s   z _nonlin_line_search..phics0t|d}||dd||S)NrF)rh)abs)rcds)rmrdiffs_normr!r"derphisz#_nonlin_line_search..derphirEg{Gz?)Zxtolaminr:)rsg?)T)rrr) r]r%r_r^Z search_typerpZsminrrrcZphi1Zphi0r`r!) r^r]rmrprqrjrkrlr%r"rRus(      rRc@s.eZdZdZdddddefddZddZdS)rJz 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(epsr2r=r>r;r<rr?f0_normrX)selfr;r<r=r>r?rr!r!r"__init__s zTerminationCondition.__init__cCs|jd7_||}||}||}|jdkr<||_|dkrHdS|jdk rbd|j|jkSt||jko||j|jko||jko||j|kS)NrrrG) rXrrvr?intr;r<r=r>)rwfr%r^Zf_normZx_normdx_normr!r!r"rPs       zTerminationCondition.check)rrr r5r&rxrPr!r!r!r"rJs rJc@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@|D]4\}}||kr.)itemsrOsetattrhasattr __array__)rwkwnamesnamevaluer!)rwr"rxs    zJacobian.__init__cCst|S)N)InverseJacobian)rwr!r!r"aspreconditionerszJacobian.aspreconditionerrcCstdS)N)NotImplementedError)rwr3rFr!r!r"r szJacobian.solvecCsdS)Nr!)rwr%rBr!r!r"rSszJacobian.updatecCs:||_|j|jf|_|j|_|jjtjkr6|||dS)N)r]rNr+r' __class__rLr|rS)rwr%rBr]r!r!r"rLs zJacobian.setupN)r) rrr r5rxrr rSrLr!r!r!r"r|s $  r|c@s,eZdZddZeddZeddZdS)rcCs>||_|j|_|j|_t|dr(|j|_t|dr:|j|_dS)NrLr)rYr r}rSrrLrr~)rwrYr!r!r"rx's  zInverseJacobian.__init__cCs|jjS)N)rYr+)rwr!r!r"r+0szInverseJacobian.shapecCs|jjS)N)rYr')rwr!r!r"r'4szInverseJacobian.dtypeN)rrr rxpropertyr+r'r!r!r!r"r&s rc 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. rGzarray must have rank <= 2rrzarray must be squarecs t|S)N)r)r3)Jr!r"rCIszasjacobian..cstj|S)N)rconjT)r3)rr!r"rCJscs t|S)N)r )r3)rr!r"rCKscstj|S)N)r rr)r3)rr!r"rCLs)r}r~r rr'r+zmatrix must be squarecs|S)Nr!)r3)rr!r"rCQscsj|S)N)rr)r3)rr!r"rCRscs |S)Nr!)r3)rspsolver!r"rCSscsj|S)N)rr)r3)rrr!r"rCTsr+r'r r}r~rrSrL)r}r~r rrSrLr'r+csLeZdZddZd fdd ZfddZdfdd Zfd d Zd S)zasjacobian..JaccSs ||_dS)N)r%)rwr%rBr!r!r"rSbszasjacobian..Jac.updatercsB|j}t|tjr t||Stj|r6||StddS)NzUnknown matrix type) r% isinstancer#ndarrayr scipysparse isspmatrixrO)rwr3rFm)rrr!r"r es      zasjacobian..Jac.solvecs@|j}t|tjr t||Stj|r4||StddS)NzUnknown matrix type) r%rr#rrrrrrO)rwr3r)rr!r"r}ns     zasjacobian..Jac.matveccsN|j}t|tjr&t|j|Stj |rB|j|St ddS)NzUnknown matrix type) r%rr#rr rrrrrrO)rwr3rFr)rrr!r"rws    zasjacobian..Jac.rsolvecsL|j}t|tjr&t|j|Stj |r@|j|St ddS)NzUnknown matrix type) r%rr#rrrrrrrrO)rwr3r)rr!r"r~s    zasjacobian..Jac.rmatvecN)r)r)rrr rSr r}rr~r!)rrr!r"Jacas   r)rrrrrrr9z#Cannot convert object to a JacobianN) rrlinalgrrr|inspectZisclass issubclassr#rndimrOZ atleast_2drr+r'rrr,r rstrdict BroydenFirst BroydenSecondAnderson DiagBroyden LinearMixingExcitingMixingKrylovJacobian TypeError)rrr!)rrr"rK9sZ            $   ' rKc@s$eZdZddZddZddZdS)GenericBroydencCs`t||||||_||_t|dr\|jdkr\t|}|rVdtt|d||_nd|_dS)Nalphag?rg?)r|rLlast_flast_xrrrr$)rwr-f0r]Znormf0r!r!r"rLszGenericBroyden.setupcCstdS)N)r)rwr%rzr^dfr{df_normr!r!r"_updateszGenericBroyden._updatec Cs@||j}||j}|||||t|t|||_||_dS)N)rrrr)rwr%rzrr^r!r!r"rSs   zGenericBroyden.updateN)rrr rLrrSr!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)rcsrorbr' collapsed)rwrrbr'r!r!r"rxs zLowRankMatrix.__init__c Csbtdddg|dd|g\}}}||}x0t||D]"\}} || |} ||||j| }q8W|S)Naxpyscaldotcr)rziprN) r3rrrorrrwcdar!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 rrN)r3rrrorrZc0AirjrqrZqcr!r!r"_solves"  " zLowRankMatrix._solvecCs.|jdk rt|j|St||j|j|jS)zEvaluate w = M vN)rr#rrrrrro)rwr3r!r!r"r}s zLowRankMatrix.matveccCs:|jdk rt|jj|St|t|j|j|j S)zEvaluate w = M^H vN) rr#rrrrrrror)rwr3r!r!r"r~s zLowRankMatrix.rmatvecrcCs,|jdk rt|j|St||j|j|jS)zEvaluate w = M^-1 vN)rr rrrrro)rwr3rFr!r!r"r s  zLowRankMatrix.solvecCs8|jdk rt|jj|St|t|j|j|j S)zEvaluate w = M^-H vN) rr rrrrr#rror)rwr3rFr!r!r"rs zLowRankMatrix.rsolvecCsp|jdk r<|j|dddf|dddf7_dS|j||j|t|j|jkrl|dS)N)rrrappendrorrNcollapse)rwrrr!r!r"rs .  zLowRankMatrix.appendcCsp|jdk r|jS|jtj|j|jd}xBt|j|jD]0\}}||dddf|dddf 7}q8W|S)N)r') rrr#rrbr'rrror)rwGmrrr!r!r"rs  ,zLowRankMatrix.__array__cCs"t||_d|_d|_d|_dS)z0Collapse the low-rank matrix to a full-rank one.N)r#r1rrror)rwr!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)rAssertionErrorrrro)rwrankr!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)rrrrro)rwrr!r!r" simple_reduce*s   zLowRankMatrix.simple_reduceNc Cs<|jdk rdS|}|dk r |}n|d}|jrBt|t|jd}tdt||d}t|j}||krldSt|jj}t|jj}t |dd\}}t ||j }t |ddd \} } } t |t | }t || j }xDt|D]8} |dd| f|j| <|dd| f|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 NrGrrZeconomic)modeFT)Z full_matricesZ compute_uv)rrrQrr$r#r1rror rrr r rrM) rwmax_rankZ to_retainrirrCDRUSZWHkr!r!r" svd_reduce5s0   zLowRankMatrix.svd_reduce)r)r)N)rrr r5rx staticmethodrrr}r~r rrrrrrrr!r!r!r"rs        ra alpha : float, optional Initial guess for the Jacobian is ``(-1/alpha)``. reduction_method : str or tuple, optional Method used in ensuring that the rank of the Broyden matrix stays low. Can either be a string giving the name of the method, or a tuple of the form ``(method, param1, param2, ...)`` that gives the name of the method and values for additional parameters. Methods available: - ``restart``: drop all matrix columns. Has no extra parameters. - ``simple``: drop oldest matrix column. Has no extra parameters. - ``svd``: keep only the most significant SVD components. Takes an extra parameter, ``to_retain``, which determines the number of SVD components to retain when rank reduction is done. Default is ``max_rank - 2``. max_rank : int, optional Maximum rank for the Broyden matrix. Default is infinity (ie., no rank reduction). Zbroyden_paramsc@sVeZdZdZdddZddZdd Zdd d Zd dZdddZ ddZ ddZ dS)raw Find a root of a function, using Broyden's first Jacobian approximation. This method is also known as \"Broyden's good method\". Parameters ---------- %(params_basic)s %(broyden_params)s %(params_extra)s Notes ----- This algorithm implements the inverse Jacobian Quasi-Newton update .. math:: H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df) which corresponds to Broyden's first Jacobian update .. math:: J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx References ---------- .. [1] B.A. van der Rotten, PhD thesis, \"A limited memory Broyden method to solve high-dimensional systems of nonlinear equations\". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003). https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf Nrestartcst|_d_|dkr$tj}|_t|tr:dn|dd|d}|df|dkrvfdd_ n@|dkrfdd_ n&|d krfd d_ n t d |dS) Nr!rrr cs jjS)N)rrr!) reduce_paramsrwr!r"rCsz'BroydenFirst.__init__..Zsimplecs jjS)N)rrr!)rrwr!r"rCsrcs jjS)N)rrr!)rrwr!r"rCsz"Unknown rank reduction method '%s') rrxrrr#r2rrr_reducerO)rwrZreduction_methodrr!)rrwr"rxs&   zBroydenFirst.__init__cCs.t||||t|j |jd|j|_dS)Nr)rrLrrr+r'r)rwr%rBr]r!r!r"rLszBroydenFirst.setupcCs t|jS)N)r r)rwr!r!r"rszBroydenFirst.todensercCs:|j|}t|s.||j|j|j|j|S)N) rr}r#r/r0rLrrr])rwrzrFrr!r!r"r s zBroydenFirst.solvecCs |j|S)N)rr )rwrzr!r!r"r}szBroydenFirst.matveccCs |j|S)N)rr~)rwrzrFr!r!r"rszBroydenFirst.rsolvecCs |j|S)N)rr)rwrzr!r!r"r~szBroydenFirst.rmatvecc CsD||j|}||j|}|t||} |j|| dS)N)rrr~r}rr) rwr%rzr^rr{rr3rrr!r!r"rs  zBroydenFirst._update)NrN)r)r) rrr r5rxrLrr r}rr~rr!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 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|}||d} |j|| dS)NrG)rrr}r) rwr%rzr^rr{rr3rrr!r!r"r s  zBroydenSecond._updateN)rrr r5rr!r!r!r"rsrc@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 References ---------- .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996). N{Gz?cCs2t|||_||_g|_g|_d|_||_dS)N)rrxrMr^rraw0)rwrrrr!r!r"rxKs 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|j||j|x,t|j|jkrR|jd|jdq(Wt|j}tj||f|jd}xbt |D]V} xPt | |D]B} | | kr|j d} nd} d| t |j| |j| || | f<qWq|W|t |dj 7}||_dS)Nr)r'rGr)rr^rrrpopr#rr'rrrZtriurrr) rwr%rzr^rr{rrbrrrZwdr!r!r"rs"      .zAnderson._update)Nrr)r)rrr r5rxr r}rr!r!r!r"rs  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 NcCst|||_dS)N)rrxr)rwrr!r!r"rxs zDiagBroyden.__init__cCs4t||||tj|jdf|jd|j|_dS)Nr)r')rrLr#Zonesr+r'rr)rwr%rBr]r!r!r"rLszDiagBroyden.setuprcCs | |jS)N)r)rwrzrFr!r!r"r szDiagBroyden.solvecCs | |jS)N)r)rwrzr!r!r"r}szDiagBroyden.matveccCs| |jS)N)rr)rwrzrFr!r!r"rszDiagBroyden.rsolvecCs| |jS)N)rr)rwrzr!r!r"r~szDiagBroyden.rmatveccCst|j S)N)r#diagr)rwr!r!r"rszDiagBroyden.todensecCs(|j||j|||d8_dS)NrG)r)rwr%rzr^rr{rr!r!r"rszDiagBroyden._update)N)r)r) rrr r5rxrLr r}rr~rrr!r!r!r"rs   rc@sNeZdZdZdddZdddZdd Zdd d Zd d ZddZ ddZ dS)rat 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 NcCst|||_dS)N)rrxr)rwrr!r!r"rxs zLinearMixing.__init__rcCs | |jS)N)r)rwrzrFr!r!r"r szLinearMixing.solvecCs | |jS)N)r)rwrzr!r!r"r}szLinearMixing.matveccCs| t|jS)N)r#rr)rwrzrFr!r!r"rszLinearMixing.rsolvecCs| t|jS)N)r#rr)rwrzr!r!r"r~szLinearMixing.rmatveccCstt|jdd|jS)Nr)r#rfullr+r)rwr!r!r"rszLinearMixing.todensecCsdS)Nr!)rwr%rzr^rr{rr!r!r"rszLinearMixing._update)N)r)r) rrr r5rxr r}rr~rrr!r!r!r"rs   rc@sVeZdZdZdddZddZdd d Zd d Zdd dZddZ ddZ ddZ dS)raF 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 N?cCs t|||_||_d|_dS)N)rrxralphamaxbeta)rwrrr!r!r"rx s zExcitingMixing.__init__cCs2t||||tj|jdf|j|jd|_dS)Nr)r')rrLr#rr+rr'r)rwr%rBr]r!r!r"rLszExcitingMixing.setuprcCs | |jS)N)r)rwrzrFr!r!r"r szExcitingMixing.solvecCs | |jS)N)r)rwrzr!r!r"r}szExcitingMixing.matveccCs| |jS)N)rr)rwrzrFr!r!r"rszExcitingMixing.rsolvecCs| |jS)N)rr)rwrzr!r!r"r~ szExcitingMixing.rmatveccCstd|jS)Nr)r#rr)rwr!r!r"r#szExcitingMixing.todensecCsL||jdk}|j||j7<|j|j|<tj|jd|j|jddS)Nr)out)rrrr#Zclipr)rwr%rzr^rr{rZincrr!r!r"r&szExcitingMixing._update)Nr)r)r) rrr r5rxrLr r}rr~rrr!r!r!r"rs   rc@sDeZdZdZdddZdd Zd d Zdd dZddZddZ dS)raA 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` Nlgmres c KsN||_||_ttjjjtjjjtjjjtjjj tjjj d |||_ t||jd|_ |j tjjjkr||j d<d|j d<|j ddn~|j tjjjkr|j ddn^|j tjjjkr||j d<d|j d<|j d g|j d d |j d d |j ddx@|D]4\}}|ds2td|||j |dd<qWdS)N)bicgstabgmresrcgsminres)r[rZrestrtrr[Zatolrouter_kZouter_vZprepend_outer_vTZstore_outer_AvFZinner_zUnknown parameter %s)preconditionerrprrrrrrrrrgetmethod method_kw setdefaultZgcrotmkr startswithrO) rwrprZ inner_maxiterZinner_Mrrkeyrr!r!r"rx{s6       zKrylovJacobian.__init__cCs<t|j}t|j}|jtd|td||_dS)Nr)rnr-r$rrpomega)rwZmxZmfr!r!r"_update_diff_stepsz KrylovJacobian._update_diff_stepcCslt|}|dkrd|S|j|}||j|||j|}tt|shtt|rhtd|S)Nrz$Function returned non-finite results) rrr]r-rr#r0r/rO)rwr3ZnvZscrr!r!r"r}s  zKrylovJacobian.matvecrcCsHd|jkr$|j|j|f|j\}}n |j|j|fd|i|j\}}|S)NrF)rrop)rwZrhsrFZsolrdr!r!r"r s  zKrylovJacobian.solvecCs<||_||_||jdk r8t|jdr8|j||dS)NrS)r-rrrrrS)rwr%rzr!r!r"rSs   zKrylovJacobian.updatecCs|t||||||_||_tjj||_|j dkrJt |j j d|_ ||jdk rxt|jdrx|j|||dS)Ng?rL)r|rLr-rrrrZaslinearoperatorrrpr#rtr'rurrr)rwr%rzr]r!r!r"rLs   zKrylovJacobian.setup)NrrNr)r) rrr r5rxrr}r rSrLr!r!r!r"r1sH )  rc Cst|j\}}}}tt|t| d|}ddd|D}|rNd|}ddd|D}|rn|d}d} | t|||j|d} i} | t t | | | |} |j | _ t | | S)a Construct a solver wrapper with given name and jacobian approx. It inspects the keyword arguments of ``jac.__init__``, and allows to use the same arguments in the wrapper function, in addition to the keyword arguments of `nonlin_solve` Nz, cSsg|]\}}d||fqS)z%s=%rr!).0rr3r!r!r" sz#_nonlin_wrapper..cSsg|]\}}d||fqS)z%s=%sr!)rrr3r!r!r"rsa def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw): jac = %(jac)s(%(kwkw)s **kw) return nonlin_solve(F, xin, jac, iter, verbose, maxiter, f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search, callback) )rrjacZkwkw) _getargspecrxlistrrjoinrrrSglobalsrr5r8) rrargsZvarargsZvarkwdefaultskwargsZkw_strZkwkw_strwrappernsr]r!r!r"_nonlin_wrappers$      r) r9NFNNNNNNr:NFT)r:rfrg)Dr5Z __future__rrrrTZnumpyr#Zscipy._lib.sixrrrZ scipy.linalgrr r r r r rrrZscipy.sparse.linalgrZ scipy.sparserrZscipy._lib._utilrrZ linesearchrr__all__ Exceptionrr&r)r.r4rstripr6r8rerRobjectrJr|rrKrrrrrrrrrrrrrrrrrr!r!r!r"jsp     )   ,@D`Z ], /(:*)