+ ,i^RIt^RIt^RIt^RIt^RIHtHtHt^RIH t H t H t H t H t Ht^RIt^RIt^RIHt^RIHt^RIHt^RIHtHt^RIHt^R IHt.R9Ot!RR]4tRt Rt!Rt"Rt#]$!RPK4RPK4R7t&Rt'R:Rlt(]'!](4R;Rlt)!R R!4t*!R"R#4t+!R$R4t,R%t-!R&R']+4t.!R(R)4t/R*PK4]&R+&!R,R].4t0!R-R.]04t1!R/R0].4t2!R1R2].4t3!R3R4].4t4!R5R6].4t5!R7R]+4t6R8t7]7!R ]04t8]7!R ]14t9]7!R ]24t:]7!R ]44t;]7!R]34t<]7!R]54t=]7!R]64t>R#)<N)asarraydotvdot)normsolveinvqrsvd LinAlgError)get_blas_funcs)copy_if_needed)getfullargspec_no_self)scalar_search_wolfe1scalar_search_armijo) signature)get_close_matchesbroyden1broyden2anderson linearmixing diagbroydenexcitingmixing newton_krylov BroydenFirstKrylovJacobianInverseJacobian NoConvergencec]tRt^ tRtRtR#)rzXException raised when nonlinear solver fails to converge within the specified `maxiter`.N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r8/usr/lib/python3/dist-packages/scipy/optimize/_nonlin.pyrr s r&cJ\P!V4P4#N)npabsolutemaxxs&r'maxnormr/&s ;;q>   r&c\V4p\P!VP\P4'g\V\P R7#V#)z:Return `x` as an array, of either floats or complex floatsdtype)rr* issubdtyper2inexactfloat64r-s&r' _as_inexactr6*s: A =="** - -q ++ Hr&c\P!V\P!V44p\VRVP4pV!V4#)z;Return ndarray `x` as same array subclass and shape as `x0`__array_wrap__)r*reshapeshapegetattrr8)r.x0wraps&& r' _array_liker>2s8 1bhhrl#A 2')9)9 :D 7Nr&c\P!V4P4'g%\P!\P4#\ V4#r))r*isfiniteallarrayinfr)vs&r' _safe_normrE9s5 ;;q>    xx 7Nr&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. ) params_basic params_extracfVP'dVP\,VnR#R#r))r$ _doc_parts)objs&r'_set_docrKws  {{{kkJ. r&c aaV f\MT p \WgWW:R7p\S4oVV3RlpSP4p\P !V\P 4pV!V4p\V4p\V4pVPVP4VV4Vf)Ve V^,pM^dVP^,,pV RJdRp MV RJdRp V R9d \R4hRpR pR pR p\V4EFpVPVVV4pV'dEM\VVV,4pVP!VVR 7)p\V4^8Xd \R 4hV 'd\#VVVVV 4wppppMRpVV,pV!V4p\V4pVP%VP4V4V 'd V !VV4VV^,,V^,, pVV^,,V8d\VV4pM$\V\'VVV^,,44pTpV'gEKA\(P*P-V RV !V4R RVR R24\(P*P/4EK V'd\1\3VS44h^pV 'd3RVP4RVRVRV^8HR^R^R/V,/p\3VS4V3#\3VS4#)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_rtoliterrcV<\S!\VS444P4#r))r6r>flatten)zFr<s&r'funcnonlin_solve..funcs#1[B/0199;;r&TarmijoFzInvalid line searchg?gH.?g?gMbP?)tolz[Jacobian inversion yielded zero vector. This indicates a bug in the Jacobian approximation.?z : |F(x)| = gz; step  nitfunstatussuccessmessagez0A solution was found at the specified tolerance.z:The maximum number of iterations allowed has been reached.)NrXwolfe)r/TerminationConditionr6rSr* full_likerCr asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater,sysstdoutwriteflushrr> iteration) rUr<jacobianrQverbosemaxiterrMrNrOrPtol_norm line_searchcallback full_outputraise_exception conditionrVr.dxFxFx_normgammaeta_max eta_tresholdetanr_rYs Fx_norm_neweta_Ainfos ff&&&&&&&&&&&&& r' nonlin_solver|sZ#*wH$5+0*.?I RB< A a B aB2hG(#H NN1668R&  QhG166!8nGd   33.// EGL C 7^Q+  #s7{#nnRSn) ) 8q=./ / $7aR8C%E !Aq"kABAaBr(K"%  QO Q&!3 36>L (gu%Cgs5%Q,78C 7 JJ  s,x|A.>gaU"M N JJ   SV  Ar 23 3Fy**r&6Q;A , 3% & 1b!4''1b!!r&caaaaa a aaa^.oV.o\V4^,.o\S4\S4, o R VVVVVV3Rllo V VV 3RlpVR8Xd\S VS^,RVR7wrp M'VR8Xd!\S S^,S^,)VR7wrXfRpSVS,,oVS^,8Xd S^,pMS!S4p\V4p VSW+3#) rc<VS ^,8Xd S^,#S VS,,pS!V4p\V4^,pV'dVS ^&VS^&VS^&V#r)rE) rstorextrDpr}rVtmp_Fxtmp_phitmp_sr.s && r'phi _nonlin_line_search..phis_ a=1:  2X H qM1  E!HGAJF1Ir&c<\V4S,^,S,pS!W,RR7S!V4, V, #)F)r)abs)rdsrrdiffs_norms& r'derphi#_nonlin_line_search..derphi$s7!fvo!U *AD&Q/255r&rb{Gz?)xtolaminrX)rrZ)T)rrr)rVr.r~r} search_typersminrrphi1phi0rrrrrrsff&f&f& @@@@@r'rmrms CETFBx{mG !WtBx F  6g,S&'!*26TC   &sGAJ ,02 y  AbDAE!H} AY !W2hG a r&c@a]tRtRtoRtRRRRR]3RltRtRtVt R#)rci>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 NcTVf6\P!\P4PR,pVf\PpVf\PpVf\PpW0nW@nWnW nW`n WPn RVn ^Vn R#)NgUUUUUU?) r*finfor5epsrCrOrPrMrNrrQf0_normrs)selfrMrNrOrPrQrs&&&&&&&r'__init__TerminationCondition.__init__Ks| =HHRZZ(,,6E >VVF =FFE >VVF       r&cV;P^, unVPV4pVPV4pVPV4pVPfW@nV^8Xd^#VPe!^VPVP8,#\ W@P 8*;'dSW@P , VP8*;'d,W`P8*;'dW`P, V8*4#)r) rsrrrQintrMrNrOrP)rfr.r}f_normx_normdx_norms&&&& r'rkTerminationCondition.checkcs !11))B- << !L Q; 99 23 3Fjj(;;{{*dll:;;::-::#KK/69< s) "$d40<.__array__s'$$'A%%IJJ||~%r&) rrnmatvecrmatvecrsolvematmatrr:r2NN)itemsrisetattrhasattr)rkwnamesnamevaluers&, r'rJacobian.__init__s_888:KD  #>  8>> 1 KK  2r&)r2rVr:Nr) r r!r"r#r$rrrrnrfr%rrs@r'rr~s)#J& %" r&rcJa]tRtRtoRtRt]R4t]R4tRt Vt R#)riaG A simple wrapper that inverts the Jacobian using the `solve` method. .. legacy:: class See the newer, more consistent interfaces in :mod:`scipy.optimize`. Parameters ---------- jacobian : Jacobian The Jacobian to invert. Attributes ---------- shape Matrix dimensions (M, N) dtype Data type of the matrix. cWnVPVnVPVn\ VR4'dVP Vn\ VR4'dVP VnR#R#)rfrN)rtrrrnrrfrr)rrts&&r'rInverseJacobian.__init__sR nn oo 8W % %!DJ 8X & &#??DL 'r&c.VPP#r))rtr:rs&r'r:InverseJacobian.shape}}"""r&c.VPP#r))rtr2rs&r'r2InverseJacobian.dtyperr&)rtrrrfrnN) r r!r"r#r$rpropertyr:r2r%rrs@r'rrs7(+####r&c aa\PPPo\ S\ 4'dS#\ P!S4'd\S\ 4'dS!4#\ S\P4'dSP^8d \R4h\P!\P!S44oSP^,SP^,8wd \R4h\ V3RlV3RlRV3RllRV3RllSP SPR7#\PP#S4'dpSP^,SP^,8wd \R4h\ V3R lV3R lRVV3R llRVV3R llSP SPR7#\%SR 4'd\%SR4'dv\%SR4'dd\ \'SR4\'SR4SP(\'SR4\'SR4\'SR4SP SPR7#\+S4'd!VV3RlR\ 4pV!4#\ S\,4'd;\/\0\2\4\6\8\:\<R7S,!4#\?R4h)z= Convert given object to one suitable for use as a Jacobian. zarray must have rank <= 2zarray must be squarec<\SV4#r))rrDJs&r'asjacobian..s Qr&cL<\SP4PV4#r))rconjTrs&r'rrs#affhjj!*.JacicWnR#r)r-rs&&&r'rnasjacobian..Jac.updatesr&c<S!VP4p\V\P4'd \ W14#\ P PV4'd S!W14#\R4hzUnknown matrix type) r. isinstancer*ndarrayrscipysparseissparserirrDrYmrrs&&& r'rasjacobian..Jac.solvesVdffIa,, ;&\\**1--"1=($%:;;r&c<S!VP4p\V\P4'd \ W!4#\ P PV4'd W!,#\R4hr) r.rr*rrrrrrirrDrrs&& r'rasjacobian..Jac.matvec"sSdffIa,,q9$\\**1--5L$%:;;r&cR<S!VP4p\V\P4'd%\ VP 4P V4#\PPV4'd"S!VP 4P V4#\R4hr) r.rr*rrrrrrrrirs&&& r'rasjacobian..Jac.rsolve+spdffIa,, Q//\\**1--"1668::q11$%:;;r&cR<S!VP4p\V\P4'd%\ VP 4P V4#\PPV4'd"VP 4P V,#\R4hr) r.rr*rrrrrrrrirs&& r'rasjacobian..Jac.rmatvec4smdffIa,,qvvxzz1--\\**1--668::>)$%:;;r&r-Nr) r r!r"r#rnrrrrr%r)rrrs@r'Jacrs-  < < < < < < >r&c2a]tRtRtoRtRtRtRtVtR#)GenericBroydeniMc\PWW#4W nWn\ VR4'dWVP fG\ V4pV'd+R\\ V4^4,V, VnR#RVnR#R#R#)alphaN?rZ)rrflast_flast_xrr!rr,)rr<f0rVnormf0s&&&& r'rfGenericBroyden.setupNsit*  4 ! !djj&8"XF T"Xq!11F:   '9 !r&c\hr)rrr.rr}dfrdf_norms&&&&&&&r'_updateGenericBroyden._update\rr&c W P, pWP, pVPWWC\V4\V44W nWnR#r))r#r$r,r)rr.rr*r}s&&& r'rnGenericBroyden.update_s< _ _ Q248T"X6  r&)r!r#r$N) r r!r"r#rfr,rnr%rrs@r'rrMs !"r&rca]tRtRtoRtRt]R4t]R4tRt Rt RRlt RR lt R t RR ltR tRtRtRRltRtVtR #) LowRankMatrixigz 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. cTWn.Vn.VnW nW0nRVnR#r))r!csrrr2 collapsed)rr!rr2s&&&&r'rLowRankMatrix.__init__rs&  r&c\.ROVR,V.,4wrEpW,p\W#4F!wrV!W4p V!WVPV 4pK# V#)axpyNrN)r7scaldotc)r ziprh) rDr!r3rr7r9r:wcdas &&&& r'_matvecLowRankMatrix._matveczsZ)*B*,R&A3,8D IKDAQ AQ1661%A r&c  \V4^8Xd W, #\RR.VR,V.,4wrEV^,pV\P!\V4VPR7,p\ V4F5wr\ V4F!wrWxV 3;;,V!W4, uu&K# K7 \P !\V4VPR7p \ V4FwrV!W4W&K W,p \W|4p W, p \W,4FwrV!WV PV)4p K V #)Evaluate w = M^-1 vr7r:r8r1) lenr r*identityr2 enumeratezerosrr;rh)rDr!r3rr7r:c0Air>jr=qr<qcs&&&& r'_solveLowRankMatrix._solves r7a<7N$VV$4bfslC  U BKKBrxx8 8bMDA!" A#$q*$&" HHSWBHH -bMDA:AD"  !K GZEAQ166B3'A r&c VPe"\P!VPV4#\P WP VP VP4#)zEvaluate w = M v)r4r*rr1r@r!r3rrrDs&&r'rLowRankMatrix.matvecsB >> %66$..!, ,$$Q DGGTWWEEr&c $VPe:\P!VPPP 4V4#\ P V\P!VP4VPVP4#)zEvaluate w = M^H v) r4r*rrrr1r@r!rr3rQs&&r'rLowRankMatrix.rmatvecs\ >> %66$..**//115 5$$Q (;TWWdggNNr&c VPe\VPV4#\PWPVP VP 4#)rC)r4rr1rNr!r3rrs&&&r'rLowRankMatrix.solves> >> %+ +##Azz477DGGDDr&c VPe/\VPPP4V4#\P V\ P!VP4VPVP4#)zEvaluate w = M^-H v) r4rrrr1rNr*r!rr3rs&&&r'rLowRankMatrix.rsolvesX >> %))..0!4 4##Arwwtzz':DGGTWWMMr&cvVPe?V;PVR,VR,P4,, unR#VPPV4VPPV4\ VP4VP 8dVP4R#R#)NNNNNNr[)r4rr3appendrrDrhcollapse)rr=r>s&&&r'r]LowRankMatrix.appendss >> % NNai!F)..*:: :N  q q twws&&& r'rLowRankMatrix.__array__s   MM99>~O%& (   MM99=nN%& ( >> %>> ! ZZ DFF$**= =)DA F)AfINN,, ,B* r&c r\P!V\R7VnRVnRVnRVnR#)z0Collapse the low-rank matrix to a full-rank one.)rgN)r*rBr r4r3rr!rs&r'r^LowRankMatrix.collapses)$^< r&c VPeR#V^8gQh\VP4V8dVPRVPRR#R#)z8 Reduce the rank of the matrix by dropping all vectors. Nr[r4rDr3rrranks&&r'restart_reduceLowRankMatrix.restart_reducesG >> % axx tww<$    r&c VPeR#V^8gQh\VP4V8dVP^VP^K6R#)z; Reduce the rank of the matrix by dropping oldest vectors. Nrirjs&&r' simple_reduceLowRankMatrix.simple_reducesG >> % axx$''lT!  "r&c VPeR#TpVeTpM V^, pVP'd'\V\VP^,44p\ ^\WC^, 44p\VP4pWS8dR#\ P !VP4Pp\ P !VP4Pp\VRR7wrx\WhPP44p\VRR7wrp \V\V 44p\W{PP44p\V4FOp VRV 3,P4VPV &VRV 3,P4VPV &KQ VPVR1VPVR1R#)al 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 Neconomic)modeF) full_matricesr[)r4r3rlrDr,r*rBrrr rrr rrjrg) rmax_rank to_retainrrLrCDRUSWHks &&& r' svd_reduceLowRankMatrix.svd_reducesY: >> %    AAA 777As4771:'A 3qA#;  L 5  HHTWW    HHTWW   !*% 3388: q.b 3r7O 4499; qA1Q3DGGAJ1Q3DGGAJ GGABK GGABKr&)r!r4r3rr2rrrr))r r!r"r#r$r staticmethodr@rNrrrrr]rr^rlror~r%rrs@r'r1r1gsr6F O E N "  ??r&r1a 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). broyden_paramsc`a]tRtRtoRtR RltRtRtRRltRt RR lt R t R t R t VtR#)riKa 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://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]) Ncaa\PS4VSnRSnVf\P pVSn\V\4'dR oMVR,oV^,pV^, 3S,oVR8XdVV3RlSn R#VR8XdVV3RlSn R#VR8XdVV3RlSn R#\RV R 24h) N:rNNr c8<SPP!S!#r))rdr~ reduce_paramsrsr'r'BroydenFirst.__init__..s477#5#5}#Er&simplec8<SPP!S!#r))rdrorsr'rrs477#8#8-#Hr&restartc8<SPP!S!#r))rdrlrsr'rrs477#9#9=#Ir&zUnknown rank reduction method ''r) rrr!rdr*rCrurr_reduceri)rr!reduction_methodrursf&&&@r'rBroydenFirst.__init__s%   vvH  & , ,M,R0M/2 !A-7 u $EDL  )HDL  *IDL>?O>PPQRS Sr&c\PWW#4\VP)VP^,VP 4VnR#)rN)rrfr1r!r:r2rdrs&&&&r'rfBroydenFirst.setups4Ta. TZZ]DJJGr&c,\VP4#r))rrdrs&r'rBroydenFirst.todenses477|r&c(VPPV4p\P!V4P 4'gMVP VP VPVP4VPPV4#V#r)) rdrr*r@rArfr$r#rV)rrrYrs&&& r'rBroydenFirst.solves_ GGNN1 {{1~!!## JJt{{DKK ;77>>!$ $r&c8VPPV4#r))rdrrrs&&r'rBroydenFirst.matvecsww}}Qr&c8VPPV4#r))rdrrrrYs&&&r'rBroydenFirst.rsolveswwq!!r&c8VPPV4#r))rdrrs&&r'rBroydenFirst.rmatvecsww~~a  r&cVP4VPPV4pW0PPV4, pV\ WG4, p VPP W4R#r))rrdrrrr] rr.rr}r*rr+rDr=r>s &&&&&&& r'r,BroydenFirst._updatesO  GGOOB  # # R O qr&)rdrr!ru)NrNr)r r!r"r#r$rrfrrrrrr,r%rrs@r'rrKs:2hT2H "!r&c*a]tRtRtoRtRtRtVtR#)ria 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]) cVP4TpW0PPV4, pWv^,, p VPPW4R#N)rrdrr]rs &&&&&&& r'r,BroydenSecond._updates@   # #  N qr&rN)r r!r"r#r$r,r%rrs@r'rrs0dr&rcDa]tRtRtoRtR RltR RltRtRtRt Vt R#) riaz 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]) Nc~\PV4WnW0n.Vn.VnRVnW nR#r))rrr!Mr}r*rw0)rr!rrs&&&&r'rAnderson.__init__Bs4%  r&cZVP)V,p\VP4pV^8XdV#\P!WAP R7p\ V4F"p\VPV,V4WV&K$ \VPV4p\ T4FSpY7T,TPT,TPTPT,,,,, pKU T# \d TPRTPRTu#i;i)rr1r[) r!rDr}r*emptyr2rjrr*rr?r ) rrrYr}rdf_fr}rrs &&& r'rAnderson.solveKsjj[] L 6Ixx)qA4771:q)DG $&&$'EqA (DGGAJDGGAJ)>>? ?B    I  sD'D*)D*c V)VP, p\VP4pV^8XdV#\P!W1P R7p\ V4F"p\VPV,V4WE&K$ \P!W33VP R7p\ V4Fp\ V4Fp\VPV,VPV,4WgV3&Wx8XgK=VP^8wgKPWgV3;;,\VPV,VPV,4VP^,,VP,,uu&K K \Wd4p \ V4FSp W)V ,VPV ,VPV ,VP, ,,, pKU V#)rr1) r!rDr}r*rr2rjrr*rr) rrr}rrr}brJrKrrs && r'rAnderson.matvecbsOR ] L 6Ixx)qA4771:q)DG HHaV177 +qA1Xdggaj$''!*5A#6dgglcFd4771:twwqz:477A:EdjjPPF aqA (DGGAJDJJ)>>? ?B r&c8VP^8XdR#VPPV4VPPV4\ VP4VP8d9VPP ^4VPP ^4K\\ VP4p\ P!Ww3VPR7p\V4Fqp \W4F_p W8XdVP^,p M^p ^V ,\VPV ,VPV ,4,WV 3&Ka Ks V\ P!V^4PP4, pWnR#)rNr1)rr}r]r*rDpopr*rGr2rjrrtriurrr?) rr.rr}r*rr+rr?rJrKwds &&&&&&& r'r,Anderson._updateys  66Q;  r r$''lTVV# GGKKN GGKKN L HHaV177 +qA1[6!BBB$TWWQZ <<A# ! RWWQ]__ ! ! ##r&)rr?r!r*r}rr)Nrr) r r!r"r#r$rrrr,r%rrs@r'rrs$+L..r&rc`a]tRtRtoRtR RltRtRRltRtRRlt R t R t R t R t VtR#)ria 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]) Nc<\PV4WnR#r)rrr!rr!s&&r'rDiagBroyden.__init__% r&c\PWW#4\P!VP^,3^VP , VP R7VnR#rr1N)rrfr*fullr:r!r2r>rs&&&&r'rfDiagBroyden.setups=Ta.$**Q-)1tzz>Lr&c*V)VP, #r)r>rs&&&r'rDiagBroyden.solverDFF{r&c*V)VP,#r)rrs&&r'rDiagBroyden.matvecrr&cFV)VPP4, #r)r>rrs&&&r'rDiagBroyden.rsolverDFFKKM!!r&cFV)VPP4,#r)rrs&&r'rDiagBroyden.rmatvecrr&cD\P!VP)4#r))r*diagr>rs&r'rDiagBroyden.todenseswwwr&cV;PW@PV,,V,V^,, ,unR#rrr)s&&&&&&&r'r,DiagBroyden._updates( 2r >2%gqj00r&)r!r>r)rr r!r"r#r$rrfrrrrrr,r%rrs@r'rrs9&PM"" 11r&rcZa]tRtRtoRtR RltR RltRtR RltRt R t R t R t Vt R#)ria 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. Nc<\PV4WnR#r)rrs&&r'rLinearMixing.__init__rr&c*V)VP,#r)r!rs&&&r'rLinearMixing.solver$**}r&c*V)VP, #r)rrs&&r'rLinearMixing.matvecrr&cRV)\P!VP4,#r)r*rr!rs&&&r'rLinearMixing.rsolver"''$**%%%r&cRV)\P!VP4, #r)rrs&&r'rLinearMixing.rmatvecrr&c\P!\P!VP^,RVP, 44#)r)r*rrr:r!rs&r'rLinearMixing.todenses,wwrwwtzz!}bm<==r&cR#r)rr)s&&&&&&&r'r,LinearMixing._updaterr&rr)r)r r!r"r#r$rrrrrrr,r%rrs@r'rrs2,&&>  r&rc`a]tRtRtoRtR RltRtRRltRtRRlt R t R t R t R t VtR#)ri a 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 NcV\PV4WnW nRVnR#r))rrr!alphamaxbeta)rr!rs&&&r'rExcitingMixing.__init__$s!%   r&c\PWW#4\P!VP^,3VP VP R7VnR#r)rrfr*rr:r!r2rrs&&&&r'rfExcitingMixing.setup*s9Ta.GGTZZ],djj K r&c*V)VP,#r)rrs&&&r'rExcitingMixing.solve.r$))|r&c*V)VP, #r)rrs&&r'rExcitingMixing.matvec1rr&cFV)VPP4,#r)rrrs&&&r'rExcitingMixing.rsolve4r$)).."""r&cFV)VPP4, #r)rrs&&r'rExcitingMixing.rmatvec7rr&cP\P!RVP, 4#)rr)r*rrrs&r'rExcitingMixing.todense:swwr$))|$$r&c&W P,^8pVPV;;,VP, uu&VPVPV(&\P!VP^VP VPR7R#)r)outN)r#rr!r*clipr)rr.rr}r*rr+incrs&&&&&&& r'r,ExcitingMixing._update=sZ}q  $4::%:: 4%  1dmm;r&)r!rr)NrZrrrs@r'rr s84 L##%<>> 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]) Nc W@nWn\\PP P \PP P\PP P\PP P\PP P\PP PR7PW"4Vn \W0PR7VnVP\PP PJd<W0PR&^VPR&VPPR^4EMVVP\PP P \PP P \PP P39dVPPR^4MVP\PP PJdWPPR&^VPR&VPPR.4VPPRR4VPPR R 4VPPR^4\#VP4P$Uu.uFpVR9gK VNK ppVP'4FwrV P)R 4'g\+R V 24hV R ,V9d[\-V R ,V^R7p V 'dRV ^, R2p MRp \.P0!RV RV R2V ,^\2R7KWPV R ,&K R#uupi))bicgstabgmreslgmrescgsminrestfqmr)rvrrrvatolouter_kouter_vprepend_outer_vTstore_outer_AvFinner_zUnknown parameter :NN)rz Did you mean 'z'?zOption 'z#' is invalid for the inner method: zO. It will be ignored.Please check inner method documentation for valid options.)racategoryN)rargskwargs)preconditionerrrrrrrrrrr r getmethod method_kw setdefaultgcrotmkr parametersr startswithrirrbrc UserWarning) rrr inner_maxiterinner_Mr rr}valid_inner_paramskeyrinner_param_suggestionssuggestion_msgs &&&&&&, r'rKrylovJacobian.__init__s% \\((11,,%%++<<&&-- ##''<<&&--,,%%++ c&! m7J7JK ;;%,,--33 3(5NN9 %()DNN9 % NN % %fa 0 [[U\\0088"\\0099"\\004466 NN % %fa 0 [[ELL//66 6(/NN9 %()DNN9 % NN % %i 4 NN % %&7 > NN % %&6 > NN % %fa 0!-88 8!22 A8  ((*JC>>(++ #5cU!;<<2w00*;CG?+A'+(7)@)C(DB'HN&(N se#FvhOQQ%% !( &+NN3r7 #7%  s = O Oc\VP4P4p\VP4P4pVP\^V4,\^V4, VnR#)rN)rr<r,r%romega)rmxmfs& r'_update_diff_step KrylovJacobian._update_diff_stepsO \    \   ZZ#a*,s1bz9 r&c\V4pV^8Xd ^V,#VPV, pVPVPW1,,4VP, V, p\ P !\ P!V44'g<\ P !\ P!V44'd \R4hV#)rz$Function returned non-finite results) rr'rVr<r%r*rAr@ri)rrDnvscrs&& r'rKrylovJacobian.matvecs !W 7Q3J ZZ"_ YYtww~ & 0B 6vvbkk!n%%"&&Q*@*@CD Dr&cRVP9d-VP!VPV3/VPBwr4V#VP!VPV3RV/VPBwr4V#)rtol)rrop)rrhsrYsolrs&&& r'rKrylovJacobian.solve s_ T^^ # DGGSCDNNCIC  DGGSMsMdnnMIC r&cWnW nVP4VPe<\ VPR4'dVPP W4R#R#R#)Nrn)r<r%r*rrrn)rr.rs&&&r'rnKrylovJacobian.updatesW      *t**H55##**106 +r&c\PWW#4WnW n\P P PV4VnVPf7\P!VP4PR,Vn VP4VPe=\!VPR4'dVPPWV4R#R#R#)Nrfr")rrfr<r%rrraslinearoperatorr2rr*rr2rr*rr)rr.rrVs&&&&r'rfKrylovJacobian.setupst(,,%%66t< :: !''*..48DJ      *t**G44##))!55 +r&)r%rrr'r2rrr<)NrN r) r r!r"r#r$rr*rrrnrfr%rrs@r'rrHs1cJK,Z: 166r&c \VP4pVwr4rVrxp \\V\ V4)RV44p RP V U U u.uF wrV RV : 2NK up p 4p V 'd RV ,p RP V U U u.uF wrV RV 2NK up p 4pV'd VR,pV'd\ RV 24hRpV\W VPVR7,p/pVP\44\VV4VV,pVPVn \V4V#uup p iuup p i)z 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, =zUnexpected signature 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) )rrjackwkw)_getfullargspecrlistr;rDjoinrirr rnglobalsexecr$rK)rr?rrvarargsvarkwdefaults kwonlyargs kwdefaults_rr}rDkw_strkwkw_strwrappernsrVs&& r'_nonlin_wrapperrP0s*  -I@I=D5JA #dCM>?+X6 7F YY81#Qqe 8 9F yy8QCq*89Hd?0 <==G$s||"*,,G BIIgi" d8D;;DL TN K;99s D; E ) rrrrrrrrrrr) rNFNNNNNNrXNFT)rXg:0yE>r)?rrorbnumpyr*rrr scipy.linalgrrrr r r scipy.sparse.linalgr scipy.sparser scipy._lib._utilr rrA _linesearchrrrdifflibr__all__ Exceptionrr/r6r>rErstriprIrKrrmrcrrrerr1rrrrrrrrPrrrrrrrrr&r'r[s $$??'+FC% J I    (P a1 h/ O"d *Z9<9<@EEP$#$#NY?@X4IIX * + 0m>m`9L9@U~UxA1.A1H+ >+ \8<^8<~a6Xa6P)X :| 4 :} 5 :x 0~|< m[9  !1>B@ r&