a j @s&ddlZddlZddlZddlZddlmZmZmZddlm Z m Z m Z m Z m Z mZddlZddlZddlmZddlmZddlmZddlmZmZgd ZGd d d eZd d ZddZddZddZ e!d"d"dZ#ddZ$dHddZ%e$e%dId!d"Z&Gd#d$d$Z'Gd%d&d&Z(Gd'd(d(Z)d)d*Z*Gd+d,d,e(Z+Gd-d.d.Z,d/"e#d0<Gd1d2d2e+Z-Gd3d4d4e-Z.Gd5d6d6e+Z/Gd7d8d8e+Z0Gd9d:d:e+Z1Gd;d<dd>e(Z3d?d@Z4e4dAe-Z5e4dBe.Z6e4dCe/Z7e4dDe1Z8e4dEe0Z9e4dFe2Z:e4dGe3Z;dS)JN)asarraydotvdot)normsolveinvqrsvd LinAlgError)get_blas_funcs)copy_if_needed)getfullargspec_no_self)scalar_search_wolfe1scalar_search_armijo) broyden1broyden2anderson linearmixing diagbroydenexcitingmixing newton_krylov BroydenFirstKrylovJacobianInverseJacobian NoConvergencec@seZdZdZdS)rz\Exception raised when nonlinear solver fails to converge within the specified `maxiter`.N)__name__ __module__ __qualname____doc__r r 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")r _doc_parts)objr r r!_set_docusr<krylovFarmijoTc sh| dur tn| } t|||| || d}tfdd}}t|tj}||}t|}t|}| | |||dur|dur|d}nd|j d}| durd} n | d urd} | d vrt 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,r2flatten)zFr0r r!funcsznonlin_solve..funcrdTr>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.)rrM)nitZfunstatussuccessmessage)r(TerminationConditionr,rDr#Z full_liker6r asjacobiansetupcopysize ValueErrorrangecheckminr_nonlin_line_searchupdater%sysstdoutwriteflushrr2 iteration) rGr0jacobianrCverbosemaxiterr?r@rArBZtol_normZ line_searchcallbackZ full_outputZraise_exception conditionrHr'dxFxFx_normgammaZeta_maxZ eta_tresholdetanrOrKsZ Fx_norm_newZeta_Ainfor rFr! nonlin_solvezs-       ro:0yE>{Gz?c sdg|gt|dgttdfdd fdd}|dkrxt|dd |d \}} } n&|d krtdd |d \}} |durd }||dkr̈d}n}t|} ||| fS)NrrMTcsT|dkrdS|}|}t|d}|rP|d<|d<|d<|S)NrrM)r9)rmstoreZxtr8p)rgrHtmp_Fxtmp_phitmp_sr'r r!phis   z _nonlin_line_search..phics0t|d}||dd||S)NrF)rr)abs)rmds)rwrdiffs_normr r!derphi#sz#_nonlin_line_search..derphirJrq)Zxtolaminr>)r}rL)T)rrr) rHr'rhrgZ search_typerzZsminr|rmZphi1Zphi0rir ) rgrHrwrzr{rtrurvr'r!r[s,      r[c@s.eZdZdZdddddefddZddZdS)rRz 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|durttjjd}|dur(tj}|dur6tj}|durDtj}||_||_||_||_||_ ||_ d|_ d|_ dS)NgUUUUUU?r) r#finfor+epsr6rArBr?r@rrCf0_normra)selfr?r@rArBrCrr r r!__init__Js zTerminationCondition.__init__cCs|jd7_||}||}||}|jdur<||_|dkrHdS|jdurbd|j|jkSt||jko||j|jko||jko||j|kS)NrrrM) rarrrCintr?r@rArB)rfr'rgZf_normZx_normdx_normr r r!rYbs         zTerminationCondition.check)rrrrr(rrYr r r r!rR=s   rRc@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 cKs^gd}|D]4\}}||vr,td||durt||||qt|drZddd}dS)N) rr\matvecrmatvecrsolveZmatmattodenser.r*zUnknown keyword argument %srcSs|durtd||S)Nz`dtype` must be None, was )rWr)rr*rUr r r! __array__sz$Jacobian.__init__..__array__)NN)itemsrWsetattrhasattr)rkwnamesnamevaluerr r r!rs  zJacobian.__init__cCst|Sr")rrr r r!aspreconditionerszJacobian.aspreconditionerrcCstdSr"NotImplementedErrorrr8rKr r r!rszJacobian.solvecCsdSr"r rr'rGr r r!r\szJacobian.updatecCs:||_|j|jf|_|j|_|jjtjur6|||dSr")rHrVr.r* __class__rTrr\rr'rGrHr r r!rTs zJacobian.setupN)r) rrrrrrrr\rTr r r r!r}s % rc@s,eZdZddZeddZeddZdS)rcCs>||_|j|_|j|_t|dr(|j|_t|dr:|j|_dS)NrTr)rbrrr\rrTrr)rrbr r r!rs  zInverseJacobian.__init__cCs|jjSr")rbr.rr r r!r.szInverseJacobian.shapecCs|jjSr")rbr*rr r r!r*szInverseJacobian.dtypeN)rrrrpropertyr.r*r r r r!rs   rc stjjjttrStr2ttr2Stt j rj dkrPt dt t jdjdkr|t dtfddfdddfd d d fd d jjd Stjr"jdjdkrt d tfd dfddd!fdd d"fdd jjd StdrtdrtdrttdtdjtdtdtdjjdStrGfdddt}|SttrttttttttdStddS)#zE Convert given object to one suitable for use as a Jacobian. rMzarray must have rank <= 2rrzarray must be squarecs t|Sr")rr7Jr r!zasjacobian..cstj|Sr")rconjTr7rr r!rrcs t|Sr")rr8rKrr r!rrcstj|Sr")rrrrrr r!rr)rrrrr*r.zmatrix must be squarecs|Sr"r r7rr r!rrcsj|Sr"rrr7rr r!rrcs |Sr"r rrspsolver r!rrcsj|Sr"rrrr r!rrr.r*rrrrr\rT)rrrrr\rTr*r.csLeZdZddZd fdd ZfddZdfdd Zfd d Zd S)zasjacobian..JaccSs ||_dSr"r&rr r r!r\szasjacobian..Jac.updatercsB|j}t|tjr t||Stj|r6||StddSNzUnknown matrix type) r' isinstancer#ndarrayrscipysparseissparserWrr8rKmrr r!rs      zasjacobian..Jac.solvecs@|j}t|tjr t||Stj|r4||StddSr) r'rr#rrrrrrWrr8rrr r!r s     zasjacobian..Jac.matveccsN|j}t|tjr&t|j|Stj |rB|j|St ddSr) r'rr#rrrrrrrrWrrr r!rs    zasjacobian..Jac.rsolvecsL|j}t|tjr&t|j|Stj |r@|j|St ddSr) r'rr#rrrrrrrrWrrr r!rs    zasjacobian..Jac.rmatvecN)r)r)rrrr\rrrrr rr r!Jacs   r)rrrrrrr=z#Cannot convert object to a JacobianN)r)r)r)r) rrlinalgrrrinspectisclass issubclassr#rndimrWZ atleast_2drr.r*rrr/rcallablestrdictr BroydenSecondAnderson DiagBroyden LinearMixingExcitingMixingr TypeError)rrr rr!rSsf          $  ' rSc@s$eZdZddZddZddZdS)GenericBroydencCs`t||||||_||_t|dr\|jdur\t|}|rVdtt|d||_nd|_dS)Nalpha?rrL)rrTlast_flast_xrrrr%)rr0f0rHZnormf0r r r!rT9szGenericBroyden.setupcCstdSr"rrr'rrgdfrdf_normr r r!_updateGszGenericBroyden._updatec Cs@||j}||j}|||||t|t|||_||_dSr")rrrr)rr'rrrgr r r!r\Js   zGenericBroyden.updateN)rrrrTrr\r r r r!r8src@seZdZdZddZeddZeddZdd Zd d Z dd dZ dddZ ddZ d 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")rcsryrlr* collapsed)rrrlr*r r r!r]s zLowRankMatrix.__init__c Cs\tgd|dd|g\}}}||}t||D]"\}} || |} ||||j| }q4|S)N)axpyscaldotcr)r ziprV) r8rrryrrrwcdar r r!_matveces  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 vrrrNrr)) lenr r#identityr* enumeratezerosrrrV)r8rrryrrZc0AirjrqrZqcr r r!_solveos"   zLowRankMatrix._solvecCs.|jdurt|j|St||j|j|jS)zEvaluate w = M vN)rr#rrrrrryrr8r r r!rs zLowRankMatrix.matveccCs:|jdurt|jj|St|t|j|j|j S)zEvaluate w = M^H vN) rr#rrrrrrryrrr r r!rs zLowRankMatrix.rmatvecrcCs,|jdurt|j|St||j|j|jS)rN)rrrrrrryrr r r!rs  zLowRankMatrix.solvecCs8|jdurt|jj|St|t|j|j|j S)zEvaluate w = M^-H vN) rrrrrrr#rryrrr r r!rs zLowRankMatrix.rsolvecCsp|jdur<|j|dddf|dddf7_dS|j||j|t|j|jkrl|dSr")rrrappendryrrVcollapse)rrrr r r!rs .  zLowRankMatrix.appendNcCs|durtjd|ddd|dur|jdurdS|dksJt|j|kr:|jd=|jd=qdS)zK Reduce the rank of the matrix by dropping oldest vectors. Nrrrr r r! simple_reduces   zLowRankMatrix.simple_reducec Cs6|jdurdS|}|dur |}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 NrMrrZeconomic)modeF)Z full_matrices)rrrZrr%r#r5rryrrrr rrXrU) rmax_rankZ to_retainrsrrCDRUSZWHkr r r! svd_reduces0    zLowRankMatrix.svd_reduce)r)r)NN)N)rrrrr staticmethodrrrrrrrrrrrrr r r r!rRs         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_|dur$tj}|_t|tr:dn|dd|d}|df|dkrvfdd_ n@|dkrfdd_ n&|d krfd d_ n t d |dS) Nr rrr cs jjSr")rrr Z reduce_paramsrr r!r}rz'BroydenFirst.__init__..simplecs jjSr")rrr rr r!rrrcs jjSr")rrr rr r!rrz"Unknown rank reduction method '%s') rrrrr#r6rrr_reducerW)rrZreduction_methodrr rr!rls(   zBroydenFirst.__init__cCs.t||||t|j |jd|j|_dS)Nr)rrTrrr.r*rrr r r!rTszBroydenFirst.setupcCs t|jSr")rrrr r r!rszBroydenFirst.todensercCs>|j|}t|s:||j|j|j|j|S|Sr") rrr#r3r4rTrrrH)rrrKrr r r!rs   zBroydenFirst.solvecCs |j|Sr")rrrrr r r!rszBroydenFirst.matveccCs |j|Sr")rrrrrKr r r!rszBroydenFirst.rsolvecCs |j|Sr")rrrr r r!rszBroydenFirst.rmatvecc CsD||j|}||j|}|t||} |j|| dSr")rrrrrr rr'rrgrrrr8rrr r r!rs  zBroydenFirst._update)NrN)r)r) rrrrrrTrrrrrrr r r r!r6s5   rc@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|| dSNrM)rrrrrr r r!rs  zBroydenSecond._updateN)rrrrrr r r r!rs2rc@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]) NrqcCs2t|||_||_g|_g|_d|_||_dSr")rrrMrgrrjw0)rrrrr r r!r/s zAnderson.__init__rc Cs|j |}t|j}|dkr"|Stj||jd}t|D]}t|j||||<q:zt |j |}Wn.t y|jdd=|jdd=|YS0t|D]*}||||j||j|j|7}q|SNrr)) rrrgr#emptyr*rXrrrrr ) rrrKrgrldf_frrjrr r r!r8s         (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)rM) rrrgr#rr*rXrrrr) rrrgrlrrbrrrjrr r r!rOs"     :  (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)rMr)rrgrrrpopr#rr*rXrrZtriurrr) rr'rrgrrrrlrrrwdr r r!rfs"       *zAnderson._update)Nrqr)r)rrrrrrrrr r r r!rs 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"rrrrrr r r!rs zDiagBroyden.__init__cCs6t||||tj|jdfd|j|jd|_dS)Nrrr))rrTr#fullr.rr*rrr r r!rTszDiagBroyden.setuprcCs | |jSr"rrr r r!rszDiagBroyden.solvecCs | |jSr"r rr r r!rszDiagBroyden.matveccCs| |jSr"rrrr r r!rszDiagBroyden.rsolvecCs| |jSr"r rr r r!rszDiagBroyden.rmatveccCst|j Sr")r#diagrrr r r!rszDiagBroyden.todensecCs(|j||j|||d8_dSrr rr r r!rszDiagBroyden._update)N)r)r rrrrrrTrrrrrrr r r r!rs(   rc@sNeZdZdZdddZdddZdd Zdd d Zd d ZddZ ddZ dS)ra Find a root of a function, using a scalar Jacobian approximation. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. Parameters ---------- %(params_basic)s alpha : float, optional The Jacobian approximation is (-1/alpha). %(params_extra)s See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='linearmixing'`` in particular. NcCst|||_dSr"rrr r r!rs zLinearMixing.__init__rcCs | |jSr"rrr r r!rszLinearMixing.solvecCs | |jSr"r rr r r!rszLinearMixing.matveccCs| t|jSr"r#rrrr r r!rszLinearMixing.rsolvecCs| t|jSr"rrr r r!rszLinearMixing.rmatveccCstt|jdd|jS)Nr)r#r rr.rrr r r!rszLinearMixing.todensecCsdSr"r rr r r!rszLinearMixing._update)N)r)r) rrrrrrrrrrrr r r r!rs   rc@sVeZdZdZdddZddZdd d Zd d Zdd dZddZ ddZ ddZ dS)ra Find a root of a function, using a tuned diagonal Jacobian approximation. The Jacobian matrix is diagonal and is tuned on each iteration. .. warning:: This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem. See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method='excitingmixing'`` in particular. Parameters ---------- %(params_basic)s alpha : float, optional Initial Jacobian approximation is (-1/alpha). alphamax : float, optional The entries of the diagonal Jacobian are kept in the range ``[alpha, alphamax]``. %(params_extra)s NrLcCs t|||_||_d|_dSr")rrralphamaxbeta)rrrr r r!rs zExcitingMixing.__init__cCs2t||||tj|jdf|j|jd|_dSr)rrTr#rr.rr*rrr r r!rTszExcitingMixing.setuprcCs | |jSr"rrr r r!rszExcitingMixing.solvecCs | |jSr"rrr r r!rszExcitingMixing.matveccCs| |jSr"rrrr r r!r!szExcitingMixing.rsolvecCs| |jSr"rrr r r!r$szExcitingMixing.rmatveccCstd|jS)Nr)r#r rrr r r!r'szExcitingMixing.todensecCsL||jdk}|j||j7<|j|j|<tj|jd|j|jddS)Nr)out)rrrr#Zclipr)rr'rrgrrrincrr r r!r*szExcitingMixing._update)NrL)r)rr r r r r!rs   rc@sDeZdZdZdddZdd Zd d Zdd dZddZddZ dS)ra Find a root of a function, using Krylov approximation for inverse Jacobian. This method is suitable for solving large-scale problems. Parameters ---------- %(params_basic)s rdiff : float, optional Relative step size to use in numerical differentiation. method : 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 tjjjur||jd<d|jd<|jddn|j tjjjtjjjtjjj fvr|jddn^|j tjjjur"||jd<d|jd<|jd g|jd d |jd d |jdd|D]4\}}|dsJtd|||j|dd<q*dS)N)bicgstabgmresrcgsminrestfqmr)rdrrrrdZatolrouter_kZouter_vZprepend_outer_vTZstore_outer_AvFZinner_zUnknown parameter %s)preconditionerrzrrrrrrrrrrgetmethod method_kw setdefaultZgcrotmkr startswithrW) rrzr"Z inner_maxiterZinner_Mrrkeyrr r r!rsB        zKrylovJacobian.__init__cCs<t|j}t|j}|jtd|td||_dS)Nr)rxr0r%rrzomega)rZmxmfr r r!_update_diff_stepsz KrylovJacobian._update_diff_stepcCslt|}|dkrd|S|j|}||j|||j|}tt|shtt|rhtd|S)Nrz$Function returned non-finite results) rr'rHr0rr#r4r3rW)rr8nvscrr r r!rs  zKrylovJacobian.matvecrcCsLd|jvr(|j|j|fi|j\}}n |j|j|fd|i|j\}}|S)NZrtol)r#r"op)rrhsrKZsolrnr r r!rs  zKrylovJacobian.solvecCs<||_||_||jdur8t|jdr8|j||dS)Nr\)r0rr)r rr\)rr'rr r r!r\s   zKrylovJacobian.updatecCs|t||||||_||_tjj||_|j durJt |j j d|_ ||jdurxt|jdrx|j|||dS)NrrT)rrTr0rrrrZaslinearoperatorr,rzr#r~r*rr)r r)rr'rrHr r r!rTs   zKrylovJacobian.setup)NrrNr)r) rrrrrr)rrr\rTr r r r!r5se /  rcCst|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|qS=r .0rr8r r r! rz#_nonlin_wrapper..cSsg|]\}}|d|qSr.r r0r r r!r2rzUnexpected 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)_getfullargspecrlistrrjoinrWrrr\globalsexecrr<)rr3 signatureargsvarargsvarkwdefaults kwonlyargs kwdefaults_kwargsZkw_strZkwkw_strwrappernsrHr r r!_nonlin_wrappers,     rDrrrrrrr) r=NFNNNNNNr>NFT)r>rprq)sr      4  -@H`M r@D.?K,