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Editing: _root.cpython-311.pyc
� d�ctn � �� � d Z dgZddlZg d�ZddlmZ ddlmZm Z m Z ddl m Z m Z dd lmZ dd lmZ d d �Zd� Z d d�Z d d�Zd� Zd� Zd� Zd� Zd� Zd� Z d � Z dS ) zp Unified interfaces to root finding algorithms. Functions --------- - root : find a root of a vector function. �root� N) �hybr�lm�broyden1�broyden2�anderson� linearmixing� diagbroyden�excitingmixing�krylov�df-sane)�warn� )� MemoizeJac�OptimizeResult�_check_unknown_options)� _root_hybr�leastsq)� _root_df_sane)�_nonlin� r c � � t |t � � s|f}|� � � }|�i }|� |dv rt d|z t � � t |� � s,|dv r(t |� � rt | � � } | j }nd}|��t |� � }|dv r|� d|� � n�|dv r|� d|� � nz|dv rv|� d|� � |� dt j � � |� dt j � � |� d t j � � |d k rt | |f||d �|��} nz|d k rt | |f||d �|��} nb|d k r"t ||� � t! | |f||d�|��} n:|dv r$t ||� � t# | |f||||d�|��} nt% d|z � � �| S )ar Find a root of a vector function. Parameters ---------- fun : callable A vector function to find a root of. x0 : ndarray Initial guess. args : tuple, optional Extra arguments passed to the objective function and its Jacobian. method : str, optional Type of solver. Should be one of - 'hybr' :ref:`(see here) <optimize.root-hybr>` - 'lm' :ref:`(see here) <optimize.root-lm>` - 'broyden1' :ref:`(see here) <optimize.root-broyden1>` - 'broyden2' :ref:`(see here) <optimize.root-broyden2>` - 'anderson' :ref:`(see here) <optimize.root-anderson>` - 'linearmixing' :ref:`(see here) <optimize.root-linearmixing>` - 'diagbroyden' :ref:`(see here) <optimize.root-diagbroyden>` - 'excitingmixing' :ref:`(see here) <optimize.root-excitingmixing>` - 'krylov' :ref:`(see here) <optimize.root-krylov>` - 'df-sane' :ref:`(see here) <optimize.root-dfsane>` jac : bool or callable, optional If `jac` is a Boolean and is True, `fun` is assumed to return the value of Jacobian along with the objective function. If False, the Jacobian will be estimated numerically. `jac` can also be a callable returning the Jacobian of `fun`. In this case, it must accept the same arguments as `fun`. tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. 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. For all methods but 'hybr' and 'lm'. options : dict, optional A dictionary of solver options. E.g., `xtol` or `maxiter`, see :obj:`show_options()` for details. Returns ------- sol : OptimizeResult The solution represented as a ``OptimizeResult`` object. Important attributes are: ``x`` the solution array, ``success`` a Boolean flag indicating if the algorithm exited successfully and ``message`` which describes the cause of the termination. See `OptimizeResult` for a description of other attributes. See also -------- show_options : Additional options accepted by the solvers Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is *hybr*. Method *hybr* uses a modification of the Powell hybrid method as implemented in MINPACK [1]_. Method *lm* solves the system of nonlinear equations in a least squares sense using a modification of the Levenberg-Marquardt algorithm as implemented in MINPACK [1]_. Method *df-sane* is a derivative-free spectral method. [3]_ Methods *broyden1*, *broyden2*, *anderson*, *linearmixing*, *diagbroyden*, *excitingmixing*, *krylov* are inexact Newton methods, with backtracking or full line searches [2]_. Each method corresponds to a particular Jacobian approximations. - Method *broyden1* uses Broyden's first Jacobian approximation, it is known as Broyden's good method. - Method *broyden2* uses Broyden's second Jacobian approximation, it is known as Broyden's bad method. - Method *anderson* uses (extended) Anderson mixing. - Method *Krylov* uses Krylov approximation for inverse Jacobian. It is suitable for large-scale problem. - Method *diagbroyden* uses diagonal Broyden Jacobian approximation. - Method *linearmixing* uses a scalar Jacobian approximation. - Method *excitingmixing* uses a tuned diagonal Jacobian approximation. .. warning:: The algorithms implemented for methods *diagbroyden*, *linearmixing* and *excitingmixing* may be useful for specific problems, but whether they will work may depend strongly on the problem. .. versionadded:: 0.11.0 References ---------- .. [1] More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom. 1980. User Guide for MINPACK-1. .. [2] C. T. Kelley. 1995. Iterative Methods for Linear and Nonlinear Equations. Society for Industrial and Applied Mathematics. <https://archive.siam.org/books/kelley/fr16/> .. [3] W. La Cruz, J.M. Martinez, M. Raydan. Math. Comp. 75, 1429 (2006). Examples -------- The following functions define a system of nonlinear equations and its jacobian. >>> import numpy as np >>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]] >>> def jac(x): ... return np.array([[1 + 1.5 * (x[0] - x[1])**2, ... -1.5 * (x[0] - x[1])**2], ... [-1.5 * (x[1] - x[0])**2, ... 1 + 1.5 * (x[1] - x[0])**2]]) A solution can be obtained as follows. >>> from scipy import optimize >>> sol = optimize.root(fun, [0, 0], jac=jac, method='hybr') >>> sol.x array([ 0.8411639, 0.1588361]) **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 ``method='krylov'`` solver: >>> from scipy import optimize >>> # 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 = np.zeros_like(P) ... d2y = np.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*np.cosh(P).mean()**2 >>> guess = np.zeros((nx, ny), float) >>> sol = optimize.root(residual, guess, method='krylov') >>> print('Residual: %g' % abs(residual(sol.x)).max()) Residual: 5.7972e-06 # may vary >>> import matplotlib.pyplot as plt >>> x, y = np.mgrid[0:1:(nx*1j), 0:1:(ny*1j)] >>> plt.pcolormesh(x, y, sol.x, shading='gouraud') >>> plt.colorbar() >>> plt.show() N)r r z#Method %s does not accept callback.�xtol)r �ftol�r r r r r r r �xatol�fatolr )�args�jacr r )r �callback)r r �_method� _callbackzUnknown solver %s)� isinstance�tuple�lowerr �RuntimeWarning�callable�boolr � derivative�dict� setdefault�np�infr � _root_leastsq�_warn_jac_unusedr �_root_nonlin_solve� ValueError) �fun�x0r �methodr �tolr �options�meth�sols �6/usr/lib/python3/dist-packages/scipy/optimize/_root.pyr r su � �b �d�E� "� "� ��w�� �<�<�>�>�D��������� 6� 6� � 2�V� ;� � � � � �C�=�=� �T�^� 3� 3� ��9�9� � �S�/�/�C��.�C�C��C� ���w�-�-�� �>� !� !� � � �v�s� +� +� +� +� �\� !� !� � � �v�s� +� +� +� +� � A� A� A� � � �v�s� +� +� +� � � �w��� /� /� /� � � �v�r�v� .� .� .� � � �w��� /� /� /� �v�~�~���b�@�t��@�@��@�@��� �� � ��C��C�$�C�C�C�7�C�C��� �� � ���f�%�%�%��C�� '�$�� '� '� %�'� '��� � =� =� =���f�%�%�%� ��b� ,�t��)-��,� ,�#*�,� ,��� �,�v�5�6�6�6� �J� c �@ � | �t d|�d�t � � d S d S )NzMethod z! does not use the jacobian (jac).)r r& )r r4 s r9 r/ r/ � s8 � � �� ��V�V�V� E� � � � � � � �r: �J�� P>� �d c �� � t | � � t | |||d|||||| | | �� � \ } }}}}t | |||dv ||� d� � �� � }|� |� � |S )a� Solve for least squares with Levenberg-Marquardt Options ------- col_deriv : bool non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation). ftol : float Relative error desired in the sum of squares. xtol : float Relative error desired in the approximate solution. gtol : float Orthogonality desired between the function vector and the columns of the Jacobian. maxiter : int The maximum number of calls to the function. If zero, then 100*(N+1) is the maximum where N is the number of elements in x0. epsfcn : float A suitable step length for the forward-difference approximation of the Jacobian (for Dfun=None). If epsfcn is less than the machine precision, it is assumed that the relative errors in the functions are of the order of the machine precision. factor : float A parameter determining the initial step bound (``factor * || diag * x||``). Should be in interval ``(0.1, 100)``. diag : sequence N positive entries that serve as a scale factors for the variables. T) r �Dfun� full_output� col_derivr r �gtol�maxfev�epsfcn�factor�diag)r � � � �fvec)�x�message�status�success�cov_xr2 )r r r �pop�update)r2 r3 r r rB r r rC �maxiter�epsrF rG �unknown_optionsrL rP �info�msg�ierr8 s r9 r. r. s� � �D �?�+�+�+� &�s�B�T��37�1:��,0�t�.5�c�.4�4� A� A� A� �A�u�d�C�� �1�c�#�!$� �!4�E� !�X�X�f� -� -� /� /� /�C� �J�J�t���� �Jr: F�armijoc �� � �� t |� � | }| }| }| }|}|�t � � }t j t j t j t j t j t j t j d�| }�r|r�� fd�}n �� fd�}n� }t j || |di |��|||||||| ||dd�� � \ }}t |�� � }|� |� � |S ) Nr c �" �� �| g��R � d S )Nr r �rL r r2 s ��r9 �fz _root_nonlin_solve.<locals>.fO s �� ��s�1�}�t�}�}�}�Q�'�'r: c � �� �| g��R � S )Nr r\ s ��r9 r] z _root_nonlin_solve.<locals>.fR s �� ��s�1�}�t�}�}�}�$r: TF) �jacobian�iter�verboserS �f_tol�f_rtol�x_tol�x_rtol�tol_norm� line_searchr rA �raise_exception)rL r ) r r* �nonlin� BroydenFirst� BroydenSecond�Anderson� LinearMixing� DiagBroyden�ExcitingMixing�KrylovJacobian� nonlin_solver rR )r2 r3 r r r" r! �nit�disprS r r r r rf rg � jac_optionsrU rb rc rd re ra r_ r] rL rV r8 s ` ` r9 r0 r0 4 sS ��� � �?�+�+�+� �E� �F� �E� �F��G����f�f� � "� /� "� 0� "�O� &� 3� %� 1�"(�"7� �/� � � ��H� � � � %� (� (� (� (� (� (� (� %� %� %� %� %� %� %� ���!�!�R�(�(�2I�2I�[�2I�2I�'*�G�*1��)/�u�)/�(�.9�+4�$�27�9� 9� 9�G�A�t� �1� � � �C��J�J�t���� �Jr: c � � dS )a Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. 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. Extra parameters: - ``to_retain`` 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). Examples -------- >>> def func(x): ... return np.cos(x) + x[::-1] - [1, 2, 3, 4] ... >>> from scipy import optimize >>> res = optimize.root(func, [1, 1, 1, 1], method='broyden1', tol=1e-14) >>> x = res.x >>> x array([4.04674914, 3.91158389, 2.71791677, 1.61756251]) >>> np.cos(x) + x[::-1] array([1., 2., 3., 4.]) Nr r r: r9 �_root_broyden1_docrv c s � �X �Dr: c � � dS )a. Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. 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. Extra parameters: - ``to_retain`` 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). Nr r r: r9 �_root_broyden2_docrx � � � �~ �Dr: c � � dS )a� Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. 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. Nr r r: r9 �_root_anderson_docr{ � s � �P �Dr: c � � dS )a� Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. alpha : float, optional initial guess for the jacobian is (-1/alpha). Nr r r: r9 �_root_linearmixing_docr} � � �F �Dr: c � � dS )a� Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. alpha : float, optional initial guess for the jacobian is (-1/alpha). Nr r r: r9 �_root_diagbroyden_docr� A r~ r: c � � dS )a> Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. 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]``. Nr r r: r9 �_root_excitingmixing_docr� f s � �L �Dr: c � � dS )a� Options ------- nit : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. disp : 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. ftol : float, optional Relative tolerance for the residual. If omitted, not used. fatol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. xtol : float, optional Relative minimum step size. If omitted, not used. xatol : 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. 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'. jac_options : dict, optional Options for the respective Jacobian approximation. 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_M : LinearOperator or InverseJacobian Preconditioner for the inner Krylov iteration. Note that you can use also inverse Jacobians as (adaptive) preconditioners. For example, >>> jac = BroydenFirst() >>> kjac = KrylovJacobian(inner_M=jac.inverse). 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. Nr r r: r9 �_root_krylov_docr� � ry r: )r r NNNN) r Nr r< r< r= r r= r> N)r NNNNFNNNNNNrY N) �__doc__�__all__�numpyr, � ROOT_METHODS�warningsr � _optimizer r r � _minpack_pyr r � _spectralr � r ri r r/ r. r0 rv rx r{ r} r� r� r� r r: r9 �<module>r� s� ��� � �(�� � � � �� � � � � � � � � � I� I� I� I� I� I� I� I� I� I� ,� ,� ,� ,� ,� ,� ,� ,� $� $� $� $� $� $� � � � � � � HL��c� c� c� c�L � � � )-�6A�AE�-� -� -� -�` .2�/3�59�?C�HL� -� -� -� -�^L � L � L �\? � ? � ? �B( � ( � ( �T# � # � # �J# � # � # �J& � & � & �P? � ? � ? � ? � ? r:
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