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Editing: _minpack_py.cpython-311.pyc
� d�cĕ � � � d dl Z ddlmZ d dlZd dlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZ d dlmZ d dlmZmZmZmZm Z d dl m Z m Z d dl m Z! dd l"m#Z#m$Z$m%Z% dd l&m'Z' dd l(m)Z) d d l*m+Z+ ej, Z,g d �Z- d'd�Z. d(d�Z/ d)d�Z0g d�Z1g d�Z2 d*d�Z3d� Z4d� Z5d� Z6dddd ej ej fddfdd �d �Z7d+d �Z8d � Z9d!� Z:d"� Z;d,d&�Z<dS )-� N� )�_minpack)� atleast_1d�dot�take�triu�shape�eye� transpose�zeros�prod�greater�asarray�inf�finfo�inexact� issubdtype�dtype)�linalg)�svd�cholesky�solve_triangular� LinAlgError�inv)�_asarray_validated� _lazywhere)�getfullargspec_no_self)�OptimizeResult�_check_unknown_options�OptimizeWarning)� least_squares)�prepare_bounds)�Bounds)�fsolve�leastsq� fixed_point� curve_fitc � � t ||d |� f|z � � � }|��t |� � |k r�|d dk rt |� � dk r|d dk rt |� � S | �d|�d�}t |dd � � } | r |d| z z }n|dz }|d|�d t |� � �d�z }t |� � �t |j t � � r|j } nt t � � } t |� � | fS ) Nr r zA: there is a mismatch between the input and output shape of the 'z ' argument�__name__z '%s'.�.zShape should be z but it is ) r r �len�getattr� TypeErrorr r r �float) �checker�argname�thefunc�x0�args� numinputs� output_shape�res�msg� func_name�dts �</usr/lib/python3/dist-packages/scipy/optimize/_minpack_py.py� _check_funcr; s$ � � �W�W��:�I�:�� 0�4� 7�9� :� :�C�� �u�S�z�z�\�'A�'A� ��O�q� � ��<� � �1�$�$� ��?�a�'�'� ��:�:�%�29�'�'�7�7�7�D�C� ���T�:�:�I�� ��x�)�+�+����s� �� �C� � � �u�S�z�z�z�z�R� R�C��C�.�.� ��#�)�W�%�%� � �Y��� �5�\�\�� ��:�:�r�>�� � �J�� P>�d c � �� ||||| | | d�} t | ||fd|i| ���|r@�d } t �fd�dD � � � � }�d |d< | |�d �d fS �d }�d }|d k rt |� � �|d k rn.|d v rt j |t � � nt |� � ��d S ) a� Find the roots of a function. Return the roots of the (non-linear) equations defined by ``func(x) = 0`` given a starting estimate. Parameters ---------- func : callable ``f(x, *args)`` A function that takes at least one (possibly vector) argument, and returns a value of the same length. x0 : ndarray The starting estimate for the roots of ``func(x) = 0``. args : tuple, optional Any extra arguments to `func`. fprime : callable ``f(x, *args)``, optional A function to compute the Jacobian of `func` with derivatives across the rows. By default, the Jacobian will be estimated. full_output : bool, optional If True, return optional outputs. col_deriv : bool, optional Specify whether the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation). xtol : float, optional The calculation will terminate if the relative error between two consecutive iterates is at most `xtol`. maxfev : int, optional 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`. band : tuple, optional If set to a two-sequence containing the number of sub- and super-diagonals within the band of the Jacobi matrix, the Jacobi matrix is considered banded (only for ``fprime=None``). epsfcn : float, optional A suitable step length for the forward-difference approximation of the Jacobian (for ``fprime=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, optional A parameter determining the initial step bound (``factor * || diag * x||``). Should be in the interval ``(0.1, 100)``. diag : sequence, optional N positive entries that serve as a scale factors for the variables. Returns ------- x : ndarray The solution (or the result of the last iteration for an unsuccessful call). infodict : dict A dictionary of optional outputs with the keys: ``nfev`` number of function calls ``njev`` number of Jacobian calls ``fvec`` function evaluated at the output ``fjac`` the orthogonal matrix, q, produced by the QR factorization of the final approximate Jacobian matrix, stored column wise ``r`` upper triangular matrix produced by QR factorization of the same matrix ``qtf`` the vector ``(transpose(q) * fvec)`` ier : int An integer flag. Set to 1 if a solution was found, otherwise refer to `mesg` for more information. mesg : str If no solution is found, `mesg` details the cause of failure. See Also -------- root : Interface to root finding algorithms for multivariate functions. See the ``method='hybr'`` in particular. Notes ----- ``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms. Examples -------- Find a solution to the system of equations: ``x0*cos(x1) = 4, x1*x0 - x1 = 5``. >>> import numpy as np >>> from scipy.optimize import fsolve >>> def func(x): ... return [x[0] * np.cos(x[1]) - 4, ... x[1] * x[0] - x[1] - 5] >>> root = fsolve(func, [1, 1]) >>> root array([6.50409711, 0.90841421]) >>> np.isclose(func(root), [0.0, 0.0]) # func(root) should be almost 0.0. array([ True, True]) )� col_deriv�xtol�maxfev�band�eps�factor�diag�jac�xc 3 �N �K � | ] }|�v �|�� |� � fV � � d S �N)�get)�.0�kr6 s �r: � <genexpr>zfsolve.<locals>.<genexpr>� sF �� � � � O� O��Q�#�X�X� ����� � �O�EM�X�X�X�O� Or< )�nfev�njev�fjac�r�qtf�fun�fvec�status�messager r )� � � � )� _root_hybr�dictr- �warnings�warn�RuntimeWarning)�funcr2 r3 �fprime� full_outputrA rB rC rD �epsfcnrF rG �optionsrI �inforW r7 r6 s @r: r$ r$ 0 s" �� �V &�� �� � �� � �G� �T�2�t� ;� ;�� ;�7� ;� ;�C�� � ��H��� O� O� O� O� A�O� O� O� O� O���5�z��V� ��$��H� �s�9�~�5�5��X� ���)�n�� �Q�;�;��C�.�.� � �q�[�[� � �|� #� #� �M�#�~� .� .� .� .��C�.�.� ��3�x�r< c �L � t | � � |} t |� � � � � }t |� � } t |t � � s|f}t dd| ||| | f� � \ }}| �t |� � j } |}|�B|�d\ }}n |dd� \ }}|dk rd| dz z }t j | ||d||||| | | � � }nBt dd |||| | | f� � |dk rd | dz z }t j | |||d|||| | � � }|d |d }}d d d|z d|z dddd�}|d }|� d� � |d< t ||dk |�� � }|� |� � || |d< n# t $ r |d |d< Y nw xY w|S )a� Find the roots of a multivariate function using MINPACK's hybrd and hybrj routines (modified Powell method). Options ------- col_deriv : bool Specify whether the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation). xtol : float The calculation will terminate if the relative error between two consecutive iterates is at most `xtol`. maxfev : 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`. band : tuple If set to a two-sequence containing the number of sub- and super-diagonals within the band of the Jacobi matrix, the Jacobi matrix is considered banded (only for ``fprime=None``). eps : float A suitable step length for the forward-difference approximation of the Jacobian (for ``fprime=None``). If `eps` 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 the interval ``(0.1, 100)``. diag : sequence N positive entries that serve as a scale factors for the variables. r$ rb N)���ri rY r �� r rc r? ���z'Improper input parameters were entered.zThe solution converged.z8The number of calls to function has reached maxfev = %d.�Wxtol=%f is too small, no further improvement in the approximate solution is possible.zuThe iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.zjThe iteration is not making good progress, as measured by the improvement from the last ten iterations.zAn error occurred.)r r rY rZ r[ r\ �unknownrV rU )rI �successrW rX rm )r r �flattenr+ � isinstance�tupler; r rE r �_hybrd�_hybrj�popr �update�KeyError)rb r2 r3 rH rA rB rC rD rE rF rG �unknown_optionsre �nr r �Dfun�ml�mu�retvalrI rW �errorsrg �sols r: r] r] � s. � �L �?�+�+�+� �F� ���� � � � �B� �B���A� �d�E� "� "� ��w�� �x���r�4��Q�D�I�I�L�E�5� �~��u� � �!�� �D� �|� �<� �F�B����"�1�"�X�F�B�� �Q�;�;��A��E�]�F�� ��r�4��D�&�!#�R����?� ?��� �H�h��b�$��A�q�6�B�B�B� �a�K�K��A��E�]�F�� ��t�R��q�!*�D�&�&�$�H� H�� �q� �6�"�:�v�A�:�*�)�+1�2�!�#'�(�*�$� .� /� /�F� �!�9�D��(�(�6�"�"�D��K� �1�v��{�F� C� C� C�C��J�J�t����+� �� ��I���� � +� +� +� � �*��I����+���� �Js �= F � F!� F!�r rY rZ r[ )r\ � � � � c � � t |� � � � � }t |� � } t |t � � s|f}t dd| ||| � � \ }}|d }| |k rt d| � d|� �� � �| �t |� � j } |�-| dk rd| dz z } t j | ||||||| | | | � � }n^|rt dd |||| | |f� � nt dd |||| || f� � | dk rd | dz z } t j | ||||||||| | | � � }d t gd |z dgd |z dgd||fz dgd|z dgd| z t gd|z t gd|z t gd|z t gd� }|d }|r�d}|t v r�|d d dz }t |� � } t t |d d � � d| �dd�f � � }t! j d|f� � } ||� � \ }}|dk rt% d|� �� � �|� � � ||< ||j z }n# t$ t f$ r Y nw xY w|d |f|dd� z || d |fz S |t* v r't- j || d t0 � � n)|dk r# || d || d � � �|d |fS )a: Minimize the sum of squares of a set of equations. :: x = arg min(sum(func(y)**2,axis=0)) y Parameters ---------- func : callable Should take at least one (possibly length ``N`` vector) argument and returns ``M`` floating point numbers. It must not return NaNs or fitting might fail. ``M`` must be greater than or equal to ``N``. x0 : ndarray The starting estimate for the minimization. args : tuple, optional Any extra arguments to func are placed in this tuple. Dfun : callable, optional A function or method to compute the Jacobian of func with derivatives across the rows. If this is None, the Jacobian will be estimated. full_output : bool, optional non-zero to return all optional outputs. col_deriv : bool, optional non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation). ftol : float, optional Relative error desired in the sum of squares. xtol : float, optional Relative error desired in the approximate solution. gtol : float, optional Orthogonality desired between the function vector and the columns of the Jacobian. maxfev : int, optional The maximum number of calls to the function. If `Dfun` is provided, then the default `maxfev` is 100*(N+1) where N is the number of elements in x0, otherwise the default `maxfev` is 200*(N+1). epsfcn : float, optional A variable used in determining a suitable step length for the forward- difference approximation of the Jacobian (for Dfun=None). Normally the actual step length will be sqrt(epsfcn)*x If epsfcn is less than the machine precision, it is assumed that the relative errors are of the order of the machine precision. factor : float, optional A parameter determining the initial step bound (``factor * || diag * x||``). Should be in interval ``(0.1, 100)``. diag : sequence, optional N positive entries that serve as a scale factors for the variables. Returns ------- x : ndarray The solution (or the result of the last iteration for an unsuccessful call). cov_x : ndarray The inverse of the Hessian. `fjac` and `ipvt` are used to construct an estimate of the Hessian. A value of None indicates a singular matrix, which means the curvature in parameters `x` is numerically flat. To obtain the covariance matrix of the parameters `x`, `cov_x` must be multiplied by the variance of the residuals -- see curve_fit. infodict : dict a dictionary of optional outputs with the keys: ``nfev`` The number of function calls ``fvec`` The function evaluated at the output ``fjac`` A permutation of the R matrix of a QR factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated. ``ipvt`` An integer array of length N which defines a permutation matrix, p, such that fjac*p = q*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix. ``qtf`` The vector (transpose(q) * fvec). mesg : str A string message giving information about the cause of failure. ier : int An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise, the solution was not found. In either case, the optional output variable 'mesg' gives more information. See Also -------- least_squares : Newer interface to solve nonlinear least-squares problems with bounds on the variables. See ``method='lm'`` in particular. Notes ----- "leastsq" is a wrapper around MINPACK's lmdif and lmder algorithms. cov_x is a Jacobian approximation to the Hessian of the least squares objective function. This approximation assumes that the objective function is based on the difference between some observed target data (ydata) and a (non-linear) function of the parameters `f(xdata, params)` :: func(params) = ydata - f(xdata, params) so that the objective function is :: min sum((ydata - f(xdata, params))**2, axis=0) params The solution, `x`, is always a 1-D array, regardless of the shape of `x0`, or whether `x0` is a scalar. Examples -------- >>> from scipy.optimize import leastsq >>> def func(x): ... return 2*(x-3)**2+1 >>> leastsq(func, 0) (array([2.99999999]), 1) r% rb r z+Improper input: func input vector length N=z- must not exceed func output vector length M=Nrj r ry r? zImproper input parameters.zTBoth actual and predicted relative reductions in the sum of squares are at most %fzAThe relative error between two consecutive iterates is at most %fz�Both actual and predicted relative reductions in the sum of squares are at most %f and the relative error between two consecutive iterates is at most %fzhThe cosine of the angle between func(x) and any column of the Jacobian is at most %f in absolute valuez4Number of calls to function has reached maxfev = %d.zOftol=%f is too small, no further reduction in the sum of squares is possible.rl zbgtol=%f is too small, func(x) is orthogonal to the columns of the Jacobian to machine precision.) r r rY rZ r[ r\ r� r� r� rk �ipvtrR �trtriztrtri returned info )r ro r+ rp rq r; r- r rE r �_lmdif�_lmder� ValueError�LEASTSQ_SUCCESSr r r �get_lapack_funcsr �copy�T�LEASTSQ_FAILUREr_ r` ra )rb r2 r3 ry rd rA �ftolrB �gtolrC re rF rG rx r r �mr| r} rg �cov_x�permrS �inv_triu�invR� trtri_infos r: r% r% s� � �~ ���� � � � �B� �B���A� �d�E� "� "� ��w�� �y�&�$��D�!�D�D�L�E�5� �a��A��1�u�u�� G�a� G� G�CD�G� G� H� H� H� �~��u� � �!�� �|� �Q�;�;��!�a�%�[�F�� ��r�4��d�D�!%�v�v�v�t�E� E��� � F� � �6�4��T�1�q�!�f� E� E� E� E� � �6�4��T�1�q�!�f� E� E� E� �Q�;�;��A��E�]�F�� ��t�R��{�!*�D�$��f�!'��/� /�� /� �:�<�>B�C�DH�J�+�-1�2�37�9�/� 37�� �>� @D�E�#�%)�*�+/�1�!�#)�*�+5�7�:�<@�A� � �=�?C�D� � � �!%�&�'1�3�)4� 4�F�2 �"�:�D�� ��� �?� "� "� �!�9�V�$�q�(�D��D� � �A��Y�v�a�y�� 0�1�1�"�1�"�a�a�a�%�8�9�9�A� �.�w���=�=�H� �#+�8�A�;�;� ��j� ��?�?�%�&I�Z�&I�&I�J�J�J� !�Y�Y�[�[��T� � �t�v� ���� ��,� � � ��� �����q� �5�!�F�1�R�4�L�0�F�4�L��O�T�3J�J�J� �?� "� "� �M�&��,�q�/�>� :� :� :� :� �Q�Y�Y�!�&��,�q�/�&��,�q�/�2�2� 2��a�y�$�� s �7AH? �?I�Ic �Z � ���� ��� ��fd�}n �j dk r � ���fd�}n� ���fd�}|S )Nc � �� ��g| �R � �z S rK r= )�paramsrb �xdata�ydatas ���r: � func_wrappedz _wrap_func.<locals>.func_wrapped� s �� ��4��'��'�'�'�%�/� /r< r c �" �� � ��g| �R � �z z S rK r= �r� rb � transformr� r� s ����r: r� z _wrap_func.<locals>.func_wrapped� s% �� � ���U� 4�V� 4� 4� 4�u� <�=� =r< c �< �� t � ��g| �R � �z d�� � S �NT��lower)r r� s ����r: r� z _wrap_func.<locals>.func_wrapped s0 �� �#�I�t�t�E�/C�F�/C�/C�/C�e�/K�SW�X�X�X� Xr< ��ndim)rb r� r� r� r� s ```` r: � _wrap_funcr� � s� ����� ��� 0� 0� 0� 0� 0� 0� 0� 0� � �1� � � >� >� >� >� >� >� >� >� >� Y� Y� Y� Y� Y� Y� Y� Y� �r< c �R � ��� ��� �fd�}n�j dk r� ��fd�}n� ��fd�}|S )Nc � �� ��g| �R � S rK r= )r� rH r� s ��r: � jac_wrappedz _wrap_jac.<locals>.jac_wrapped s �� ��3�u�&�v�&�&�&� &r< r c �h �� �d d �t j f t j ��g| �R � � � z S rK )�np�newaxisr �r� rH r� r� s ���r: r� z _wrap_jac.<locals>.jac_wrapped s9 �� � �Q�Q�Q�� �]�+�b�j���U�9L�V�9L�9L�9L�.M�.M�M� Mr< c �Z �� t �t j ��g| �R � � � d�� � S r� )r r� r r� s ���r: r� z _wrap_jac.<locals>.jac_wrapped s5 �� �#�I�r�z�#�#�e�:M�f�:M�:M�:M�/N�/N�VZ�[�[�[� [r< r� )rH r� r� r� s ``` r: � _wrap_jacr� s� ���� ��� '� '� '� '� '� '� '� � �1� � � N� N� N� N� N� N� N� N� \� \� \� \� \� \� \� �r< c � � t j | � � }t j | � � }t j |� � }||z }d| | || z z ||< || z }| | dz ||< | |z }|| dz ||< |S )N� �?r )r� � ones_like�isfinite)�lb�ub�p0� lb_finite� ub_finite�masks r: �_initialize_feasibler� s� � � � �b� � �B�� �B���I�� �B���I� �y� �D��b��h��D��)�*�B�t�H� � �z� !�D��$�x�!�|�B�t�H� �:� � !�D��$�x�!�|�B�t�H� �Ir< FT)rd c �$ � |�Kt | � � } | j } t | � � dk rt d� � �t | � � dz }nt j |� � }|j }t |t � � r|j |j }}nt ||� � \ }}|�t ||� � }t j |t j k |t j k z � � }|�|rd}nd}|dk r|rt d� � �|rt j |t � � }nt j |t � � }t |t$ t& t j f� � r7|rt j |t � � }nt j |t � � }|j dk rt d � � �|��t j |� � }|j |j fk rd |z }n]|j |j |j fk r5 t- |d � � � }n3# t. $ r}t d � � |�d}~ww xY wt d� � �d}t1 | |||� � }t3 | � � rt5 | ||� � } n | �|dk rd} d| v rt d� � �|dk r�|j dk r%||j k rt7 d|� d|j � �� � �t9 ||f| dd�| ��}|\ }}}}}t |d � � }t j |d dz � � } |dvrt= d|z � � ��n1d| vr| � dd� � | d< tA ||f| ||d�| ��}|j! st= d|j" z � � �tG |j$ |j% �� � }|j& }|j" }t |j% � � }d|j' z } |j( }tS |j* d � � � \ } } } t j+ t � � j, t[ |j* j � � z | d z } | | | k } | d| j � } t j. | j/ | dz z | � � }d }!|�Ota t |� � t |� � ft � � � }|�1 t � � d }!n<|s:||j k r| ||j z z }"||"z }n |�1 t � � d }!|!rte j3 d th � � � | r|||||fS ||fS )!a�$ Use non-linear least squares to fit a function, f, to data. Assumes ``ydata = f(xdata, *params) + eps``. Parameters ---------- f : callable The model function, f(x, ...). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments. xdata : array_like The independent variable where the data is measured. Should usually be an M-length sequence or an (k,M)-shaped array for functions with k predictors, and each element should be float convertible if it is an array like object. ydata : array_like The dependent data, a length M array - nominally ``f(xdata, ...)``. p0 : array_like, optional Initial guess for the parameters (length N). If None, then the initial values will all be 1 (if the number of parameters for the function can be determined using introspection, otherwise a ValueError is raised). sigma : None or M-length sequence or MxM array, optional Determines the uncertainty in `ydata`. If we define residuals as ``r = ydata - f(xdata, *popt)``, then the interpretation of `sigma` depends on its number of dimensions: - A 1-D `sigma` should contain values of standard deviations of errors in `ydata`. In this case, the optimized function is ``chisq = sum((r / sigma) ** 2)``. - A 2-D `sigma` should contain the covariance matrix of errors in `ydata`. In this case, the optimized function is ``chisq = r.T @ inv(sigma) @ r``. .. versionadded:: 0.19 None (default) is equivalent of 1-D `sigma` filled with ones. absolute_sigma : bool, optional If True, `sigma` is used in an absolute sense and the estimated parameter covariance `pcov` reflects these absolute values. If False (default), only the relative magnitudes of the `sigma` values matter. The returned parameter covariance matrix `pcov` is based on scaling `sigma` by a constant factor. This constant is set by demanding that the reduced `chisq` for the optimal parameters `popt` when using the *scaled* `sigma` equals unity. In other words, `sigma` is scaled to match the sample variance of the residuals after the fit. Default is False. Mathematically, ``pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)`` check_finite : bool, optional If True, check that the input arrays do not contain nans of infs, and raise a ValueError if they do. Setting this parameter to False may silently produce nonsensical results if the input arrays do contain nans. Default is True. bounds : 2-tuple of array_like or `Bounds`, optional Lower and upper bounds on parameters. Defaults to no bounds. There are two ways to specify the bounds: - Instance of `Bounds` class. - 2-tuple of array_like: Each element of the tuple must be either an array with the length equal to the number of parameters, or a scalar (in which case the bound is taken to be the same for all parameters). Use ``np.inf`` with an appropriate sign to disable bounds on all or some parameters. method : {'lm', 'trf', 'dogbox'}, optional Method to use for optimization. See `least_squares` for more details. Default is 'lm' for unconstrained problems and 'trf' if `bounds` are provided. The method 'lm' won't work when the number of observations is less than the number of variables, use 'trf' or 'dogbox' in this case. .. versionadded:: 0.17 jac : callable, string or None, optional Function with signature ``jac(x, ...)`` which computes the Jacobian matrix of the model function with respect to parameters as a dense array_like structure. It will be scaled according to provided `sigma`. If None (default), the Jacobian will be estimated numerically. String keywords for 'trf' and 'dogbox' methods can be used to select a finite difference scheme, see `least_squares`. .. versionadded:: 0.18 full_output : boolean, optional If True, this function returns additioal information: `infodict`, `mesg`, and `ier`. .. versionadded:: 1.9 **kwargs Keyword arguments passed to `leastsq` for ``method='lm'`` or `least_squares` otherwise. Returns ------- popt : array Optimal values for the parameters so that the sum of the squared residuals of ``f(xdata, *popt) - ydata`` is minimized. pcov : 2-D array The estimated covariance of popt. The diagonals provide the variance of the parameter estimate. To compute one standard deviation errors on the parameters use ``perr = np.sqrt(np.diag(pcov))``. How the `sigma` parameter affects the estimated covariance depends on `absolute_sigma` argument, as described above. If the Jacobian matrix at the solution doesn't have a full rank, then 'lm' method returns a matrix filled with ``np.inf``, on the other hand 'trf' and 'dogbox' methods use Moore-Penrose pseudoinverse to compute the covariance matrix. infodict : dict (returned only if `full_output` is True) a dictionary of optional outputs with the keys: ``nfev`` The number of function calls. Methods 'trf' and 'dogbox' do not count function calls for numerical Jacobian approximation, as opposed to 'lm' method. ``fvec`` The function values evaluated at the solution. ``fjac`` A permutation of the R matrix of a QR factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated. Method 'lm' only provides this information. ``ipvt`` An integer array of length N which defines a permutation matrix, p, such that fjac*p = q*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix. Method 'lm' only provides this information. ``qtf`` The vector (transpose(q) * fvec). Method 'lm' only provides this information. .. versionadded:: 1.9 mesg : str (returned only if `full_output` is True) A string message giving information about the solution. .. versionadded:: 1.9 ier : int (returnned only if `full_output` is True) An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise, the solution was not found. In either case, the optional output variable `mesg` gives more information. .. versionadded:: 1.9 Raises ------ ValueError if either `ydata` or `xdata` contain NaNs, or if incompatible options are used. RuntimeError if the least-squares minimization fails. OptimizeWarning if covariance of the parameters can not be estimated. See Also -------- least_squares : Minimize the sum of squares of nonlinear functions. scipy.stats.linregress : Calculate a linear least squares regression for two sets of measurements. Notes ----- Users should ensure that inputs `xdata`, `ydata`, and the output of `f` are ``float64``, or else the optimization may return incorrect results. With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm through `leastsq`. Note that this algorithm can only deal with unconstrained problems. Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to the docstring of `least_squares` for more information. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.optimize import curve_fit >>> def func(x, a, b, c): ... return a * np.exp(-b * x) + c Define the data to be fit with some noise: >>> xdata = np.linspace(0, 4, 50) >>> y = func(xdata, 2.5, 1.3, 0.5) >>> rng = np.random.default_rng() >>> y_noise = 0.2 * rng.normal(size=xdata.size) >>> ydata = y + y_noise >>> plt.plot(xdata, ydata, 'b-', label='data') Fit for the parameters a, b, c of the function `func`: >>> popt, pcov = curve_fit(func, xdata, ydata) >>> popt array([2.56274217, 1.37268521, 0.47427475]) >>> plt.plot(xdata, func(xdata, *popt), 'r-', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) Constrain the optimization to the region of ``0 <= a <= 3``, ``0 <= b <= 1`` and ``0 <= c <= 0.5``: >>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5])) >>> popt array([2.43736712, 1. , 0.34463856]) >>> plt.plot(xdata, func(xdata, *popt), 'g--', ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.legend() >>> plt.show() NrY z-Unable to determine number of fit parameters.r �trf�lmzQMethod 'lm' only works for unconstrained problems. 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Given a function of one or more variables and a starting point, find a fixed point of the function: i.e., where ``func(x0) == x0``. Parameters ---------- func : function Function to evaluate. x0 : array_like Fixed point of function. args : tuple, optional Extra arguments to `func`. xtol : float, optional Convergence tolerance, defaults to 1e-08. maxiter : int, optional Maximum number of iterations, defaults to 500. method : {"del2", "iteration"}, optional Method of finding the fixed-point, defaults to "del2", which uses Steffensen's Method with Aitken's ``Del^2`` convergence acceleration [1]_. The "iteration" method simply iterates the function until convergence is detected, without attempting to accelerate the convergence. References ---------- .. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80 Examples -------- >>> import numpy as np >>> from scipy import optimize >>> def func(x, c1, c2): ... return np.sqrt(c1/(x+c2)) >>> c1 = np.array([10,12.]) >>> c2 = np.array([3, 5.]) >>> optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2)) array([ 1.4920333 , 1.37228132]) TF)r � iteration)� as_inexact)r r )rb r2 r3 rB r� r� r� s r: r& r& � sA � �T �E�2�2�6�:�I� �B�4� 0� 0� 0�B� �t�R��t�W�i� H� H�Hr< rK ) r= Nr r r> r NNr? N) r= Nr r> r NNr? N) r= Nr r r> r> r� r Nr? 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