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_numdiff.cpython-35.opt-1.pyc
�B�W�R � @ s d Z d d l m Z d d l Z d d l m Z m Z m Z m Z m Z d d l m Z m Z e j e j � j Z d d � Z d d � Z d d � Z d d d � Z d d d e j e j f d f i d d � Z d d � Z d d � Z e j e j f f i d d � Z d S)z'Routines for numerical differentiation.� )�divisionN� )�issparse� csc_matrix� csr_matrix� coo_matrix�find� )� group_dense� group_sparsec C s] | d k r$ t j | d t �} n? | d k rW t j | � } t j | d t �} n t d � � t j | t j k | t j k @� r� | | f S| | } | j � } | | } | | } | d k rf| | } | | k | | k B} t j | � t j | | � k } | | | @d 9<| | k | @} | | | | | <| | k | @} | | | | | <n� | d k rS| | k | | k @} | | k | @} t j | | d | | | � | | <d | | <| | k | @} t j | | d | | | � | | <d | | <t j | | � | } | t j | � | k @} | | | | <d | | <| | f S) a� Adjust final difference scheme to the presence of bounds. Parameters ---------- x0 : ndarray, shape (n,) Point at which we wish to estimate derivative. h : ndarray, shape (n,) Desired finite difference steps. num_steps : int Number of `h` steps in one direction required to implement finite difference scheme. For example, 2 means that we need to evaluate f(x0 + 2 * h) or f(x0 - 2 * h) scheme : {'1-sided', '2-sided'} Whether steps in one or both directions are required. In other words '1-sided' applies to forward and backward schemes, '2-sided' applies to center schemes. lb : ndarray, shape (n,) Lower bounds on independent variables. ub : ndarray, shape (n,) Upper bounds on independent variables. Returns ------- h_adjusted : ndarray, shape (n,) Adjusted step sizes. Step size decreases only if a sign flip or switching to one-sided scheme doesn't allow to take a full step. use_one_sided : ndarray of bool, shape (n,) Whether to switch to one-sided scheme. Informative only for ``scheme='2-sided'``. z1-sided�dtypez2-sidedz(`scheme` must be '1-sided' or '2-sided'.r g �?TF���) �npZ ones_like�bool�absZ zeros_like� ValueError�all�inf�copy�maximumZminimum)�x0�hZ num_steps�scheme�lb�ub� use_one_sidedZh_totalZ h_adjustedZ lower_distZ upper_dist�xZviolatedZfittingZforwardZbackwardZcentralZmin_distZadjusted_central� r � /_numdiff.py�_adjust_scheme_to_bounds sH & ! ! r c C s� | d k rc | d k r% t d } n> | d k r> t d } n% | d k rW t d } n t d � � | d k j t � d d } | | t j d t j | � � S) Nz2-pointg �?z3-pointr � �csz(`method` must be '2-point' or '3-point'.r r g �?gUUUUUU�?)�EPSr �astype�floatr r r )�rel_stepr �methodZsign_x0r r r �_compute_absolute_step\ s r' c C sk d d � | D� \ } } | j d k r= t j | | j � } | j d k ra t j | | j � } | | f S)Nc S s% g | ] } t j | d t �� q S)r )r �asarrayr$ )�.0�br r r � <listcomp>l s z#_prepare_bounds.<locals>.<listcomp>r )�ndimr Zresize�shape)�boundsr r r r r r �_prepare_boundsk s r/ c C s> t | � r t | � } n' t j | � } | d k j t j � } | j d k r] t d � � | j \ } } | d k s� t j | � r� t j j | � } | j | � } n- t j | � } | j | f k r� t d � � | d d � | f } t | � rt | | | j | j � } n t | | | � } | j � | | <| S)a� Group columns of a 2-d matrix for sparse finite differencing [1]_. Two columns are in the same group if in each row at least one of them has zero. A greedy sequential algorithm is used to construct groups. Parameters ---------- A : array_like or sparse matrix, shape (m, n) Matrix of which to group columns. order : int, iterable of int with shape (n,) or None Permutation array which defines the order of columns enumeration. If int or None, a random permutation is used with `order` used as a random seed. Default is 0, that is use a random permutation but guarantee repeatability. Returns ------- groups : ndarray of int, shape (n,) Contains values from 0 to n_groups-1, where n_groups is the number of found groups. Each value ``groups[i]`` is an index of a group to which i-th column assigned. The procedure was helpful only if n_groups is significantly less than n. References ---------- .. [1] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. r r z`A` must be 2-dimensional.Nz `order` has incorrect shape.)r r r � atleast_2dr# Zint32r, r r- Zisscalar�randomZ RandomStateZ permutationr( r �indicesZindptrr r )�A�order�m�n�rng�groupsr r r � group_columnsv s& r9 z3-pointc sX | d k r t d | � � t j | � } | j d k rF t d � � t | | � \ } } | j | j k s | j | j k r� t d � � � � � f d d � } | d k r� | | � } n* t j | � } | j d k r� t d � � t j | | k | | k B� rt d � � t | | | � } | d k rUt | | d d | | � \ } } nB | d k r�t | | d d | | � \ } } n | d k r�d } | d k r�t | | | | | | � St | � r�t | � d k r�| \ } } n | } t | � } t | � rt | � } n t j | � } t j | � } t | | | | | | | | � Sd S)a� Compute finite difference approximation of the derivatives of a vector-valued function. If a function maps from R^n to R^m, its derivatives form m-by-n matrix called the Jacobian, where an element (i, j) is a partial derivative of f[i] with respect to x[j]. Parameters ---------- fun : callable Function of which to estimate the derivatives. The argument x passed to this function is ndarray of shape (n,) (never a scalar even if n=1). It must return 1-d array_like of shape (m,) or a scalar. x0 : array_like of shape (n,) or float Point at which to estimate the derivatives. Float will be converted to a 1-d array. method : {'3-point', '2-point'}, optional Finite difference method to use: - '2-point' - use the fist order accuracy forward or backward difference. - '3-point' - use central difference in interior points and the second order accuracy forward or backward difference near the boundary. - 'cs' - use a complex-step finite difference scheme. This assumes that the user function is real-valued and can be analytically continued to the complex plane. Otherwise, produces bogus results. rel_step : None or array_like, optional Relative step size to use. The absolute step size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `h` is ignored. If None (default) then step is selected automatically, see Notes. f0 : None or array_like, optional If not None it is assumed to be equal to ``fun(x0)``, in this case the ``fun(x0)`` is not called. Default is None. bounds : tuple of array_like, optional Lower and upper bounds on independent variables. Defaults to no bounds. Each bound must match the size of `x0` or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation. sparsity : {None, array_like, sparse matrix, 2-tuple}, optional Defines a sparsity structure of the Jacobian matrix. If the Jacobian matrix is known to have only few non-zero elements in each row, then it's possible to estimate its several columns by a single function evaluation [3]_. To perform such economic computations two ingredients are required: * structure : array_like or sparse matrix of shape (m, n). A zero element means that a corresponding element of the Jacobian identically equals to zero. * groups : array_like of shape (n,). A column grouping for a given sparsity structure, use `group_columns` to obtain it. A single array or a sparse matrix is interpreted as a sparsity structure, and groups are computed inside the function. A tuple is interpreted as (structure, groups). If None (default), a standard dense differencing will be used. Note, that sparse differencing makes sense only for large Jacobian matrices where each row contains few non-zero elements. args, kwargs : tuple and dict, optional Additional arguments passed to `fun`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)``. Returns ------- J : ndarray or csr_matrix Finite difference approximation of the Jacobian matrix. If `sparsity` is None then ndarray with shape (m, n) is returned. Although if m=1 it is returned as a gradient with shape (n,). If `sparsity` is not None, csr_matrix with shape (m, n) is returned. See Also -------- check_derivative : Check correctness of a function computing derivatives. Notes ----- If `rel_step` is not provided, it assigned to ``EPS**(1/s)``, where EPS is machine epsilon for float64 numbers, s=2 for '2-point' method and s=3 for '3-point' method. Such relative step approximately minimizes a sum of truncation and round-off errors, see [1]_. A finite difference scheme for '3-point' method is selected automatically. The well-known central difference scheme is used for points sufficiently far from the boundary, and 3-point forward or backward scheme is used for points near the boundary. Both schemes have the second-order accuracy in terms of Taylor expansion. Refer to [2]_ for the formulas of 3-point forward and backward difference schemes. For dense differencing when m=1 Jacobian is returned with a shape (n,), on the other hand when n=1 Jacobian is returned with a shape (m, 1). Our motivation is the following: a) It handles a case of gradient computation (m=1) in a conventional way. b) It clearly separates these two different cases. b) In all cases np.atleast_2d can be called to get 2-d Jacobian with correct dimensions. References ---------- .. [1] W. H. Press et. al. "Numerical Recipes. The Art of Scientific Computing. 3rd edition", sec. 5.7. .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. .. [3] B. Fornberg, "Generation of Finite Difference Formulas on Arbitrarily Spaced Grids", Mathematics of Computation 51, 1988. Examples -------- >>> import numpy as np >>> from scipy.optimize import approx_derivative >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> approx_derivative(f, x0, args=(1, 2)) array([[ 1., 0.], [-1., 0.]]) Bounds can be used to limit the region of function evaluation. In the example below we compute left and right derivative at point 1.0. >>> def g(x): ... return x**2 if x >= 1 else x ... >>> x0 = 1.0 >>> approx_derivative(g, x0, bounds=(-np.inf, 1.0)) array([ 1.]) >>> approx_derivative(g, x0, bounds=(1.0, np.inf)) array([ 2.]) �2-point�3-pointr! zUnknown method '%s'. r z#`x0` must have at most 1 dimension.z,Inconsistent shapes between bounds and `x0`.c s: t j � | � � � � } | j d k r6 t d � � | S)Nr z-`fun` return value has more than 1 dimension.)r � atleast_1dr, � RuntimeError)r �f)�args�fun�kwargsr r � fun_wrappedJ s z&approx_derivative.<locals>.fun_wrappedNz&`f0` passed has more than 1 dimension.z `x0` violates bound constraints.z1-sidedz2-sidedFr )r: r; r! )r r r< r, r/ r- �anyr' r �_dense_differencer �lenr9 r r0 �_sparse_difference)r@ r r& r% �f0r. �sparsityr? rA r r rB r r � structurer8 r )r? r@ rA r �approx_derivative� sJ � $ ! ! rJ c C s� | j } | j } t j | | f � } t j | � } x�t | j � D]�} | d k r� | | | } | | | | } | | � | } n/| d k r| | r| | | } | d | | } | | | | } | | � } | | � } d | d | | } n� | d k rn| | rn| | | } | | | } | | | | } | | � } | | � } | | } nL | d k r�| | | | d � } | j } | | | f } n t d � � | | | | <qF W| d k r�t j | � } | j S) Nz2-pointz3-pointr g @� r! y �?zNever be here.r g �) �sizer �emptyZdiag�range�imagr= �ravel�T)r@ r rG r r r& r5 r6 Z J_transposedZh_vecs�ir �dx�df�x1�x2�f1�f2r r r rD y s@ rD c ! C sq | j } | j } g } g } g } t j | � d } x�t | � D]�} t j | | � } | | } | d k r� | | } | | } | | � | } t j | � \ } t | d d � | f � \ } } } | | } n| d k r`| j � } | j � } | | @} | | | | 7<| | d | | 7<| | @} | | | | 8<| | | | 7<t j | � } | | | | | | <| | | | | | <| | � } | | � } t j | � \ } t | d d � | f � \ } } } | | } | | } t j | � } | | } d | | d | | | | | | <| | } | | | | | | <n | d k r�| | | d � } | j } | } t j | � \ } t | d d � | f � \ } } } | | } n t d � � | j | � | j | � | j | | | | � qD Wt j | � } t j | � } t j | � } t | | | f f d | | f �} t | � S) Nr z2-pointz3-pointr r rK r! y �?zNever be here.r- ���)rL r �maxrN ZequalZnonzeror r ZzerosrM rO r �appendZhstackr r )!r@ r rG r r rI r8 r& r5 r6 Z row_indicesZ col_indicesZ fractionsZn_groups�group�eZh_vecr rS rT ZcolsrR �j�_rU rV Zmask_1Zmask_2rW rX �maskZrows�Jr r r rF � sn % % & % $rF c C s | | | | � } t | � r� t | | d | d | d | d | �} t | � } | | } t | � \ } } } t j | | | f � j � } t j t j | � t j d t j | � � � St | | d | d | d | �} t j | | � } t j | t j d t j | � � � Sd S)aK Check correctness of a function computing derivatives (Jacobian or gradient) by comparison with a finite difference approximation. Parameters ---------- fun : callable Function of which to estimate the derivatives. The argument x passed to this function is ndarray of shape (n,) (never a scalar even if n=1). It must return 1-d array_like of shape (m,) or a scalar. jac : callable Function which computes Jacobian matrix of `fun`. It must work with argument x the same way as `fun`. The return value must be array_like or sparse matrix with an appropriate shape. x0 : array_like of shape (n,) or float Point at which to estimate the derivatives. Float will be converted to 1-d array. bounds : 2-tuple of array_like, optional Lower and upper bounds on independent variables. Defaults to no bounds. Each bound must match the size of `x0` or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation. args, kwargs : tuple and dict, optional Additional arguments passed to `fun` and `jac`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)`` and the same for `jac`. Returns ------- accuracy : float The maximum among all relative errors for elements with absolute values higher than 1 and absolute errors for elements with absolute values less or equal than 1. If `accuracy` is on the order of 1e-6 or lower, then it is likely that your `jac` implementation is correct. See Also -------- approx_derivative : Compute finite difference approximation of derivative. Examples -------- >>> import numpy as np >>> from scipy.optimize import check_derivative >>> >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> def jac(x, c1, c2): ... return np.array([ ... [np.sin(c1 * x[1]), c1 * x[0] * np.cos(c1 * x[1])], ... [np.cos(c2 * x[1]), -c2 * x[0] * np.sin(c2 * x[1])] ... ]) ... >>> >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> check_derivative(f, jac, x0, args=(1, 2)) 2.4492935982947064e-16 r. rH r? rA r N) r rJ r r r r( rP rZ r r ) r@ Zjacr r. r? rA Z J_to_testZJ_diffZabs_errrR r^ Z abs_err_dataZ J_diff_datar r r �check_derivative� s = rb )�__doc__Z __future__r Znumpyr Zsparser r r r r Z_group_columnsr r ZfinfoZfloat64Zepsr" r r' r/ r9 r rJ rD rF rb r r r r �<module> s ( O = � ( P
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__init__.cpython-35.opt-1.pyc
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__init__.cpython-35.pyc
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_basinhopping.cpython-35.opt-1.pyc
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_basinhopping.cpython-35.pyc
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_differentialevolution.cpython-35.opt-1.pyc
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_differentialevolution.cpython-35.pyc
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_hungarian.cpython-35.opt-1.pyc
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_linprog.cpython-35.opt-1.pyc
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_linprog.cpython-35.pyc
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_minimize.cpython-35.opt-1.pyc
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_minimize.cpython-35.pyc
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_numdiff.cpython-35.opt-1.pyc
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_numdiff.cpython-35.pyc
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_root.cpython-35.opt-1.pyc
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_spectral.cpython-35.opt-1.pyc
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_trustregion.cpython-35.opt-1.pyc
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_trustregion_dogleg.cpython-35.opt-1.pyc
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_trustregion_ncg.cpython-35.opt-1.pyc
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_trustregion_ncg.cpython-35.pyc
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_tstutils.cpython-35.opt-1.pyc
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_tstutils.cpython-35.pyc
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cobyla.cpython-35.opt-1.pyc
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cobyla.cpython-35.pyc
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lbfgsb.cpython-35.opt-1.pyc
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lbfgsb.cpython-35.pyc
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linesearch.cpython-35.opt-1.pyc
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linesearch.cpython-35.pyc
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minpack.cpython-35.opt-1.pyc
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minpack.cpython-35.pyc
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nnls.cpython-35.opt-1.pyc
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nnls.cpython-35.pyc
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nonlin.cpython-35.opt-1.pyc
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nonlin.cpython-35.pyc
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optimize.cpython-35.opt-1.pyc
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optimize.cpython-35.pyc
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setup.cpython-35.opt-1.pyc
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setup.cpython-35.pyc
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slsqp.cpython-35.opt-1.pyc
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tnc.cpython-35.opt-1.pyc
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tnc.cpython-35.pyc
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zeros.cpython-35.opt-1.pyc
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zeros.cpython-35.pyc
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