ftj LdZddlZddlZddlmZddlmZddlm Z m Z m Z m Z m Z mZddlmZmZdd lmZdd lmZdd lmZd Zejd ZdZdZddZddddej ejfddddddf dZ dZ!dZ"dZ#GddZ$ej ejfddfdZ%dS)z'Routines for numerical differentiation.N)norm)LinearOperator)issparse isspmatrixfind csc_array csr_array csr_matrix) group_dense group_sparse)array_namespace) MapWrapper)array_api_extrac2|dkrtj|t}nE|dkr0tj|}tj|t}nt dtj|tj k|tjkzr||fS||z}|}||z } ||z } |dkr||z} | |k| |kz} tj|tj | | k} || | zxxdzcc<| | k| z}| ||z ||<| | k| z}| | |z ||<n|dkr| |k| |kz}| | k|z}tj ||d| |z|z ||<d||<| | k|z}tj ||d| |z|z  ||<d||<tj | | |z }|tj||kz}||||<d||<||fS) aAdjust 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 absolute 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 absolute 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'``. 1-sideddtype2-sidedz(`scheme` must be '1-sided' or '2-sided'.?TF) np ones_likeboolabs zeros_like ValueErrorallinfcopymaximumminimum)x0h num_stepsschemelbub use_one_sidedh_total h_adjusted lower_dist upper_distxviolatedfittingforwardbackwardcentralmin_distadjusted_centrals U/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/optimize/_numdiff.py_adjust_scheme_to_boundsr8s> Qd333 9   F1II at444 CDDD vrbfW}rv.// -)mGJbJbJ  LFq2v&&//RZ J%G%GG8g%&&&",&&&+x7(1I= 7+x7 *8 44y@ 8 9  (Z7-BC+x7 j gJj11I=?? 7!% g+x7 " hKz(33i?!A!A A 8"& h:j*55 A$Hz(:(:h(FG'/0@'A #$*/ &' } $$ctjtjj}d}tj|tjr4tj|j}tj|j}d}tj|tjr:tj|j}|r||krtj|j}|dvr|dzS|dvr|dzStd)a Calculates relative EPS step to use for a given data type and numdiff step method. Progressively smaller steps are used for larger floating point types. Parameters ---------- f0_dtype: np.dtype dtype of function evaluation x0_dtype: np.dtype dtype of parameter vector method: {'2-point', '3-point', 'cs'} Returns ------- EPS: float relative step size. May be np.float16, np.float32, np.float64 Notes ----- The default relative step will be np.float64. However, if x0 or f0 are smaller floating point types (np.float16, np.float32), then the smallest floating point type is chosen. FT)2-pointcsr)3-pointgUUUUUU?zBUnknown step method, should be one of {'2-point', '3-point', 'cs'}) rfinfofloat64eps issubdtypeinexactritemsize RuntimeError)x0_dtypef0_dtypemethodEPSx0_is_fp x0_itemsize f0_itemsizes r7_eps_for_methodrL]s< (2:   "CH }Xrz**hx  $hx((1  }Xrz**)hx((1  ) k11(8$$(C """Cx ;  Sz:;; ;r9c |dktdzdz }t|j|j|}|.||zt jdt j|z}ng||zt j|z}||z|z }t j|dk||zt jdt j|z|}|S)az Computes an absolute step from a relative step for finite difference calculation. Parameters ---------- rel_step: None or array-like Relative step for the finite difference calculation x0 : np.ndarray Parameter vector f0 : np.ndarray or scalar method : {'2-point', '3-point', 'cs'} Returns ------- h : float The absolute step size Notes ----- `h` will always be np.float64. However, if `x0` or `f0` are smaller floating point dtypes (e.g. np.float32), then the absolute step size will be calculated from the smallest floating point size. rrr N?)astypefloatrLrrr"rwhere)rel_stepr$f0rGsign_x0rstepabs_stepdxs r7_compute_absolute_steprXs6Qwu%%)A-G BHbh 7 7E7?RZRVBZZ%@%@@ g%r 2H}"8B!G!GObjbfRjj.I.II$&& Or9cd|D\}}|jdkrtj||j}|jdkrtj||j}||fS)aa Prepares new-style bounds from a two-tuple specifying the lower and upper limits for values in x0. If a value is not bound then the lower/upper bound will be expected to be -np.inf/np.inf. Examples -------- >>> _prepare_bounds([(0, 1, 2), (1, 2, np.inf)], [0.5, 1.5, 2.5]) (array([0., 1., 2.]), array([ 1., 2., inf])) c3LK|]}tj|tV dS)rN)rasarrayrP).0bs r7 z"_prepare_bounds..s1 9 9Qbj%((( 9 9 9 9 9 9r9r)ndimrresizeshape)boundsr$r(r)s r7_prepare_boundsrcs`: 9& 9 9 9FB w!|| Yr28 $ $ w!|| Yr28 $ $ r6Mr9ct|rt|}n7tj|}|dktj}|jdkrtd|j\}}|tj |r5tj |}| |}n/tj |}|j|fkrtd|dd|f}t|rt|||j|j}nt#|||}|||<|S)aGroup 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 array, 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 ith 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. rrz`A` must be 2-dimensional.Nz`order` has incorrect shape.)rr r atleast_2drOint32r_rraisscalarrandom RandomState permutationr[rindicesindptrr r!)Aordermnrnggroupss r7 group_columnsrss1<{{& aLL M!   !VOOBH % %v{{5666 7DAq } E**}i##E**"" 5!! ;1$  ;<< < !!!U( A{{&aAIqx88Q1%%KKMMF5M Mr9r=Fc |dvrtd|dddi} t|}tj||d|}|j}||jdr|j}|||}|j dkrtd t||\}}|j |j ks|j |j krtd |r[tj tj|r&tj tj|std | i} t||| | }d x}}|||}d}n.tj|}|j dkrtd tj||k||kzrtd|r5|t%|j|j|}t'|||||\}}n|t)||||}n|d kt*dzdz }|}||z|z }tj|d kt%|j|j||ztjdtj|z|}|dkrt3||dd||\}}n&|dkrt3||dd||\}}n|dkrd}| pt4} t7| 5}|t9|||||||\}}nt;|st=|dkr|\}}n|}t?|}t;|r| }ntj!|}tj|}tE||||||||| \}}dddn #1swxYwY| r||z }|| d<|| fS|S)uZ*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', 'cs'}, optional Finite difference method to use: - '2-point' - use the first 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. If None (default) the absolute step size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, with `rel_step` being selected automatically, see Notes. Otherwise ``h = rel_step * sign(x0) * abs(x0)``. For ``method='3-point'`` the sign of `h` is ignored. The calculated step size is possibly adjusted to fit into the bounds. abs_step : array_like, optional Absolute step size to use, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `abs_step` is ignored. By default relative steps are used, only if ``abs_step is not None`` are absolute steps used. 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. Bounds checking is not implemented when `as_linear_operator` is True. sparsity : {None, array_like, sparse array, 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 array 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 array 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. as_linear_operator : bool, optional When True the function returns an `scipy.sparse.linalg.LinearOperator`. Otherwise it returns a dense array or a sparse array depending on `sparsity`. The linear operator provides an efficient way of computing ``J.dot(p)`` for any vector ``p`` of shape (n,), but does not allow direct access to individual elements of the matrix. By default `as_linear_operator` is False. args, kwargs : tuple and dict, optional Additional arguments passed to `fun`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)``. full_output : bool, optional If True then the function also returns a dictionary with extra information about the calculation. workers : int or map-like callable, optional Supply a map-like callable, such as `multiprocessing.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(fun, iterable)``. Alternatively, if `workers` is an int the task is subdivided into `workers` sections and the fun evaluated in parallel (uses `multiprocessing.Pool `). Supply -1 to use all available CPU cores. It is recommended that a map-like be used instead of int, as repeated calls to `approx_derivative` will incur large overhead from setting up new processes. Returns ------- J : {ndarray, sparse array, LinearOperator} Finite difference approximation of the Jacobian matrix. If `as_linear_operator` is True returns a LinearOperator with shape (m, n). Otherwise it returns a dense array or sparse array depending on how `sparsity` is defined. If `sparsity` is None then a ndarray with shape (m, n) is returned. If `sparsity` is not None returns a csr_array or csr_matrix with shape (m, n) following the array/matrix type of the incoming structure. For sparse arrays and linear operators it is always returned as a 2-D structure. For ndarrays, if m=1 it is returned as a 1-D gradient array with shape (n,). info_dict : dict Dictionary containing extra information about the calculation. The keys include: - `nfev`, number of function evaluations. If `as_linear_operator` is True then `fun` is expected to track the number of evaluations itself. This is because multiple calls may be made to the linear operator which are not trackable here. See Also -------- check_derivative : Check correctness of a function computing derivatives. Notes ----- If `rel_step` is not provided, it assigned as ``EPS**(1/s)``, where EPS is determined from the smallest floating point dtype of `x0` or `fun(x0)`, ``np.finfo(x0.dtype).eps``, 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]_. Relative steps are used by default. However, absolute steps are used when ``abs_step is not None``. If any of the absolute or relative steps produces an indistinguishable difference from the original `x0`, ``(x0 + dx) - x0 == 0``, then a automatic step size is substituted for that particular entry. 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._numdiff 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.]) We can also parallelize the derivative calculation using the workers keyword. >>> from multiprocessing import Pool >>> import time >>> def fun2(x): # import from an external file for use with multiprocessing ... time.sleep(0.002) ... return rosen(x) >>> rng = np.random.default_rng() >>> x0 = rng.uniform(high=10, size=(2000,)) >>> f0 = rosen(x0) >>> %timeit approx_derivative(fun2, x0, f0=f0) # may vary 10.5 s ± 5.91 ms per loop (mean ± std. dev. of 7 runs, 1 loop each) >>> elapsed = [] >>> with Pool() as workers: ... for i in range(10): ... t = time.perf_counter() ... approx_derivative(fun2, x0, workers=workers.map, f0=f0) ... et = time.perf_counter() ... elapsed.append(et - t) >>> np.mean(elapsed) # may vary np.float64(1.442545195999901) Create a map-like vectorized version. `x` is a generator, so first of all a 2-D array, `xx`, is reconstituted. 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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 array 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._numdiff 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])], ... 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