o Ebgn @sdZddlZddlZddlmZddlmZddlm Z m Z m Z m Z m Z ddlmZmZd d Zed d Zd dZddZd ddZddddej ejfdddif ddZddZddZddZej ejfdifddZdS)!z'Routines for numerical differentiation.N)norm)LinearOperator)issparse csc_matrix csr_matrix coo_matrixfind) group_dense group_sparsecCs|dkr tj|td}n|dkrt|}tj|td}ntdt|tj k|tjk@r3||fS||}|}||} ||} |dkr||} | |k| |kB} t|t | | k} || | @d9<| | k| @}| ||||<| | k| @}| | |||<||fS|dkr| |k| |k@}| | k|@}t ||d| ||||<d||<| | k|@}t ||d| || ||<d||<t | | |}|t||k@}||||<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) npZ ones_likeboolabs zeros_like ValueErrorallinfcopymaximumZminimum)x0hZ num_stepsZschemelbub use_one_sidedZh_totalZ h_adjustedZ lower_distZ upper_distxZviolatedZfittingZforwardZbackwardZcentralZmin_distZadjusted_centralr"9/usr/lib/python3/dist-packages/scipy/optimize/_numdiff.py_adjust_scheme_to_bounds sN   r$cCsttjj}d}t|tjrt|j}t|j}d}t|tjr7t|j}|r7||kr7t|j}|dvr?|dS|dvrG|dStd)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'}) rZfinfoZfloat64ZepsZ issubdtypeZinexactritemsize RuntimeError)Zx0_dtypeZf0_dtypemethodZEPSZx0_is_fpZ x0_itemsizeZ f0_itemsizer"r"r#_eps_for_methodZs     r+c Cs|dktdd}t|j|j|}|dur&||tdt|}|S||t|}|||}t|dk||tdt||}|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?)astypefloatr+rrrrwhere)rel_steprf0r*sign_x0Zrstepabs_stepdxr"r"r#_compute_absolute_steps  r5cCsJdd|D\}}|jdkrt||j}|jdkr!t||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])) cSsg|] }tj|tdqS)r)rasarrayr.).0br"r"r# sz#_prepare_bounds..r)ndimrZresizeshape)boundsrrrr"r"r#_prepare_boundss   r=cCst|r t|}n t|}|dktj}|jdkrtd|j\}}|dus-t |r9tj |}| |}nt |}|j|fkrHtd|dd|f}t|r^t|||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 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 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.)rrr atleast_2dr-Zint32r:rr;ZisscalarZrandomZ RandomStateZ permutationr6r indicesZindptrr r)AZordermnrnggroupsr"r"r# group_columnss&          rEr'Fr"c  sV|dvr td|t|}|jdkrtdt||\} } | j|jks+| j|jkr/td|rEtt| rAtt| sEtdfdd} |d urV| |}nt|}|jdkrdtd t|| k|| kBrstd |r|d urt |j |j |}t | ||||S|d urt ||||}n-|d k td d}|}|||}t|d kt |j |j ||tdt||}|dkrt||dd| | \}}n|dkrt||dd| | \}}n|dkrd}|d urt| |||||St|st|d kr|\}}n|}t|}t|rt|}nt|}t|}t| |||||||S)aACompute 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 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. as_linear_operator : bool, optional When True the function returns an `scipy.sparse.linalg.LinearOperator`. Otherwise it returns a dense array or a sparse matrix 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)``. Returns ------- J : {ndarray, sparse matrix, 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 matrix 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_matrix with shape (m, n). For sparse matrices 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,). 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 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.]) )r%r'r&zUnknown method '%s'. r z#`x0` must have at most 1 dimension.z,Inconsistent shapes between bounds and `x0`.z7Bounds not supported when `as_linear_operator` is True.cs2t|gRi}|jdkrtd|S)Nr z-`fun` return value has more than 1 dimension.)r atleast_1dr:r))r!fargsfunkwargsr"r# fun_wrappeds z&approx_derivative..fun_wrappedNz&`f0` passed has more than 1 dimension.z `x0` violates bound constraints.rrr,r%r r'rr&F)rrrFr:r=r;rZisinfanyr+r_linear_operator_differencer5r-r.r/rrr$_dense_differencerlenrErr>_sparse_difference)rJrr*r0r3r1r<sparsityZas_linear_operatorrIrKrrrLrr2r4r structurerDr"rHr#approx_derivatives$                   rTcsxjj}|dkrfdd}n |dkr#fdd}n|dkr1fdd}ntdt|f|S) Nr%csHt|t|rtSt|}||}|}||S)Nr array_equalrzerosr)pr4r!dfr1rJrrArr"r#matvecs     z+_linear_operator_difference..matvecr'cslt|t|rtSdt|}|d|}|d|}|}|}||}||S)NrrU)rXr4x1x2f1f2rYrJrrArr"r#r[s r&csNt|t|rtSt|}||d}|}|j}||S)N?)rrVrrWrimag)rXr4r!r^rYr`r"r#r['s  Never be here.)sizer)r)rJrr1rr*rBr[r"rZr#rN s  rNcCsv|j}|j}t||f}t|} t|jD]} |dkr2|| | } | | || } || |} nv|dkrc|| rc|| | }|d| | }|| || } ||}||}d|d||} nE|dkr|| s|| | }|| | }|| || } ||}||}||} n|dkr||| | d}|j} | | | f} ntd| | || <q|d krt|}|jS) Nr%r'rgr&rarcr ) rdremptyZdiagrangerbr)ravelT)rJrr1rr r*rArBZ J_transposedZh_vecsir!r4rYr\r]r^r_r"r"r#rO6s@       rOc!Cs|j}|j} g} g} g} t|d} t| D]}t||}||}|dkrN||}||}|||}t|\}t|dd|f\}}}||}n|dkr|}|}||@}||||7<||d||7<||@}||||8<||||7<t| }||||||<||||||<||}||}t|\}t|dd|f\}}}||}||}t |}||}d||d||||||<||}||||||<n.|dkr|||d}|j }|}t|\}t|dd|f\}}}||}nt d | || || ||||qt | } t | } t | } t| | | ff|| fd } t| S) Nr r%r'rrer&rarc)r;)rdrmaxrgZequalZnonzeror rrWrfrbrappendZhstackrr)!rJrr1rr rSrDr*rArBZ row_indicesZ col_indicesZ fractionsZn_groupsgroupeZh_vecr!r4rYZcolsrjj_r\r]Zmask_1Zmask_2r^r_maskZrowsJr"r"r#rQ^sn        $         rQc Cs||g|Ri|}t|rDt||||||d}t|}||}t|\} } } t|| | f} tt| t dt| St|||||d}t||}t|t dt|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<rRrIrKr )r<rIrK) rrTrr rr6rhrlrr) rJZjacrr<rIrKZ J_to_testZJ_diffZabs_errrjrpZ abs_err_dataZ J_diff_datar"r"r#check_derivatives"=  rt)r)__doc__ functoolsZnumpyrZ numpy.linalgrZscipy.sparse.linalgrZsparserrrrr Z_group_columnsr r r$ lru_cacher+r5r=rErrTrNrOrQrtr"r"r"r#s0  O 51 = {)(P