U |eL@s4ddlmZmZmZmZmZd ddZd dd Zd S) )arangenewaxishstackprodarraycCs||dkrtd|ddkr(tdddlm}|d?}t| |d}|ddtf}|d }td|D]}t|||g}qnttd|ddd |||}|S) a Return weights for an Np-point central derivative. Assumes equally-spaced function points. If weights are in the vector w, then derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx) Parameters ---------- Np : int Number of points for the central derivative. ndiv : int, optional Number of divisions. Default is 1. Returns ------- w : ndarray Weights for an Np-point central derivative. Its size is `Np`. Notes ----- Can be inaccurate for a large number of points. Examples -------- We can calculate a derivative value of a function. >>> def f(x): ... return 2 * x**2 + 3 >>> x = 3.0 # derivative point >>> h = 0.1 # differential step >>> Np = 3 # point number for central derivative >>> weights = _central_diff_weights(Np) # weights for first derivative >>> vals = [f(x + (i - Np/2) * h) for i in range(Np)] >>> sum(w * v for (w, v) in zip(weights, vals))/h 11.79999999999998 This value is close to the analytical solution: f'(x) = 4x, so f'(3) = 12 References ---------- .. [1] https://en.wikipedia.org/wiki/Finite_difference rz;Number of points must be at least the derivative order + 1.rz!The number of points must be odd.)linalg?Naxis) ValueErrorscipyr rrrangerrinv)ZNpZndivr hoxXkwr[/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/_lib/_finite_differences.py_central_diff_weightss/   $rr rc Cs||dkrtd|ddkr(td|dkr|dkrLtdddgd}nv|d krltdd dd dgd }nV|d krtdddddddgd}n2|dkrtdddddddddg d}n t|d}n|dkrd|dkrtdddg}n||d krtdddddgd }nZ|d kr.tdddd dddgd!}n4|dkrXtdd"d#d$d%d$d#d"dg d&}n t|d}n t||}d'}|d?}t|D],} ||| ||| ||f|7}q|t|f|dd(S))a  Find the nth derivative of a function at a point. Given a function, use a central difference formula with spacing `dx` to compute the nth derivative at `x0`. Parameters ---------- func : function Input function. x0 : float The point at which the nth derivative is found. dx : float, optional Spacing. n : int, optional Order of the derivative. Default is 1. args : tuple, optional Arguments order : int, optional Number of points to use, must be odd. Notes ----- Decreasing the step size too small can result in round-off error. Examples -------- >>> def f(x): ... return x**3 + x**2 >>> _derivative(f, 1.0, dx=1e-6) 4.9999999999217337 rzm'order' (the number of points used to compute the derivative), must be at least the derivative order 'n' + 1.rrzJ'order' (the number of points used to compute the derivative) must be odd.rg@ig(@ i-igN@ii`iiX g@@giiiigf@iiig@r r )rrrrr) funcx0dxnargsorderweightsvalrrrrr _derivativeEsJ"           *r.N)r)r rrr)numpyrrrrrrr.rrrrs A