o ֭jL@s4ddlmZmZmZmZmZd ddZd dd Zd S) )arangenewaxishstackprodarraycCs||dkr td|ddkrtdddlm}|d?}t| |d}|ddtf}|d }td|D] }t|||g}q7ttd|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)Npndivr hoxXkwr[/var/www/html/CropPilot/venv/lib/python3.10/site-packages/scipy/_lib/_finite_differences.py_central_diff_weightss /  $rr rc Cs~||dkr td|ddkrtd|dkrR|dkr%tgdd}nn|d kr2tgd d }na|d kr?tgd d}nT|dkrLtgdd}nGt|d}nA|dkr|dkratgd}n2|d krntgdd }n%|d kr{tgdd}n|dkrtgdd}n t|d}nt||}d}|d?}t|D]} ||| ||| ||g|R7}q|t|f|ddS)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.r)rrg@)rirrg(@)r ir-rgN@r!) rii`riiX g@@)rgr)rir'r)rir)r(rgf@) r#ir,r+r*r#g@r r )rrrrr) funcx0dxnargsorderweightsvalrrrrr _derivativeEsJ "      *r5N)r)r rrr)numpyrrrrrrr5rrrrs A