(phL4SSKJrJrJrJrJr SSjrSSjrg))arangenewaxishstackprodarraycJXS-:a [S5eUS-S:Xa [S5eSSKJn US- n[U*US-5nUSS2[4nUS -n[ SU5Hn[ XTU-/5nM [[SUS-5SS 9URU5U-nU$) a# Return weights for an Np-point central derivative. Assumes equally-spaced function points. If weights are in the vector w, then derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx) Parameters ---------- Np : int Number of points for the central derivative. ndiv : int, optional Number of divisions. Default is 1. Returns ------- w : ndarray Weights for an Np-point central derivative. Its size is `Np`. Notes ----- Can be inaccurate for a large number of points. Examples -------- We can calculate a derivative value of a function. >>> def f(x): ... return 2 * x**2 + 3 >>> x = 3.0 # derivative point >>> h = 0.1 # differential step >>> Np = 3 # point number for central derivative >>> weights = _central_diff_weights(Np) # weights for first derivative >>> vals = [f(x + (i - Np/2) * h) for i in range(Np)] >>> sum(w * v for (w, v) in zip(weights, vals))/h 11.79999999999998 This value is close to the analytical solution: f'(x) = 4x, so f'(3) = 12 References ---------- .. [1] https://en.wikipedia.org/wiki/Finite_difference z;Number of points must be at least the derivative order + 1.rz!The number of points must be odd.)linalg?Naxis) ValueErrorscipyr rrrangerrinv)Npndivr hoxXkws Q/var/www/html/venv/lib/python3.13/site-packages/scipy/_lib/_finite_differences.py_central_diff_weightsrs^ 1H} I   Av{<== qBsBHA !W* A 3A 1b\ A!t9  VAtax q)FJJqM$,??A HcXSS-:a [S5eUS-S:Xa [S5eUS:XaiUS:Xa[/SQ5S- nOUS :Xa[/S Q5S - nOUS :Xa[/S Q5S- nOUS:Xa[/SQ5S- nO[US5nOwUS:XafUS:Xa[/SQ5nO]US :Xa[/SQ5S - nOFUS :Xa[/SQ5S- nO/US:Xa[/SQ5S- nO[US5nO [XS5nSnUS- n[U5Hn XvU U"XU- U--/UQ76-- nM U[ U4U-SS9- $)a Find the nth derivative of a function at a point. Given a function, use a central difference formula with spacing `dx` to compute the nth derivative at `x0`. Parameters ---------- func : function Input function. x0 : float The point at which the nth derivative is found. dx : float, optional Spacing. n : int, optional Order of the derivative. Default is 1. args : tuple, optional Arguments order : int, optional Number of points to use, must be odd. Notes ----- Decreasing the step size too small can result in round-off error. Examples -------- >>> def f(x): ... return x**3 + x**2 >>> _derivative(f, 1.0, dx=1e-6) 4.9999999999217337 r zm'order' (the number of points used to compute the derivative), must be at least the derivative order 'n' + 1.r rzJ'order' (the number of points used to compute the derivative) must be odd.)rr g@)r irr g(@)r ir-r gN@r$) rii`riiX g@@)r gr )r ir*r )r ir,r+r gf@) r&ir/r.r-r&g@r r)rrrrr) funcx0dxnargsorderweightsvalrrs r _derivativer8EsD 1u} =   qyA~    Av A:J'#-G aZ-.5G aZ67$>G aZEFNG+E15G a A:L)G aZ12T9G aZ<=EG aZJK  ,E15G'1 C !B 5\ qzD2vm!3;d;;; reaia( ((rN)r )r r r)numpyrrrrrrr8r9rrr;s66> BL)r