+ LV-jHB^RIHtHtHtHtHt^RIHtRRltRRlt R#))arangenewaxishstackprodarray)linalgcW^,8d \R4hV^,^8Xd \R4hV^, p\V)VR,4pVR\3,pVR,p\^V4Fp\ WCV,.4pK \ \^V^,4^R7\ P!V4V,,pV#)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.z!The number of points must be odd.?:NNNaxis) ValueErrorrrrangerrrinv)NpndivhoxXkws&& p/Volumes/fast/ai/experiments/ui-tars-smoke/.venv/lib/python3.14/site-packages/scipy/stats/_finite_differences.py_central_diff_weightsrs^ 1H} I   Av{<== qBsBHA !W* A 3A 1b\ A!t9  VAtax q)FJJqM$,??A HcRWS^,8d \R4hV^,^8Xd \R4hV^8Xd~V^8Xd\.R O4R, pMV^8Xd\.R O4R, pMV^8Xd\.R O4R, pMV^ 8Xd\.RO4R, pM\V^4pMV^8XdwV^8Xd\.RO4pMlV^8Xd\.RO4R, pMPV^8Xd\.RO4R, pM4V^ 8Xd\.RO4R, pM\V^4pM \WS4pR pV^, p\V4F5p WvV ,V!WV, V,,.VO5!,, pK7 V\ V3V,^R 7, #)a Find the nth derivative of a function at a point. Given a function, use a central difference formula with spacing `dx` to compute the nth derivative at `x0`. Parameters ---------- func : function Input function. x0 : float The point at which the nth derivative is found. dx : float, optional Spacing. n : int, optional Order of the derivative. Default is 1. args : tuple, optional Arguments order : int, optional Number of points to use, must be odd. Notes ----- Decreasing the step size too small can result in round-off error. Examples -------- >>> def f(x): ... return x**3 + x**2 >>> _derivative(f, 1.0, dx=1e-6) 4.9999999999217337 zm'order' (the number of points used to compute the derivative), must be at least the derivative order 'n' + 1.zJ'order' (the number of points used to compute the derivative) must be odd.g@g(@gN@g@@gf@g@r r )r)rirr)r ir-r) ii`riiX )rgr)rir&r)ir)r(r') r!ir,r+r*r!)rrrrr) funcx0dxnargsorderweightsvalrrs &&&&&& r _derivativer5EsD 1u} =   qyA~    Av A:J'#-G aZ-.5G aZ67$>G aZEFNG+E15G a A:L)G aZ12T9G aZ<=EG aZJK  ,E15G'1 C !B 5\ qzD2vm!3;d;;; reaia( ((rN)r)r rr") numpyrrrrrscipyrrr5r6rrr9s66= @L)r