Refactor iterative Lie derivatives for numerical evaluation

[ManojVS06]

Refactor iter_lie to iter_lie_numeric for numerical computation of Lie derivatives. Introduced helper functions for gradient and derivative calculations.

[iperezav]

Hello @ManojVS06,

Thanks for putting effort on this package. I did some checking. I took the system example in
https://arxiv.org/pdf/2509.19908

and defined it in your code:

if __name__ == "__main__":
    # define state functions and vector fields
    def h(z):
        # output function h = x2
        return float(z[1])

    # vector fields as callables: g0, g1, g2 producing arrays of length 2
    def g0(z):
        x1, x2 = z
        return np.array([-x1 + (1-x1)*sp.exp(x2/(1+x2)), -2*x2 + (1-x1)*sp.exp(x2/(1+x2))], dtype=float)

    def g1(z):
        x1, x2 = z
        return np.array([0.0, 1.0], dtype=float)  # example; adapt to your system

    #def g2(z):
    #    x1, x2 = z
    #    return np.array([0.0, -x2], dtype=float)

    g_list = [g0, g1]

    z0 = np.array([0.0, 0.0], dtype=float)
    Ntrunc = 2

    funcs, vals = iter_lie_numeric(h, g_list, z0, Ntrunc)
    print("Total Lie derivatives:", vals.size)
    print("Values at z0:\n", vals)

checked it with the package:

import numpy as np
import sympy as sp
from CFS import iter_lie

# Define the symbolic variables
x1, x2 = sp.symbols('x1 x2')
x = sp.Matrix([x1, x2])


# Define the system symbolically
g = sp.transpose(sp.Matrix([[-x1 + (1-x1)*sp.exp(x2/(1+x2)), -2*x2 + (1-x1)*sp.exp(x2/(1+x2))], [0, 1]]))

# Define the output symbolically
h = x2

# The truncation of the length of the words that index the Chen-Fliess series
Ntrunc = 2

# Coefficients of the Chen-Fliess series evaluated at the initial state
Ceta = np.array(iter_lie(h,g,x,Ntrunc).subs([(x[0], 0),(x[1], 0)]))

print(Ceta)

and they don't coincide