Computes Hermite coefficients of the tanh function using symbolic differentation (SymPy)

hermiteCoeffs_tanh.py

import numpy as np
import sympy as sym
import time

def hermiteCoeffsTanh(nCoeffs):
    """
    Computes the first nCoeffs Hermite coefficients of the tanh function. The r-th
    Hermite coefficients is defined by

        H_r(g) = int_{-oo}^{oo}g(y)h_r(y) * 1/sqrt(2 * pi) * exp(-x^2/2) dy

    where h_r is the r-th normalized probablistic Hermite polynomial.


    Params
    -----------------
    nCoeffs: number of Coefficients to return


    Returns
    -----------------
    Hermite coefficients in a numpy array

    # WARNING: Slow code!
    """
    x = sym.symbols('x')
    coeffs = np.zeros(nCoeffs)
    gaussWeight = sym.exp((-x**2)/2)
    fun = sym.tanh(x)
    for i in range(nCoeffs):
        t = time.time()
        if i > 0:
            der = sym.diff(der, x)
        else:
            der = fun
        elapsed = time.time() - t
        print(f"Computed {i}-th derivative in time : {elapsed}")
        if i % 2 == 0:
            coeffs[i] = 0.0
            print(f"{i}-th derivative has coefficient: {coeffs[i]}")
            continue
        t = time.time()
        test = sym.integrate(sym.expand(der * 1/sym.sqrt(2 * sym.pi) * sym.exp(-sym.Rational(1,2) * (x**2))), (x, -sym.oo, sym.oo))
        coeffs[i] = 1.0/np.sqrt(np.math.factorial(i)) * float(test.n())
        elapsed = time.time() - t
        print(f"{i}-th derivative has coefficient: {coeffs[i]}")
        print(f"Used time: {elapsed}")
    return coeffs