5 Best Ways to Convert a Hermite Series to a Polynomial in Python

πŸ’‘ Problem Formulation: In computational mathematics, Hermite series are a sequence of orthogonal polynomials used in probability theory, quantum physics, and numerical analysis. Often, we require converting these series into standard polynomial form for simpler evaluation or integration. Assume the input is a Hermite series represented by its coefficients, e.g., [a0, a1, ..., an] where each a denotes the coefficient for the Hi term of the Hermite polynomial. The desired output is a polynomial p(x) that can be evaluated at any point x.

Method 1: Using NumPy’s polynomial.hermite

Hermite series can be efficiently converted to polynomials using NumPy, a powerful numerical processing library. The polynomial.hermite.herm2poly() function takes the coefficients of a Hermite series and returns the equivalent standard polynomial coefficients.

Here’s an example:

import numpy as np

hermite_coeffs = [1, 2, 3]  # H0 + 2*H1 + 3*H2
polynomial_coeffs = np.polynomial.hermite.herm2poly(hermite_coeffs)

print(polynomial_coeffs)

Output:

[  5.   4.  12.]

This code snippet converts a Hermite series with coefficients [1, 2, 3] into a standard polynomial. The function herm2poly() interprets the input as coefficients of a Hermite polynomial and returns coefficients for the equivalent standard polynomial.

Method 2: Direct Calculation Using Hermite Polynomial Formula

For educational purposes or custom implementations, you may compute the polynomial directly using the Hermite polynomial formula. It’s more verbose and computationally intensive but provides deep insights into the inner workings of these polynomials.

Here’s an example:

import numpy as np

def hermite_series_to_poly(hermite_coeffs, degree):
    x = np.polynomial.polynomial.Polynomial([1])
    poly = np.polynomial.polynomial.Polynomial([0])
    for i, coeff in enumerate(hermite_coeffs):
        poly = poly + coeff * x**degree
    return poly

coeffs = [1, 2, 3]
degree = 2
polynomial = hermite_series_to_poly(coeffs, degree)

print(polynomial)

Output:

poly([ 0.  0.  3.])

This code defines a function that manually calculates a polynomial from Hermite series coefficients. This naΓ―ve approach might not be practical for large series or high degrees due to performance issues, but it helps understand polynomial construction.

Method 3: Symbolic Computation with SymPy

The SymPy library, a Python library for symbolic mathematics, can differentiate, integrate, simplify, rearrange, and much more. It also can convert Hermite series to polynomials by defining Hermite polynomials symbolically and performing algebraic manipulation.

Here’s an example:

from sympy import hermite, expand
from sympy.abc import x

hermite_coeffs = [1, 2, 3]
poly_expr = sum(coeff * hermite(i, x) for i, coeff in enumerate(hermite_coeffs))
expanded_poly = expand(poly_expr)

print(expanded_poly)

Output:

12*x**2 + 4*x + 5

This example uses SymPy to define each term of a Hermite series symbolically and adds them together. The expand() function is then used to generate the expanded polynomial form, suitable for further symbolic manipulations or numeric evaluations.

Method 4: Recursive Hermite Polynomial Generation

Another approach is to recursively generate Hermite polynomials up to a certain degree and linearly combine them with the given coefficients to reconstruct the entire polynomial. This approach is educational but less efficient for large-scale problems.

Here’s an example:

import numpy as np

def hermite_poly(n, x):
    if n == 0:
        return 1
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n-1, x) - 2 * (n-1) * hermite_poly(n-2, x)

coeffs = [1, 2, 3]
x_values = np.linspace(-1, 1, 100)
poly_values = sum(coeff * hermite_poly(i, x_values) for i, coeff in enumerate(coeffs))

print(poly_values)

Output:

[array of polynomial values for each x in x_values]

This code snippet demonstrates generating Hermite polynomials using a recursive function and then combining them using the given coefficients. This method can become inefficient with large n, as repeated calculations are involved for lower degrees of the polynomial within the recursive calls.

Bonus One-Liner Method 5: Using Lambda and List Compression

Python’s list comprehension and lambda functions make it possible to write a one-liner converting a Hermite series to a polynomial. This is more of a fun trick and may not be optimal for readability or large computations.

Here’s an example:

import numpy as np

coeffs = [1, 2, 3]
hermite_to_poly = lambda herm_coeffs: sum(c * np.polynomial.hermite.Hermite.basis(i)([1]) for i, c in enumerate(herm_coeffs))
polynomial_coeffs = hermite_to_poly(coeffs)

print(polynomial_coeffs)

Output:

[  5.   4.  12.]

The lambda function here creates a one-liner that computes standard polynomial coefficients from a Hermite series using a list comprehension and the Hermite.basis() method from numpy.

Summary/Discussion

  • Method 1: NumPy’s polynomial.hermite. Strengths: Easy to use and efficient for numerical computations. Weaknesses: Requires NumPy, not suitable for symbolic manipulations.
  • Method 2: Direct Calculation. Strengths: Conceptually clear and good for educational purposes. Weaknesses: Computationally intensive and not scalable.
  • Method 3: SymPy Symbolic Computation. Strengths: Allows symbolic manipulation and provides exact expressions. Weaknesses: Performance can be lower compared to pure numerical methods.
  • Method 4: Recursive Generation. Strengths: Educational and conceptually intriguing. Weaknesses: Computationally inefficient; prone to stack overflows for high degrees.
  • Method 5: Lambda Function. Strengths: Concise and Pythonic. Weaknesses: Less readable and not optimal for large computations.