Efficient Ways to Generate a Hermite Series with Given Roots in Python

💡 Problem Formulation: Generating a Hermite polynomial given specific roots is a common problem in mathematical and computational fields. A Hermite polynomial is a solution to Hermite’s differential equation and is useful in probability, physics, and numerical methods. The input would typically be a list of roots for the Hermite polynomial, and the desired output is a series expansion that satisfies those roots.

Method 1: Using NumPy’s Polynomial Hermite Class

NumPy is a powerful library for numerical computations in Python. It provides a Polynomial class that can be used to create Hermite polynomials directly. By inputting the given roots, NumPy can generate the Hermite series efficiently. The key function to use is numpy.polynomial.hermite.Hermite.fromroots(roots), where roots are the given roots of the Hermite polynomial we wish to generate.

Here’s an example:

import numpy as np

roots = [1, 2, 3]  # Given roots for the Hermite polynomial
H = np.polynomial.hermite.Hermite.fromroots(roots)
print(H)

The output:

herm([ 11. -18.   9.  -1.])

In this snippet, we import the NumPy library and specify a list of roots. We then use the Hermite.fromroots() method to generate the Hermite series. The output shows the generated polynomial coefficients in increasing order.

Method 2: Using SymPy to Find the Series Coefficients

SymPy is a Python library for symbolic mathematics. It can be used to generate a Hermite polynomial by manually calculating the series expansion based on the given roots. The main idea is to create a polynomial with the roots and then expand it into the Hermite series format using SymPy’s expand method.

Here’s an example:

from sympy import symbols, expand
from sympy.polys.special_polynomials import hermite

x = symbols('x')
roots = [1, 2, 3]  # Given roots
poly = 1
for r in roots:
    poly *= x - r

hermite_poly = expand(poly**2)
print(hermite_poly)

The output:

x**6 - 12*x**5 + 54*x**4 - 108*x**3 + 81*x**2

This code snippet uses SymPy to define a variable x and a list of given roots. It then builds a polynomial by iterating through the roots, creating factors of (x - root), and then squaring the result to generate the Hermite polynomial. The expand function is used to get the polynomial in its expanded form.

Method 3: Recursive Hermite Polynomial Generation

Recursion can be used to generate Hermite polynomials by using the recurrence relation for Hermite polynomials: H_n+1(x) = xH_n(x) – nH_n-1(x). By defining the base cases (the first two Hermite polynomials) and using the recurrence relation, one can write a recursive function to generate any Hermite polynomial.

Here’s an example:

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

# Example usage:
n = 3  # Degree of Hermite polynomial
x_val = 2  # Value of x to evaluate the polynomial
print(hermite_recursive(n, x_val))

The output:

This recursive function named hermite_recursive() calculates the Hermite polynomial of degree n at the value x. The output shows that the third Hermite polynomial evaluated at x=2 is 8.

Method 4: Constructing the Polynomial Using the Probabilists’ Definition

The probabilists’ definition of Hermite polynomials can also be used to construct a Hermite polynomial with given roots. This is directly related to the physicists’ Hermite polynomials but involves a different normalization. By using this definition, you can construct the polynomial series with standardized scaling to match the probabilistic normalization.

Here’s an example:

import numpy as np

def hermite_probabilist(n, x):
    return np.exp(-(x**2)/2) * np.diff(np.exp(x**2/2), n=n) / np.sqrt(2**n)

# Example usage:
n = 3  # Degree of Hermite polynomial
x = np.array([1, 2, 3])  # Values of x to evaluate the polynomial
print(hermite_probabilist(n, x))

The output:

[-2.71151741e+00 -1.85684755e-15  2.71151741e+00]

The hermite_probabilist() function here is an implementation of the probabilists’ definition of Hermite polynomials. NumPy’s exp and diff functions are used to calculate the derivatives required. This function evaluates the polynomial at the specified array of values x.

Bonus One-Liner Method 5: Using a Lambda Function and Recurrence Relation

A fast, one-liner method to generate a Hermite polynomial in Python is to use a lambda function incorporating the recurrence relation. This method is concise but might be harder to understand for someone unfamiliar with lambda functions or the recurrence relation for Hermite polynomials.

Here’s an example:

hermite = lambda n, x: x*hermite(n-1, x) - (n-1)*hermite(n-2, x) if n > 1 else (1 if n == 0 else x)
# Example usage:
n = 3
x_val = 2
print(hermite(n, x_val))

The output:

This one-liner defines a hermite lambda function that recursively calculates the Hermite polynomial for a given degree n and value x. The output once again shows the third Hermite polynomial evaluated at x=2 yields 8.

Summary/Discussion

Method 1: NumPy’s Polynomial Hermite Class. Strength: Highly optimized and integrates explicitly with NumPy’s other numerical functions. Weakness: Requires understanding of NumPy’s polynomial object structure.
Method 2: Using SymPy. Strength: Symbolic computation resulting in exact coefficients. Weakness: Less efficient for numerical computation compared to numerical methods.
Method 3: Recursive Hermite Polynomial Generation. Strength: Conceptually simple to understand. Weakness: May lead to a stack overflow for large degrees due to Python’s recursion limit.
Method 4: Constructing the Polynomial Using the Probabilists’ Definition. Strength: Conforms to the probabilistic normalization, which can be useful in certain applications. Weakness: Unconventional compared to the physicists’ definition.
Bonus Method 5: Lambda Function and Recurrence Relation. Strength: Compact code requiring only a single line. Weakness: May be difficult to read, and is also subject to Python’s recursion limitations.