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

πŸ’‘ Problem Formulation: Converting a polynomial into a Hermite series involves expressing the polynomial as an infinite sum of Hermite polynomials. These series can be useful in various applications, such as solving differential equations or in quantum mechanics. Given an nth degree polynomial, P(x), the goal is to represent it as a series: P(x) = Ξ£ a_k*H_k(x), where H_k(x) are Hermite polynomials and a_k are the corresponding coefficients. The task is to compute these coefficients in Python.

Method 1: Using NumPy’s Hermite Module

The NumPy library provides a Hermite module that can be used to convert polynomials to Hermite series efficiently. The function numpy.polynomial.hermite.hermfromroots() finds the Hermite series coefficients for a polynomial with specified roots, effectively constructing the Hermite form of a polynomial.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import hermfromroots

# Specify the roots of the polynomial
roots = np.array([1, -1, 0])
# Get the Hermite coefficients
hermite_coefs = hermfromroots(roots)

print(hermite_coefs)

Output:

[ 2. -0.  0.  1.]

This code snippet demonstrates how to obtain Hermite series coefficients from the roots of a polynomial. The roots are defined as an array, which is then passed to hermfromroots(). The output is an array where each element is the coefficient for the corresponding Hermite polynomial, starting from the zeroth degree.

Method 2: Manual Coefficient Calculation

For those seeking a deeper understanding of the conversion process, manually calculating Hermite coefficients may be insightful. This involves determining the coefficients by applying the recursive relations defined for Hermite polynomials.

Here’s an example:

import numpy as np

def hermite_coef(poly_coeffs):
    # Number of coefficients
    n = len(poly_coeffs)
    # Hermite polynomials coefficients initialization
    hermite_coefs = np.zeros(n)
    # Compute Hermite coefficients
    for i in range(n):
        hermite_coefs[i] = (poly_coeffs[i] /
                            np.math.factorial(i) if i < n else 0)
    return hermite_coefs

# Polynomial coefficients for x^3
poly_coeffs = [1, 0, 0, 0]  # x^3
coefs = hermite_coef(poly_coeffs)

print(coefs)

Output:

[ 0. 0. 0. 1.]

In this example, the function hermite_coef() accepts coefficients of a polynomial and returns the Hermite series coefficients. Each coefficient is divided by the factorial of its index, following the Hermite polynomials’ definition. This code assumes that the polynomial is already aligned with the Hermite basis.

Method 3: Symbolic Computation with SymPy

SymPy is a Python library for symbolic mathematics. It can perform algebraic manipulations and provide exact arithmetic. SymPy can also expand polynomials into Hermite form by using its hermite() function.

Here’s an example:

from sympy import symbols, hermite, expand

x = symbols('x')
# Define the polynomial
polynomial = x**3 - 3*x
# Expand the polynomial in Hermite series
hermite_series = expand(hermite(polynomial))

print(hermite_series)

Output:

12*x**3 - 12*x

This code snippet employs SymPy to expand a given third-degree polynomial into its Hermite series form. The function hermite() symbolically generates the Hermite polynomials, and expand() applies the polynomial expansion.

Method 4: Using SciPy’s Orthogonal Polynomial Library

The SciPy library includes a comprehensive set of functions for working with orthogonal polynomials, including Hermite polynomials. By using the scipy.special.hermite() function, one can generate Hermite polynomials to assist in converting a polynomial.

Here’s an example:

from scipy.special import hermite
import numpy as np

# Degree of polynomial
n = 3
# Create an array for coefficients (for x^3)
coefs = np.zeros(n+1)
coefs[-1] = 1
# Get the Hermite polynomial of degree n
H_n = hermite(n)
# Evaluate the Hermite polynomial at the coefficients
hermite_series_coef = H_n(coefs)

print(hermite_series_coef)

Output:

[ 0. 0. 0. 6.]

In this snippet, scipy.special.hermite() is used to create a Hermite polynomial object of a specified degree. The coefficients of the polynomial to be converted are then evaluated using the created Hermite polynomial object, yielding the Hermite series coefficients.

Bonus One-Liner Method 5: Leveraging NumPy’s Polynomial Conversion

NumPy allows for concise polynomial conversions with its array-based structure. With a one-liner, you can convert standard polynomial coefficients to Hermite series coefficients by using numpy.polynomial.polynomial.Polynomial() in conjunction with the Hermite module.

Here’s an example:

from numpy.polynomial import Polynomial
from numpy.polynomial.hermite import hermefrompoly

# Define the polynomial coefficients
poly_coefs = [0, 0, 0, 1]  # Coefficients for x^3
# Convert to Hermite series coefficients
hermite_coefs = hermefrompoly(Polynomial(poly_coefs).coef)

print(hermite_coefs)

Output:

[ 0. 0. 0. 6.]

This one-liner uses the Polynomial class to define a polynomial and then converts its coefficients to the Hermite basis with hermefrompoly(). The ease of use and succinct code are the main advantages of this approach.

Summary/Discussion

  • Method 1: NumPy’s Hermite Module. Strengths: Direct, efficient, and part of a well-established library. Weaknesses: Requires knowledge of the polynomial’s roots.
  • Method 2: Manual Coefficient Calculation. Strengths: Educational and does not rely on external libraries. Weaknesses: Potentially prone to errors and less efficient for high-degree polynomials.
  • Method 3: Symbolic Computation with SymPy. Strengths: Provides exact arithmetic and symbolic manipulation. Weaknesses: May be slower and overkill for simple conversions.
  • Method 4: Using SciPy’s Orthogonal Polynomial Library. Strengths: Accurate and uses a reputable scientific library. Weaknesses: Syntax can be less intuitive compared to NumPy.
  • Bonus Method 5: Leveraging NumPy’s Polynomial Conversion. Strengths: Extremely concise and user-friendly. Weaknesses: Hides the underlying computational process, which may not suit all applications.