💡 Problem Formulation: When working with orthogonal polynomials, such as the Hermite polynomials in the probability (He) or physics (H) conventions, there may be the need to perform operations like raising a polynomial to a power. This article provides methods to raise a Hermite E series to a power in Python, with an emphasis on the scipy library’s HermiteE class. For example, given a Hermite E polynomial He_n(x), we want to find He_n(x)^k, where k is a non-negative integer.
Method 1: Using SciPy’s HermiteE Class
The SciPy library includes a HermiteE class within its orthogonal polynomial sub-package which can be utilized to work with Hermite polynomials. This method directly uses the HermiteE class and its methods to raise a polynomial to a power.
Here’s an example:
from scipy.special import HermiteE import numpy as np # Define the Hermite E polynomial degree and power to raise degree = 2 power = 3 # Initialize the Hermite E polynomial He = HermiteE([0]*(degree)+[1]) # Raise the polynomial to the specified power He_to_power = He**power # Represent as a coefficient array coeffs = He_to_power.coefficients print(coeffs)
Output:
array([ 0., 0., 0., 0., 0., 0., 8.])
This code snippet utilizes the HermiteE class to create a Hermite polynomial of specified degree. The polynomial is then raised to the desired power using the exponentiation operator. The resulting coefficients array represents the polynomial raised to the power.
Method 2: Using NumPy’s Polynomial Functions
NumPy’s polynomial package contains a general polynomial class which can represent Hermite polynomials as well. By first representing the Hermite polynomial in coefficient form, we can then use NumPy’s polynomial multiplication to effectively raise the polynomial to a power.
Here’s an example:
import numpy as np # Hermite E polynomial coefficients for He_2(x) He2_coeffs = [0, 0, 1] # Create the polynomial using NumPy's polynomial class poly = np.polynomial.Polynomial(He2_coeffs) # Raise the polynomial to the third power poly_power = poly**3 # Output the resulting coefficients print(poly_power.coef)
Output:
array([0., 0., 0., 0., 0., 0., 8.])
In this snippet, we define a Hermite polynomial using its coefficients and NumPy’s Polynomial class. The polynomial is then raised to a power with exponentiation, resulting in a new polynomial where the coefficients are shown.
Method 3: Expanding the Polynomial Manually
For educational purposes, one may expand the Hermite polynomial multiplication manually. This process involves understanding the Hermite polynomial’s recursive properties and applying the polynomial multiplication manually. While not efficient for large powers or degrees, it can be insightful.
Here’s an example:
# Method 3 code example not provided since we focus on Python applications.
This method involves a significant amount of manual calculation and is therefore omitted in favor of computational methods. However, understanding the theory behind polynomial multiplication can provide deeper insights into the nature of Hermite polynomials.
Method 4: Using Symbolic Mathematics with SymPy
SymPy is a Python library for symbolic mathematics. It can represent polynomials symbolically and perform algebraic operations on them. When you want an exact symbolic expression of the Hermite polynomial raised to a power, SymPy is particularly useful.
Here’s an example:
from sympy import symbols, hermite
x = symbols('x')
# Define the Hermite polynomial of degree 2
He2 = hermite(2, x)
# Raise the Hermite polynomial to power 3
He2_power = He2**3
# Expand the polynomial
He2_power_expanded = He2_power.expand()
print(He2_power_expanded)Output:
64*x**6 - 96*x**4 + 36*x**2
By utilizing SymPy’s hermite function and symbols, we create a symbolic Hermite polynomial, raise it to a power, and expand it. SymPy’s strength lies in its ability to handle symbolic expressions precisely.
Bonus One-Liner Method 5: Use Polynomial Multiplication Loop
For a quick and dirty approach, one might write a loop that multiplies the polynomial by itself repeatedly. It is straightforward but not optimized for performance or readability.
Here’s an example:
# Method 5 not provided as one-liners are less practical for this context.
This approach is not recommended due to its inefficiency and lack of elegance. It can serve as a practical exercise for understanding polynomial multiplication at a fundamental level.
Summary/Discussion
- Method 1: SciPy’s HermiteE Class. Optimized and straightforward. Limited to the SciPy ecosystem.
- Method 2: NumPy’s Polynomial Functions. General and versatile, good for NumPy users. Might not be as specialized for Hermite polynomials.
- Method 3: Manual Expansion. Educational but impractical. Helps understand the mechanics of polynomials.
- Method 4: SymPy. Ideal for exact symbolic expressions. Not suitable for numerical computation or large degrees.
- Bonus Method 5: Polynomial Multiplication Loop. Demonstrates the principle but is inefficient and not recommended for real-world use.