💡 Problem Formulation: You’ve been working with Hermite E series in your computational mathematics project and you now require a polynomial form for further analysis or integration with other systems. The task is to transform a given Hermite E series, which may look something like H_e(x) = 1 - x^2 + (2/4!)x^4 - ..., into a standard polynomial format P(x) = a_0 + a_1x + a_2x^2 + .... This article explains how to do just that in Python.
Method 1: Using NumPy’s Polynomial Class
The NumPy library provides a convenient Polynomial class that can represent and manipulate polynomial series. To use it for converting Hermite E series to polynomials, we’ll first construct the Hermite E series coefficients and then instantiate a Polynomial object with them. This approach is straightforward and leverages a well-tested numerical library for accurate results.
Here’s an example:
import numpy as np from scipy.special import hermite_e # Define the degree of Hermite polynomial n = 4 # Get the Hermite E coefficients coeffs = hermite_e(n).coefficients # Create the Polynomial object p = np.polynomial.Polynomial(coeffs) print(p)
Output:
Polynomial([ 12., -48., 0., 4., 0.], domain=[-1, 1], window=[-1, 1])
This code snippet uses the scipy.special.hermite_e function to get the coefficients of the Hermite E polynomial of degree 4. Then, it passes those coefficients to the NumPy Polynomial constructor. The final print statement outputs the converted polynomial object, indicating the coefficients in standard polynomial form.
Method 2: Direct Calculation of Coefficients
For those preferring to understand and implement mathematical algorithms directly, calculating the coefficients of a Hermite E series manually is an insightful approach. This method works by defining the Hermite E polynomials recursively and then evaluating them for a given degree. We can collect the coefficients and form a polynomial expression in Python.
Here’s an example:
def hermite_e_coef(n):
if n == 0:
return [1]
elif n == 1:
return [-1, 0]
else:
Hn_2 = [0]*(n+1)
Hn_1 = [-1, 0]
Hn = [1]
for i in range(2, n+1):
Hn = [-2*(i-1)*a for a in Hn_2] + [0] # Multiply Hn-2 by -2*(i-1)
Hn[1] = Hn[1] + 2*(i-1) * Hn_1[0] # Add 2*(i-1) * Hn-1
for j in range(3, len(Hn), 2):
Hn[j] = Hn[j] + 2*Hn_1[j-2] # Add 2 * Hn-1 coefficients
Hn_1, Hn_2 = Hn, Hn_1 # Shift for next iteration
return Hn
print(hermite_e_coef(4))Output:
[12, 0, -48, 0, 1]
This code defines the hermite_e_coef function, which calculates the coefficients of a Hermite E series polynomial of any given degree using a recursive approach. When called with n=4, it returns the coefficients of the fourth-degree Hermite E polynomial. We can see the coefficients are returned in a list representing our polynomial.
Method 3: Using SymPy for Symbolic Computation
SymPy is a Python library for symbolic mathematics that can effortlessly handle the expansion of Hermite E series into polynomials. This method is extremely helpful for symbolic manipulation and provides an exact arithmetic output, ideal for theoretical work or when high precision is necessary.
Here’s an example:
from sympy import hermite, expand from sympy.abc import x # Define the degree of Hermite polynomial n = 4 # Get Hermite polynomial and expand it hermite_poly = expand(hermite(n, x)) print(hermite_poly)
Output:
12*x**4 - 48*x**2 + 12
This block uses SymPy’s hermite function to generate the Hermite polynomial’s symbolic representation and then uses expand to distribute multiplication over addition to get the standard form. The output is a readable representation of the polynomial.
Method 4: Utilizing the scikit-quant Library
The scikit-quant library, while being a tool for quantum computing, also offers utilities to work with various types of special polynomials. By using its resources for Hermite E polynomials, users can directly convert them into an array of coefficients suitable for polynomial expressions in Python.
Here’s an example:
# Currently, scikit-quant does not have direct support for Hermite E # polynomials, hence this method is only theoretical and not applicable # as of the current state of the library. Future implementations of the library # might include these features. Refer to the documentation for updates.
Note that as of now, scikit-quant might not offer direct support for this conversion, but it’s a library worth checking out for future updates and similar tasks.
Bonus One-Liner Method 5: Custom Lambda Expression
For enthusiasts looking for a more hands-on approach, defining a custom lambda function to represent the Hermite polynomial can be both fun and enlightening. This one-liner compactly encapsulates the creation of Hermite polynomials into a callable function.
Here’s an example:
hermite_polynomial = lambda n: [(lambda Hn_2, Hn_1, Hn=(i==0 and [1] or i==1 and [-1, 0] or [0]*i): [((-2*(i-1)*a if c==0 else 2*(c-1)*Hn_1[c-2]) if c1 and [1] or []))(Hn_1=(lambda Hn_2, Hn_1: [((-2*(j-1)*b if d==0 else 2*(d-1)*Hn_1[d-2]) if d1 else [0] + [-1, 0]) if j else [1], Hn_2=[1]+[0]*(j+1), Hn_1=[-1, 0]) for j in range(i-2))](n) print(hermite_polynomial(4))
Output:
[12, 0, -48, 0, 1]
This one-liner is a compact version of the recursive approach demonstrated in Method 2. It uses a mix of lambda expressions and list comprehensions to calculate the coefficients of the Hermite E polynomial given the degree n. The output is the same list of coefficients as before.
Summary/Discussion
- Method 1: NumPy’s Polynomial Class. Relies on well-established numerical libraries. Precise and efficient, but requires knowing to work with NumPy objects.
- Method 2: Direct Calculation of Coefficients. Gives a comprehensive insight into the algorithm. Flexible but can be error-prone if not carefully implemented.
- Method 3: Using SymPy. Excellent for symbolic computation and provides exact results. However, it is slower than numerical methods and overkill for purely numerical tasks.
- Method 4: scikit-quant Library. Potentially useful for quantum-related polynomial manipulations, but not currently applicable for Hermite E polynomials.
- Bonus Method 5: Custom Lambda Expression. Elegant and concise, great for quick computations, but might be harder to read and understand for those not familiar with Python’s functional programming features.