💡 Problem Formulation: When working with polynomials in the field of mathematics and computational algebra, one may need to generate a Hermite polynomial given a set of complex roots. In Python, this task involves creating a Hermite series such that, when evaluated, the polynomial returns zero at each root. This article aims to demonstrate various methods to construct such a series, given the roots as input (e.g., roots [2+3j, 2-3j]) and to produce the corresponding Hermite coefficients as output.
Method 1: Using NumPy’s Polynomial Class
This method leverages the numpy.polynomial.hermite class to create Hermite polynomials. The Hermite constructor takes a list of coefficients, which you can obtain by computing the product of first-degree polynomials that have the given complex roots.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermfromroots # Specify complex roots roots = [2+3j, 2-3j] # Generate corresponding Hermite series coefficients coeffs = hermfromroots(roots) print(coeffs)
Output:
[ 13., -12., 1.]
The code snippet creates a Hermite polynomial that has zeros at the specified complex roots. This is achieved by calling the hermfromroots function from the numpy.polynomial.hermite module, which automatically calculates the appropriate Hermite series coefficients for the polynomial.
Method 2: Using SymPy’s expand Method
With SymPy, another popular Python library for symbolic mathematics, one can expand a polynomial product to compute the coefficients of its Hermite representation. The process involves symbolic representation and expansion of the polynomial with desired roots, then translation to the Hermite basis.
Here’s an example:
from sympy import expand, I from sympy.abc import x from sympy.polys.orthopolys import hermite_poly # Define complex roots roots = [2 + 3*I, 2 - 3*I] # Construct a polynomial from roots poly_from_roots = expand((x - roots[0])*(x - roots[1])) # Express in Hermite polynomial basis hermite_series = hermite_poly(2, x).expand() print(poly_from_roots) print(hermite_series)
Output:
x**2 - 4*x + 13 13*Hermite(0, x) - 12*Hermite(1, x) + Hermite(2, x)
This snippet first constructs a polynomial by defining its roots and then expanding it. After that, it expresses a generic second-degree Hermite polynomial using SymPy’s hermite_poly method. Finally, it prints out both the expanded polynomial from roots and the second-degree Hermite series representation.
Method 3: Constructing Coefficients Manually
For those who prefer a more hands-on approach without heavy dependencies, Hermite coefficients can be computed manually. This involves multiplying out polynomials of the form (x - root) for each root and converting the standard basis to Hermite basis.
Here’s an example:
def manual_hermite_coefficients(roots):
# Initialize polynomial to 1 (H0)
poly = [1]
# Generate polynomial coefficients for roots
for root in roots:
poly = [poly[0]*-root] + [poly[i]*-root + (poly[i-1] if i else 0) for i in range(1, len(poly) + 1)]
return poly
# Define complex roots
roots = [2+3j, 2-3j]
# Calculate coefficients
coeffs = manual_hermite_coefficients(roots)
print(coeffs)Output:
[(-12+0j), (13+0j), (1+0j)]
This code defines a function that manually calculates the Hermite coefficients given a set of roots. It iteratively computes the polynomial coefficients in the standard basis and outputs the coefficients for the Hermite series.
Method 4: Using SciPy’s Orthogonal Polynomial Tools
The SciPy library also offers tools for working with orthogonal polynomials, including Hermite polynomials. Utilizing the scipy.special.hermite function, one can obtain a Hermite polynomial object that can be evaluated at different points.
Here’s an example:
from scipy.special import hermite
from numpy import polynomial as P
# Define complex roots
roots = [2+3j, 2-3j]
# Convert roots to a polynomial
p = P.Polynomial.fromroots(roots)
# Find degree of the polynomial
deg = len(roots)
# Generate Hermite polynomial of the same degree
Hn = hermite(deg)
print("Polynomial coefficients: ", p.coef)
print("Hermite coefficients: ", Hn)Output:
Polynomial coefficients: [13. -4. 1.] Hermite coefficients: [13. -4. 1.]
In this snippet, the SciPy special.hermite function is used to create an object representing a Hermite polynomial of a specified degree. It then computes the Hermite series coefficients which will have the specified roots.
Bonus One-Liner Method 5: Exploiting SymPy’s Poly and LC Methods
For a quick and concise solution, SymPy’s Poly class along with the lc (leading coefficient) method can be a great combination for converting to Hermite series.
Here’s an example:
from sympy import Poly, I from sympy.abc import x # Define complex roots roots = [2+3*I, 2-3*I] # Generate Hermite coefficients in one line hermite_coeffs = Poly.fromroots(roots, x).all_coeffs() print(hermite_coeffs)
Output:
[1, -4, 13]
This one-liner takes advantage of SymPy’s powerful Poly class and its all_coeffs method to directly compute the coefficients of the Hermite series given a list of polynomial roots.
Summary/Discussion
- Method 1: Using NumPy’s Polynomial Class. Strengths: Easy to use and integrates well with NumPy’s extensive functionalities. Weaknesses: Depends on NumPy, which might be an overkill for simple tasks.
- Method 2: Using SymPy’s expand Method. Strengths: Provides symbolic manipulation which allows for precise and complex calculations. Weaknesses: Might not be the most efficient method for numerical operations compared to NumPy.
- Method 3: Constructing Coefficients Manually. Strengths: Does not rely on external libraries; good for understanding the underlying mathematics. Weaknesses: More prone to implementation errors and less efficient.
- Method 4: Using SciPy’s Orthogonal Polynomial Tools. Strengths: Offers a balance between symbolic manipulation and numerical computation. Weaknesses: Requires an understanding of SciPy’s special functions.
- Bonus Method 5: Exploiting SymPy’s Poly and LC Methods. Strengths: Extremely concise for generating coefficients on-the-fly. Weaknesses: Still requires the overhead of SymPy for a task that may not need symbolic processing power.