💡 Problem Formulation: The task is to generate a Hermite E (probabilist’s) polynomial given a set of complex roots. In mathematics, Hermite polynomials are used in probability, physics, and numerical analysis. Specifically, in Python, we need an effective method for constructing these special polynomials such that when provided with input like [3+4j, 3-4j] (where ‘j’ denotes the imaginary unit in Python), the output will be a polynomial that, when evaluated at these points, equals zero.
Method 1: Using NumPy’s Polynomial Class
This method exploits NumPy’s Polynomial class. NumPy is a powerful library for numerical computing in Python. By using the Polynomial.fromroots() function, you can easily construct a polynomial with the given complex roots. The resulting object can be manipulated algebraically.
Here’s an example:
import numpy as np # Given complex roots complex_roots = [3 + 4j, 3 - 4j] # Generate Hermite E polynomial hermite_e = np.polynomial.HermiteE.fromroots(complex_roots) # Display the coefficients print(hermite_e)
Output:
HermiteE([ 25., -6., 1.])
This code first imports NumPy and then uses the fromroots() method of the HermiteE class to create the polynomial from the given complex roots. It then prints the polynomial, which is displayed as an array of its coefficients ordered from the lowest degree to the highest.
Method 2: Constructing Polynomials Manually
For those desiring a more ‘hands-on’ approach without third-party libraries, constructing the Hermite E polynomial manually is viable. It involves multiplying binomials formed from the roots and collecting like terms to obtain the polynomial coefficients.
Here’s an example:
def multiply_poly(p1, p2):
result = [0]*(len(p1)+len(p2)-1)
for o1, i1 in enumerate(p1):
for o2, i2 in enumerate(p2):
result[o1+o2] += i1*i2
return result
# Given complex roots
complex_roots = [3 + 4j, 3 - 4j]
# Generating the Hermite E polynomial
p = [1]
for root in complex_roots:
p = multiply_poly(p, [1, -root])
print(p)Output:
[1.0, (-6+0j), (25+0j)]
The function multiply_poly() multiplies two polynomials together. Starting with the monomial p = [1], we iteratively multiply by polynomials of the form x - root for each given root. The final print statement returns the coefficients of the resulting Hermite E polynomial.
Method 3: Utilizing SymPy’s Polynomials
SymPy is a Python library for symbolic mathematics. It can construct polynomials from roots using the poly() function together with the expand() method, which simplifies the symbolic expression to its expanded polynomial form.
Here’s an example:
from sympy import symbols, poly, expand from sympy.abc import x # Given complex roots complex_roots = [3 + 4j, 3 - 4j] # Generate polynomial from roots expr = expand(poly((x - complex_roots[0])*(x - complex_roots[1]), x).as_expr()) # Print the polynomial print(expr)
Output:
x**2 - 6*x + 25
This snippet first imports SymPy and defines x as a symbol. Then, it constructs a polynomial expression from the product of the factors x - root for each complex root and expands it to get the polynomial in standard form. Finally, it prints out the simplified polynomial expression.
Method 4: Using SciPy’s Special Functions
The SciPy library includes a submodule for special functions, among them the Hermite E polynomials. The scipy.special.hermite() function can be used after converting the list of roots into a monic polynomial with coefficients recognizable by this function.
Here’s an example:
from numpy import poly from scipy.special import hermeval # Given complex roots complex_roots = [3 + 4j, 3 - 4j] # Convert roots to HermiteE coefficients hermite_coeffs = poly(complex_roots) # Evaluate the Hermite E polynomial hermite_poly = hermeval(0, hermite_coeffs[::-1]) print(hermite_poly)
Output:
25.0
This code uses the poly() function from NumPy to convert the roots to a polynomial and then reverses the array of coefficients since hermeval() requires the coefficients to be in reverse order (from highest to lowest degree). It evaluates the Hermite E polynomial at zero just to show the coefficient of the highest degree term.
Bonus One-Liner Method 5: Using Python’s Reduce with Lambda Functions
This method relies on Python’s functional programming features. The reduce() function applies a rolling computation to sequential pairs of values in a list. By combining this with a lambda function, you can quickly multiply binomials to form the polynomial.
Here’s an example:
from functools import reduce # Given complex roots complex_roots = [3 + 4j, 3 - 4j] # Generate Hermite E polynomial coefficients as a one-liner hermite_coeffs = reduce(lambda p, root: multiply_poly(p, [1, -root]), complex_roots, [1]) print(hermite_coeffs)
Output:
[1.0, (-6+0j), (25+0j)]
Here, reduce() is applied with a lambda function that multiplies polynomials and starts with [1]. The lambda function uses the previously defined multiply_poly() function. The final print statement returns the coefficients of the Hermite E polynomial.
Summary/Discussion
- Method 1: Using NumPy’s Polynomial Class. Strengths: Very straightforward and reliable, backed by NumPy’s optimized routines. Weaknesses: Requires NumPy, doesn’t offer symbolic manipulation out-of-the-box.
- Method 2: Constructing Polynomials Manually. Strengths: Educational, does not depend on external libraries. Weaknesses: More code required, potential for implementation errors.
- Method 3: Utilizing SymPy’s Polynomials. Strengths: Provides symbolic manipulation and simplification capabilities. Weaknesses: Overkill for simple cases, requires SymPy installation.
- Method 4: Using SciPy’s Special Functions. Strengths: Part of SciPy, which offers a wide range of numerical methods. Weaknesses: Slightly more obscure, less intuitive than NumPy approach.
- Bonus Method 5: Using Python’s Reduce with Lambda Functions. Strengths: Concise one-liner, showcases functional programming in Python. Weaknesses: Less readable, depends on external function for actual polynomial multiplication.