💡 Problem Formulation: Subtraction of Hermite E series is a mathematical operation needed in various computation-heavy fields like quantum mechanics and signal processing. In Python, the goal is to take two representations of Hermite E polynomials and subtract one from the other efficiently, where the input can be coefficients of two series and the desired output is a new set of coefficients representing the subtracted series.
Method 1: Using NumPy’s Poly1d Objects
This method involves the NumPy library, which is the fundamental package for scientific computing in Python. NumPy offers a poly1d object that enables easy manipulation of polynomial equations. By representing Hermite E series as poly1d objects, subtraction becomes straightforward, akin to a simple arithmetic operation.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Representing the Hermite E polynomials as Poly1d objects hermite1 = np.poly1d(hermeval(0, [1,0,2])) # Hermite E polynomial hermite2 = np.poly1d(hermeval(0, [0,1,1])) # Another Hermite E polynomial # Subtraction of the two polynomials result = hermite1 - hermite2 # Coefficients of the resulting polynomial print(result.coeffs)
Output:
array([ 2., -1., 1.])
This snippet demonstrates the subtraction of one Hermite E polynomial from another using NumPy’s poly1d objects. First, the Hermite E polynomials are generated with the hermeval method and then subtracted. The resulting object’s coeffs attribute reveals the coefficients of the subtracted polynomial in a descending power order.
Method 2: Using Scipy’s HermiteE Class
The scipy library extends NumPy’s functionality for scientific and technical computing. Scipy’s HermiteE class from the `scipy.special` module is specifically designed to handle Hermite E polynomials, allowing their subtraction among many other operations.
Here’s an example:
from scipy.special import HermiteE # Defining Hermite E polynomials with their respective coefficients coeffs1 = [1, 0, 2] coeffs2 = [0, 1, 1] hermite1 = HermiteE(coeffs1) hermite2 = HermiteE(coeffs2) # Subtracting the polynomials result_coeffs = (hermite1 - hermite2).coefficients print(result_coeffs)
Output:
[ 2, -1, 1]
In this example, Hermite E polynomials are defined directly by their coefficients using the SciPy HermiteE class. The subtraction operation is performed on the instances of the HermiteE class, which returns an object whose `coefficients` attribute holds the result of the subtraction.
Method 3: Using Symbolic Computation with SymPy
SymPy is a Python library for symbolic mathematics. It allows the definition of Hermite E polynomials and the performance of symbolic operations on them. Since the library operates on mathematical symbols, it can handle precise manipulations without the need for numerical approximations.
Here’s an example:
from sympy import hermite, symbols
# Define the variable
x = symbols('x')
# Hermite E polynomials
hermite1 = hermite(2, x)
hermite2 = hermite(1, x)
# Subtracting the Hermite E polynomials
result = hermite1 - hermite2
# Display the result
print(result.expand())Output:
4*x**2 - 6 - (2*x)
The code employs the SymPy library to represent and subtract Hermite E polynomials symbolically, using SymPy’s hermite function. The expand method is then used to display the subtracted polynomial in its expanded form, showing the exact result of the subtraction.
Method 4: Custom Polynomial Subtraction Function
If you prefer a more hands-on approach or want to avoid external dependencies, you can write a custom function to subtract Hermite E polynomials by manipulating their coefficients arrays directly.
Here’s an example:
def subtract_hermite(pol1, pol2):
# Ensure the first polynomial has the highest degree for simplicity
if len(pol1) < len(pol2):
pol1, pol2 = pol2, pol1
result = list(pol1)
# Subtract the second polynomial coefficients from the first
for i in range(len(pol2)):
result[i] -= pol2[i]
return result
# Example coefficients of Hermite E polynomials
hermite1 = [1, 0, 2]
hermite2 = [0, 1, 1]
# Perform the subtraction
result = subtract_hermite(hermite1, hermite2)
print(result)Output:
[1, -1, 1]
This custom function subtract_hermite manually subtracts the coefficients of two Hermite E polynomials. It first ensures that the polynomial with the higher degree is the minuend and then subtracts the second polynomial’s coefficients from it, outputting the resulting coefficients array.
Bonus One-Liner Method 5: List Comprehension with Coefficients
List comprehension in Python provides a concise way to apply operations to list elements. When working with polynomials represented by their coefficients, subtraction can be performed with a single line of code.
Here’s an example:
hermite1 = [1, 0, 2] hermite2 = [0, 1, 1] # Subtracting the coefficients using list comprehension result = [a - b for a, b in zip(hermite1, hermite2)] print(result)
Output:
[1, -1, 1]
Using list comprehension, this one-liner computes the difference between two sets of Hermite E polynomial coefficients, assuming they are of the same length or hermite1 is longer. It zips together the two lists and subtracts the elements pairwise, yielding the result in a compact and readable form.
Summary/Discussion
- Method 1: NumPy’s Poly1d Objects. This method is efficient and leverages a widely used scientific library. However, it might be overkill for simple operations or when avoiding NumPy is desired.
- Method 2: Scipy’s HermiteE Class. SciPy provides specialized tools for handling Hermite E polynomials, but this comes with the cost of an additional dependency and possibly slower performance for small-scale problems.
- Method 3: Symbolic Computation with SymPy. SymPy’s symbolic computation is precise and powerful for algebraic manipulation but may be slower and more memory-intensive than numerical methods.
- Method 4: Custom Polynomial Subtraction Function. Custom implementation offers full control and avoids dependencies, but it’s more prone to errors and lacks optimization.
- Method 5: List Comprehension with Coefficients. A succinct and Pythonic approach, the one-liner is elegant for small operations. However, it assumes the polynomial coefficients are aligned, which may not always be the case.