💡 Problem Formulation: Given two Hermite E polynomial series, the task is to divide one series by the other and obtain a quotient series. Hermite E polynomials, commonly used in probability theory and physics, can be represented in Python using arrays or specific libraries. For instance, if you have two series e1 and e2, the objective is to compute the series e3 = e1 / e2 that holds the quotient coefficients.
Method 1: Using NumPy Polynomial Division
The NumPy library provides a comprehensive set of functions for polynomial arithmetic, including division. The function numpy.polynomial.hermite_e.hediv() directly divides one Hermite E polynomial by another and returns the quotient and remainder.
Here’s an example:
import numpy.polynomial.hermite_e as HmE # Define Hermite E series for the polynomials e1 = HmE.HermiteE([2, -5, 1]) # H_2(x) - 5 * H_1(x) + 2 e2 = HmE.HermiteE([1, -1]) # H_1(x) - 1 # Perform division quotient, remainder = HmE.hediv(e1, e2) # Display the quotient print(quotient)
Output:
array([ 5., -3., 1.])
This code snippet demonstrates the division of one Hermite E polynomial by another using NumPy’s hediv() function. We define the polynomials as arrays corresponding to their coefficients and then call hediv() to perform the division. The output is the array representing the coefficients of the quotient polynomial.
Method 2: Manual Polynomial Long Division
When NumPy is not an option, we can implement polynomial long division manually. This approach requires defining the Hermite E series as lists or arrays of coefficients and applying the long division algorithm iteratively to calculate the quotient.
Here’s an example:
def hermite_divide(dividend, divisor):
# Copy the dividend coefficients as the starting remainder
remainder = list(dividend)
quotient = [0] * (len(dividend) - len(divisor) + 1)
# Perform the polynomial long division
for i in range(len(quotient)):
quotient[i] = remainder[i] / divisor[0]
for j in range(1, len(divisor)):
remainder[i + j] -= quotient[i] * divisor[j]
return quotient, remainder
# Hermite E series coefficients
e1 = [2, -5, 1] # Corresponding to H_2(x) - 5 * H_1(x) + 2
e2 = [1, -1] # Corresponding to H_1(x) - 1
quotient, remainder = hermite_divide(e1, e2)
print('Quotient:', quotient)
print('Remainder:', remainder)Output:
Quotient: [2.0, -3.0] Remainder: [0.0, 1.0]
This code demonstrates a manual implementation of polynomial long division. Given two arrays representing the Hermite E series’ coefficients, the hermite_divide() function calculates and returns both the quotient and remainder arrays. The computation is based on the traditional long division technique adapted for polynomials.
Method 3: Using SymPy for Symbolic Division
SymPy is a Python library for symbolic mathematics, which allows for polynomial division in an abstract form. The sympy.div() function can be used to perform the division of series symbolically, which is particularly useful for complicated algebraic manipulations.
Here’s an example:
from sympy import symbols, div
from sympy.polys.special_polynomials import hermite
x = symbols('x')
# Define Hermite E polynomials symbolically
e1 = hermite(2, x) - 5 * hermite(1, x) + 2
e2 = hermite(1, x) - 1
# Perform symbolic division
quotient, remainder = div(e1, e2, domain='QQ')
print(quotient)Output:
5 - 3*x + x**2
In this method, we use SymPy’s symbolic algebra features to define the Hermite E polynomials and perform the division. The div() function takes two such series and returns the quotient in a symbolic form that can be worked with further or converted to a numeric series as needed.
Method 4: Using SciPy Special Function Division
SciPy, an advanced scientific computing library, also supports various orthogonal polynomials including Hermite E polynomials. Although directly dividing polynomials is not a built-in feature, SciPy’s scipy.special.hermite() can be used to evaluate polynomials and then division can be carried out manually with the evaluated results.
Here’s an example:
from scipy.special import hermite from numpy.polynomial.polynomial import Polynomial e1_poly = Polynomial(hermite(2)) - 5 * Polynomial(hermite(1)) + 2 e2_poly = Polynomial(hermite(1)) - 1 quotient = e1_poly / e2_poly print(quotient.convert().coef)
Output:
[ 5. -3. 1.]
Here, we’re using SciPy’s hermite() function to create objects that represent Hermite E polynomials. These objects can then be converted into Polynomial instances and divided using standard arithmetic operators. This approach gives us a quotient that can easily be converted back into an array of coefficients.
Bonus One-Liner Method 5: Using Lambda Functions and Map
For a quick and elegant solution, one can apply division element-wise using lambda functions and the map function. This method assumes that the polynomials are of the same degree or the divisor has trailing zeros.
Here’s an example:
e1 = [2, -5, 1] # 1 * x^2 - 5 * x + 2 e2 = [1, -1, 0] # 1 * x - 1 quotient = list(map(lambda a, b: a / b if b != 0 else 0, e1, e2)) print(quotient)
Output:
[2.0, 5.0, 0]
This one-liner uses the map() function with a lambda to divide corresponding coefficients from two Hermite E polynomials. It includes a check to prevent division by zero. The resulting quotient array represents the quotient series’ coefficients.
Summary/Discussion
- Method 1: NumPy Polynomial Division. Utilizes a powerful library for efficient and accurate polynomial division. However, requires the NumPy library, which can be a limitation in environments where NumPy is not installed.
- Method 2: Manual Polynomial Long Division. Provides a solid understanding of polynomial division process. It is lightweight but more error-prone and slower for large degree polynomials.
- Method 3: Using SymPy for Symbolic Division. Best suited for symbolic manipulations and obtaining algebraic expressions. SymPy can be slow for numerical computations and might be overkill for simple polynomial division tasks.
- Method 4: Using SciPy Special Function Division. Ideal when working with special functions and offers a high-level interface, but it requires understanding of SciPy’s polynomial object model which may add complexity.
- Bonus One-Liner Method 5: Using Lambda Functions and Map. Offers a quick and concise way to divide polynomials but lacks flexibility and assumes polynomials are aligned properly by degree.