Convert a polynomial to Laguerre series in Python

Laguerre polynomials are a sequence of orthogonal polynomials widely used in physics and engineering, particularly in quantum mechanics and numerical analysis. They are solutions to Laguerre's differential equation and are used to express functions in terms of a series expansion known as a Laguerre series. In many applications, it is useful to convert a standard polynomial form into a Laguerre series.

Example

Input: Polynomial: 15 - 40*x + 22.5*x**2 - 4*x**3 + 0.20833333*x**4

Coefficients of the converted polynomial to Laguerre series: [0.99999992 2.00000032 2.99999952 4.00000032 4.99999992]

This article explores what a Laguerre series is and demonstrates two methods to perform this conversion in Python.

What is a Laguerre Series?

A Laguerre series is a representation of a function in terms of Laguerre polynomials. The generalized Laguerre polynomials L_n^{(\alpha)}(x) are defined by the Rodrigues' formula:

L_n^{(\alpha)}(x) = \frac{e^x x^{-\alpha}}{n!} \frac{d^n}{dx^n} \left( e^{-x} x^{n+\alpha} \right)

For the special case where α=0, these polynomials are referred to as the "standard" Laguerre polynomials. A function f(x) can be approximated by a finite Laguerre series:

f(x) \approx \sum_{n=0}^{N} c_n L_n^{(\alpha)}(x)

where cn are the coefficients of the series. Converting this series to a standard polynomial involves expressing the Laguerre polynomials in their expanded polynomial form and then combining like terms.

Converting a Polynomial to Laguerre Series

Now, let us see a few different methods by which we can convert a Polynomial to Laguerre Series in Python.

Using SymPy

SymPy is a Python library for symbolic mathematics. It provides tools to work with polynomial expansions and symbolic representations of mathematical expressions, including Laguerre polynomials. We can install sumpy by writing the following command in the terminal.

Example

In this example, we first imported the necessary modules and then defines symbolic variable and a polynomial. Then we generate the laguerre by setting the degree of Laguerre polynomials to be generated as 4. Then loop over the range of degrees (0 to 4) to calculate each coefficient and append it to the coefficient list.

import sympy as sp
from sympy.integrals.quadrature import gauss_laguerre

# Define the variable and the polynomial
x = sp.symbols('x')
polynomial = 15 - 40*x + 22.5*x**2 - 4*x**3 + 0.20833333*x**4

# Generate Laguerre polynomials up to a certain degree
degree = 4
laguerre_polynomials = [sp.laguerre(n, x) for n in range(degree + 1)]

# Calculate the coefficients using the orthogonality of Laguerre polynomials
coefficients = []
for n in range(degree + 1):
    coef = sp.integrate(polynomial * sp.exp(-x) * laguerre_polynomials[n], (x, 0, sp.oo))
    coefficients.append(coef / sp.factorial(n))

# Display the Laguerre series coefficients
print(coefficients)

Output:

[0.999999920000000, 2.00000031999999, 1.49999975999998, 0.666666719999967, 0.208333329999997]

Time Complexity: O(n^2) for integrating and finding coefficients using orthogonality.

Space Complexity: O(n) for storing the polynomial terms and Laguerre coefficients.

Using NumPy

NumPy, a fundamental library for scientific computing in Python, also supports operations with Laguerre polynomials. It includes the numpy.polynomial.laguerre module, which can be used to handle Laguerre series.

Example

In this example, we will first import the necessary modules. Then define the coefficient and create a polynomial object using the Polynomial() function. Then we define a range for x values using numpy's linespace() function with an array of 500 evenly spaced values between 0 and 10. The np.polynomial.laguerre.lagfit() function fits the polynomial values (y_values) to Laguerre polynomials using the lagfit function. The deg=4 argument specifies that the fitting should consider Laguerre polynomials up to degree 4.

The output will be the coefficients of the equivalent Laguerre series. The fitting process in the Laguerre class transforms the standard polynomial into a Laguerre series.

import numpy as np
from numpy.polynomial.polynomial import Polynomial
from numpy.polynomial.laguerre import lagval

# Define the coefficients of the standard polynomial (e.g., 2 + 3x + x^2)
coefficients = [15, -40, 22.5, -4, 0.20833333]

# Create a Polynomial object
poly = Polynomial(coefficients)

# Define a range for x values for approximation
x_values = np.linspace(0, 10, 500)
y_values = poly(x_values)

# Fit the polynomial values to Laguerre polynomials
laguerre_coefficients = np.polynomial.laguerre.lagfit(x_values, y_values, deg=4)

# Display the Laguerre series coefficients
print(laguerre_coefficients)

Output:

[0.99999992 2.00000032 2.99999952 4.00000032 4.99999992]

Time Complexity: O(n) for fitting the polynomial values to Laguerre polynomials.

Space Complexity: O(n) for storing the coefficients of the polynomial and Laguerre series.