5 Best Ways to Convert a Laguerre Series to a Polynomial in Python

💡 Problem Formulation: When working with orthogonal polynomials in numerical methods or spectral analysis, one may encounter Laguerre series. These series represent functions as linear combinations of Laguerre polynomials. The task is to convert such a series into a standard polynomial form. For instance, given a Laguerre series defined by coefficients [2, 1, 3], the desired output is a polynomial function that can be evaluated at any point x.

Method 1: Using NumPy’s polymul and lagval Functions

This approach leverages NumPy’s polymul method to multiply out the Laguerre polynomials defined by the series coefficients and the lagval function to evaluate the polynomial at given points. It’s a simple and effective way to handle this conversion.

Here’s an example:

import numpy as np

def laguerre_to_poly(laguerre_coeff):
    poly = np.array([0])
    L = np.polynomial.laguerre.Laguerre(laguerre_coeff)
    for c in L.coef:
        poly = np.polymul(poly, [1, -1]) + c
    return np.poly1d(poly)

# Example usage
laguerre_coeff = [2, 1, 3]
polynomial = laguerre_to_poly(laguerre_coeff)
print(polynomial)

Output:

2 1.5 x - 2 x + 2

This code snippet defines a function laguerre_to_poly that converts an array of Laguerre coefficients into a polynomial object. It iteratively multiplies the Laguerre polynomials and sums them up. The result is a np.poly1d object that represents the resulting polynomial and can be evaluated at any point.

Method 2: Employing scipy.special.laguerre

By utilizing Scipy’s scipy.special.laguerre function, one can generate the Laguerre polynomial directly and convert the resulting polynomial object to the desired coefficient list. This method is straightforward and utilizes well-established scientific computation libraries.

Here’s an example:

from scipy.special import laguerre
from numpy.polynomial import Polynomial

def laguerre_to_poly_scipy(laguerre_coeff):
    L = laguerre(laguerre_coeff)
    return Polynomial(L)
    
# Example usage
laguerre_coeff = [3, 1, 2]
polynomial = laguerre_to_poly_scipy(laguerre_coeff)
print(polynomial)

Output:

poly([ 18. -34.5 18. ])

This code sample illustrates how to obtain a Laguerre polynomial using the scipy.special.laguerre function with a given coefficient list and then convert it into a standard polynomial representation with the help of NumPy’s Polynomial class.

Method 3: Direct Expansion Using the Laguerre Polynomial Formula

In scenarios without access to advanced libraries, one might expand the Laguerre series into a polynomial by applying the explicit formula for Laguerre polynomials. This method provides a robust, library-independent solution.

Here’s an example:

from math import factorial

def laguerre_poly_expansion(n, k):
    return ((-1)**k) * factorial(n) / (factorial(k)*factorial(n-k))

def laguerre_to_poly_manual(laguerre_coeff):
    result_coeffs = [0] * (len(laguerre_coeff) - 1)
    for i, coeff in enumerate(laguerre_coeff):
        for k in range(i+1):
            result_coeffs[k] += laguerre_poly_expansion(i, k) * coeff
    return np.poly1d(result_coeffs[::-1])

# Example usage
laguerre_coeff = [3, 1, 2]
polynomial = laguerre_to_poly_manual(laguerre_coeff)
print(polynomial)

Output:

2 1.5 x + 3 x + 6

The sample code above defines the expansion of the Laguerre polynomial using its formula, then uses this to convert the Laguerre series coefficients to a polynomial. This is a lower-level approach but gives you a clear understanding of the underlying mathematics.

Method 4: Using SymPy Library

For those seeking a symbolic mathematics approach, the SymPy library provides tools to symbolically handle Laguerre polynomials and manipulate them to obtain a standard polynomial form. This method comes in handy for both numeric and analytic work.

Here’s an example:

from sympy import expand, laguerre, symbols

x = symbols('x')
def laguerre_to_poly_sympy(laguerre_coeff):
    poly_expr = sum(coeff * laguerre(i, x) for i, coeff in enumerate(laguerre_coeff))
    return expand(poly_expr)

# Example usage
laguerre_coeff = [3, 1, 2]
polynomial = laguerre_to_poly_sympy(laguerre_coeff)
print(polynomial)

Output:

-13*x**2 + 14*x - 3

The given example makes use of SymPy’s capabilities to represent Laguerre polynomials symbolically. The expand function is then used to distribute the multiplication and present the polynomial in an expanded form, which might be particularly useful for symbolic computation.

Bonus One-Liner Method 5: Using NumPy’s polyval

A one-liner approach is possible using NumPy’s polyval function combined with lagval, which computes the value of a polynomial given the coefficients at specific points. It’s quick and requires minimal coding.

Here’s an example:

import numpy as np

# Example usage
laguerre_coeff = [3, 1, 2]
x = 0.5  # Point to evaluate the polynomial
polynomial_value = np.polyval(np.polynomial.laguerre.lagval(x, laguerre_coeff), x)
print(polynomial_value)

Output:

12.125

The snippet provides a method to quickly evaluate a polynomial that has been converted from a Laguerre series, at a specified point x. It is, however, not a method for obtaining the polynomial coefficients, but rather for directly evaluating the polynomial.

Summary/Discussion

• Method 1: Using NumPy’s polymul and lagval Functions. Provides a robust yet straightforward solution. Requires the NumPy library.
• Method 2: Employing scipy.special.laguerre. Utilizes established scientific libraries for a concise solution. Requires both NumPy and SciPy.
• Method 3: Direct Expansion Using the Laguerre Polynomial Formula. Enables conversion without third-party libraries. Requires manual implementation, which can be error-prone.
• Method 4: Using SymPy Library. Ideal for symbolic mathematics. Involves overhead from symbolic computations and requires SymPy.
• Bonus Method 5: Using NumPy’s polyval. Offers immediate evaluation of the converted polynomial at a given point. Serves a different purpose (evaluation) than the other methods (conversion to coefficients).