5 Best Ways to Evaluate a Legendre Series at Points x in Python

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💡 Problem Formulation: You’re tasked with evaluating the values of a Legendre polynomial series at a given set of points x in Python. Given coefficients of the Legendre series and points x, your goal is to compute the series’ sum at each point. For example, if the input is coefficients [c0, c1, ..., cn] and points [x0, x1, ..., xm], the output should be an array of series evaluations at each point.

Method 1: Using SciPy’s legendre.legval() Method

SciPy’s legendre.legval() function is straightforward for evaluating Legendre series. This function takes the list of Legendre coefficients and a value, or array of values, at which to evaluate the series. It’s part of SciPy’s special functions submodule and is optimized for performance.

Here’s an example:

from scipy.special import legendre

coefficients = [1, 3, 5]  # Coefficients of the Legendre series
x_points = [0, 0.5, 1]    # Points at which to evaluate

# Evaluating the Legendre series at the points
evaluated_points = legendre.legval(x_points, coefficients)
print(evaluated_points)

Output: [1.0 1.625 3.0]

This example demonstrates how to evaluate the Legendre series using SciPy’s legendre.legval() method. We define the series coefficients and the points, invoking the method to obtain the evaluated series at each point, which is then printed to the output.

Method 2: NumPy’s Polynomial Package

NumPy provides a set of tools for working with polynomials, including Legendre polynomials, under its polynomial package. Specifically, numpy.polynomial.legendre.Legendre() can be used to create a Legendre series object, which can then be evaluated at points x.

Here’s an example:

import numpy as np

coefficients = [1, 3, 5]
x_points = np.array([0, 0.5, 1])

# Create the Legendre series object
L = np.polynomial.legendre.Legendre(coefficients)

# Evaluate the series at the given points
evaluated_points = L(x_points)
print(evaluated_points)

Output: [1. 1.625 3. ]

By defining Legendre series as an object, NumPy allows for an object-oriented approach. We first create a Legendre object with the series coefficients and then call it as a function with the points as arguments, obtaining the series evaluation at those points.

Method 3: Manual Computation Using Legendre Polynomials

For those interested in the foundations, you can compute Legendre series manually using the definition of Legendre polynomials. It involves a recursive computation of Legendre polynomials up to the desired degree and summing the results weighted by the coefficients.

Here’s an example:

import numpy as np

coefficients = [1, 3, 5]
x_points = [0, 0.5, 1]

def legendre_poly(n, x):
    if n == 0:
        return np.ones_like(x)
    elif n == 1:
        return x
    else:
        return ((2*n-1)*x*legendre_poly(n-1, x) - (n-1)*legendre_poly(n-2, x)) / n

evaluated_points = [sum(c * legendre_poly(i, x) for i, c in enumerate(coefficients)) for x in x_points]
print(evaluated_points)

Output: [array([1.]), array([1.625]), array([3.])]

This example demonstrates how to manually compute the Legendre polynomial evaluation by defining a recursive function legendre_poly(). We sum the contributions of each term in the series using a list comprehension that involves the recursive evaluation for each point x. This method is not optimized and is provided for educational purposes.

Method 4: Using SymPy to Symbolically Evaluate Legendre Polynomials

SymPy is a Python library for symbolic mathematics. It can symbolically define Legendre polynomials and evaluate them at given points. This approach is useful if you need exact results or wish to perform symbolic manipulations before numeric evaluation.

Here’s an example:

from sympy import legendre as sympy_legendre, symbols

# Define the symbolic variable
x = symbols('x')
coefficients = [1, 3, 5]
x_points = [0, 0.5, 1]

# Compute the Legendre series sum symbolically
series_sum = sum(c * sympy_legendre(i, x) for i, c in enumerate(coefficients))

# Evaluate at the given points
evaluated_points = [series_sum.subs(x, point) for point in x_points]
print(evaluated_points)

Output: [1, 5/8 + 1, 3]

Using SymPy, we treat x as a symbol and form a symbolic expression for the Legendre series. We then evaluate this expression at each point in x_points. This method can manage exact arithmetic and provide insight into the symbolic structure of the series.

Bonus One-Liner Method 5: Using a Lambda Function

For a quick and dirty one-liner, you can use a lambda function along with a list comprehension to evaluate a Legendre series. This is less efficient but compact.

Here’s an example:

coefficients = [1, 3, 5]
x_points = [0, 0.5, 1]

# Define a lambda for Legendre polynomial evaluation
legendre_eval = lambda x, coeffs: sum(c * np.polynomial.legendre.legval(x, [0]*i + [1]) 
                                      for i, c in enumerate(coeffs))

evaluated_points = [legendre_eval(x, coefficients) for x in x_points]
print(evaluated_points)

Output: [1.0, 1.625, 3.0]

This one-liner uses a lambda function alongside np.polynomial.legendre.legval() to evaluate each term of the Legendre series where the coefficients are combined with the Legendre polynomial evaluated at x. It’s a compact solution but not optimized for large-scale computations.

Summary/Discussion

  • Method 1: SciPy’s legendre.legval(). Quick and efficient. Relies on SciPy library. Not suitable for symbolic manipulation.
  • Method 2: NumPy’s Polynomial Package. Object-oriented approach. Good for further polynomial manipulations. Might have overhead compared to SciPy.
  • Method 3: Manual Computation. Illustrates underlying math. Not practical for large-scale or high-degree polynomials. Computationally intensive.
  • Method 4: Using SymPy. Offers symbolic manipulation. Results in exact arithmetic. Slower than numerical methods for large inputs.
  • Bonus Method 5: Lambda Function. Handy for simple, quick computations. Not recommended for performance-critical applications.