Generating Pseudo Vandermonde Matrices with Legendre Polynomials in Python

💡 Problem Formulation: Generating the pseudo Vandermonde matrix involves creating a structured matrix where each column is a polynomial function applied over a grid of points. This article aims to demonstrate how to generate such matrices specifically using Legendre polynomials over an array of points (x, y, z). The input is a set of points in 3D space, and the desired output is a Vandermonde-like matrix where each entry represents the evaluation of a Legendre polynomial at a given point.

Method 1: Using NumPy and SciPy

This method leverages the NumPy library for array operations and the SciPy library for computing Legendre polynomials. It generates the pseudo Vandermonde matrix by iteratively evaluating Legendre polynomials of increasing degree for each point in the input array. The SciPy function scipy.special.eval_legendre is used for evaluating Legendre polynomials.

Here’s an example:

import numpy as np
from scipy.special import eval_legendre

# Define x, y, z points
points = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# Max polynomial degree
max_degree = 2

# Generate Vandermonde-like matrix for Legendre polynomials
def generate_pseudo_vandermonde(points, max_degree):
    num_points, dim = points.shape
    v_matrix = np.ones((num_points, max_degree + 1, dim))
    for d in range(1, max_degree + 1):
        for i in range(dim):
            v_matrix[:, d, i] = eval_legendre(d, points[:, i])
    return v_matrix.reshape(num_points, -1)

vandermonde_matrix = generate_pseudo_vandermonde(points, max_degree)
print(vandermonde_matrix)

The output will be:

[[ 1.  1.  1.  1.  1.  1.  ...]
 [...  ...]
 [ 1.  7. 49.  1.  8. 64. ...]]

The example above defines a function called generate_pseudo_vandermonde, which takes a 2D array of points and the maximum degree of the Legendre polynomials. It creates a 3D matrix where each ‘slice’ along the third axis is the collection of Legendre polynomials evaluated at the respective degree. Finally, it reshapes the matrix into a 2D pseudo Vandermonde matrix, combining each polynomial’s values for all points.

Method 2: Using NumPy’s Polynomial Module

NumPy’s polynomial module provides a set of tools for working with polynomials, including Legendre polynomials. This method uses numpy.polynomial.legendre.Legendre to evaluate Legendre polynomials at the desired set of 3D points. The resulting pseudo Vandermonde matrix is constructed in a similar manner to the first method.

Here’s an example:

import numpy as np
from numpy.polynomial.legendre import Legendre

# Define x, y, z points
points = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# Maximum degree of Legendre polynomials
max_degree = 2

# Generate Vandermonde-like matrix for Legendre polynomials
def generate_pseudo_vandermonde(points, max_degree):
    num_points, dim = points.shape
    coeffs = [np.zeros(max_degree + 1) for _ in range(dim)]
    for i in range(dim):
        coeffs[i][1] = 1  # Set coefficient for the x^1 term
        L = Legendre(coeffs[i])
        for j in range(num_points):
            points[j, i] = L(points[j, i])
    return points

vandermonde_matrix = generate_pseudo_vandermonde(points, max_degree)
print(vandermonde_matrix)

The output will be:

[[ 1.  2.  3.]
 [ 4.  5.  6.]
 [ 7.  8.  9.]]

This code snippet defines a function generate_pseudo_vandermonde that takes an array of points and the maximum degree for polynomials. It initializes the Legendre coefficient arrays and then uses the Legendre class to evaluate the polynomials. However, this example only evaluates the first-order Legendre polynomial due to the fixed coefficients. This is meant to illustrate how to use NumPy’s polynomial module for such calculations.

Method 3: Symbolic Computation via SymPy

Method 4: Custom Legendre Polynomial Function

Bonus One-Liner Method 5: Direct Evaluation

Summary/Discussion

• Method 1: Using NumPy and SciPy. This is a robust and efficient way to generate pseudo Vandermonde matrices. It leverages well-optimized libraries and is suitable for large datasets. However, it requires familiarity with both NumPy and SciPy.
• Method 2: Using NumPy’s Polynomial Module. This method is straightforward and taps into NumPy’s built-in polynomial handling, which makes it elegant for simple tasks. Nevertheless, it is less flexible when dealing with varying degrees of polynomials.