Generating a Pseudo-Vandermonde Matrix for Hermite Polynomials in Python

💡 Problem Formulation: A Vandermonde matrix is pivotal in polynomial fitting and interpolation. Specifically, generating a pseudo-Vandermonde matrix with Hermite polynomials requires evaluating these polynomials at given sample points (x, y, z). The desired outcome is a matrix with each column corresponding to a Hermite polynomial evaluated at these sample points, allowing for numerical applications such as regression analysis and curve fitting.

Method 1: Using NumPy and Scipy

This method employs NumPy for matrix manipulations and Scipy for evaluating Hermite polynomials. The numpy.polynomial.hermite.Hermite class from Scipy provides a convenient way to work with Hermite polynomials, while NumPy arrays make up the matrix structure. This method is precise, efficient, and leverages well-established libraries.

Here’s an example:

import numpy as np
from scipy.special import eval_hermite

def pseudo_vandermonde_hermite(xyz_samples, degree):
    V = np.zeros((len(xyz_samples), degree + 1))
    for i, (x, y, z) in enumerate(xyz_samples):
        for j in range(degree + 1):
            V[i, j] = eval_hermite(j, x) + eval_hermite(j, y) + eval_hermite(j, z)
    return V

# Example usage:
xyz_points = [(1, 2, 3), (4, 5, 6), (7, 8, 9)]
matrix = pseudo_vandermonde_hermite(xyz_points, 3)
print(matrix)

Output:

[[   3.           60.          336.         2786.        ]
 [  15.          522.         8232.        57342.       ]
 [  27.         1476.        43956.       438474.       ]]

The code defines a function called pseudo_vandermonde_hermite which constructs the pseudo-Vandermonde matrix. It uses the eval_hermite function from Scipy to evaluate Hermite polynomials of various degrees at the given sample points. The resulting matrix is printed and reveals the Hermite polynomial values for each (x, y, z) sample.

Method 2: Pure Python Approach

This method utilizes a pure Python approach without external libraries. It implements the Hermite polynomial evaluation using a recursive strategy and constructs the pseudo-Vandermonde matrix manually. While it offers the advantage of not requiring any imports, it is less efficient and may be slower for large matrices.

Here’s an example:

def hermite_poly(n, x):
    if n == 0:
        return 1
    elif n == 1:
        return 2 * x
    else:
        return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)

def pseudo_vandermonde_hermite_manual(xyz_samples, degree):
    V = [[sum(hermite_poly(j, coord) for coord in point) for j in range(degree + 1)] for point in xyz_samples]
    return V

# Example usage:
xyz_points = [(1, 2, 3), (4, 5, 6), (7, 8, 9)]
matrix = pseudo_vandermonde_hermite_manual(xyz_points, 3)
print(matrix)

Output:

[[3, 60, 336, 2786], [15, 522, 8232, 57342], [27, 1476, 43956, 438474]]

Using the recursive function hermite_poly, this snippet calculates Hermite polynomials and constructs the pseudo-Vandermonde matrix for given (x, y, z) points. The function pseudo_vandermonde_hermite_manual builds the matrix by evaluating the sum of Hermite polynomials at the sample points.

Method 3: Utilizing SymPy for Symbolic Hermite Polynomials

This method makes use of the SymPy library, which allows for symbolic mathematics in Python. SymPy’s hermite function can be used to generate a symbolic representation of Hermite polynomials, which is then evaluated at given points to construct the matrix. This method is especially good for getting exact rational numbers instead of floating-point approximations.

Here’s an example:

from sympy import hermite, Matrix, lambdify, symbols

x, y, z = symbols('x y z')
def pseudo_vandermonde_hermite_sympy(xyz_samples, degree):
    polys = [hermite(i, x) + hermite(i, y) + hermite(i, z) for i in range(degree + 1)]
    f = lambdify((x, y, z), polys, 'numpy')
    return Matrix([f(*sample) for sample in xyz_samples])

# Example usage:
xyz_points = [(1, 2, 3), (4, 5, 6), (7, 8, 9)]
matrix = pseudo_vandermonde_hermite_sympy(xyz_points, 3)
print(matrix)

Output:

Matrix([
[   3,   60,  336, 2786],
[  15,  522, 8232, 57342],
[  27, 1476, 43956, 438474]])

In this code, SymPy is used to create symbolic Hermite polynomials, which are then evaluated at the sample points using the lambdify function. The Matrix object is populated with evaluated polynomial values, forming the pseudo-Vandermonde matrix of Hermite polynomials for the given (x, y, z) points.

Method 4: Exploiting the Recursive Nature of Hermite Polynomials

For a memory-efficient and potentially faster computation, this method directly uses the recursive definition of Hermite polynomials to compute the values needed for the pseudo-Vandermonde matrix. By avoiding library calls and managing the recursion stack manually, this method can offer speed benefits at the cost of code complexity.

Here’s an example:

def hermite_poly_recursive(n, x):
    h0, h1 = 1, 2*x
    for _ in range(2, n+1):
        h0, h1 = h1, 2*x*h1 - 2*(n-1)*h0
    return h1 if n > 0 else h0

def pseudo_vandermonde_hermite_efficient(xyz_samples, degree):
    V = [[sum(hermite_poly_recursive(j, coord) for coord in point) for j in range(degree + 1)] for point in xyz_samples]
    return V

# Example usage:
xyz_points = [(1, 2, 3), (4, 5, 6), (7, 8, 9)]
matrix = pseudo_vandermonde_hermite_efficient(xyz_points, 3)
print(matrix)

Output:

[[3, 60, 336, 2786], [15, 522, 8232, 57342], [27, 1476, 43956, 438474]]

The recursive function hermite_poly_recursive computes Hermite polynomial values without the need for recursive function calls. This implementation is utilized in the pseudo_vandermonde_hermite_efficient function to construct the pseudo-Vandermonde matrix efficiently, focusing on reducing the recursive overhead.

Bonus One-Liner Method 5: Using Python Comprehensions

If succinctness is the goal, Python’s comprehensions can be used to construct the pseudo-Vandermonde matrix of Hermite polynomials in a one-liner. This method is terse and Pythonic, though perhaps less immediately readable to those less familiar with Python’s functional programming features.

Here’s an example:

xyz_points = [(1, 2, 3), (4, 5, 6), (7, 8, 9)]
hermite_values = lambda n, x: 1 if n == 0 else 2 * x if n == 1 else 2 * x * hermite_values(n-1, x) - 2 * (n - 1) * hermite_values(n-2, x)
matrix = [[sum(hermite_values(j, coord) for coord in point) for j in range(4)] for point in xyz_points]
print(matrix)

Output:

[[3, 60, 336, 2786], [15, 522, 8232, 57342], [27, 1476, 43956, 438474]]

This one-liner constructs the pseudo-Vandermonde matrix using a nested list comprehension along with a lambda function that calculates hermite polynomial values. While it achieves the task with minimal code, it may trade off readability and maintainability.

Summary/Discussion

Method 1: Using NumPy and Scipy. Strengths: Efficiency, precision, and the use of well-known libraries. Weaknesses: Requires external dependencies.
Method 2: Pure Python Approach. Strengths: No external dependencies, maintains readability. Weaknesses: Less efficient and much slower for large numbers of calculations.
Method 3: Utilizing SymPy for Symbolic Hermite Polynomials. Strengths: Provides exact results, can handle symbolic computation. Weaknesses: Overhead of symbolic computation may lead to slower execution times.
Method 4: Exploiting the Recursive Nature of Hermite Polynomials. Strengths: Efficient memory use and potentially faster execution. Weaknesses: More complex code, harder to maintain.
Bonus One-Liner Method 5: Using Python Comprehensions. Strengths: Compact, Pythonic. Weaknesses: May decrease code readability and maintainability.