Evaluating 3D Hermite Series on the Cartesian Product of x, y, and z in Python

by

💡 Problem Formulation: Calculating the value of a 3D Hermite series for given x, y, z coordinates using a 2D array of coefficients poses a unique computational challenge. For example, given the 2D array coefficients of size m x n x l and a set of points (x_values, y_values, z_values), we desire to compute the 3D Hermite series sum at each (x,y,z) in the Cartesian product of these points.

Method 1: Using NumPy and Iterative Computation

This method exploits the vectorization and efficient computation of NumPy, a powerful numerical library in Python, and iterates over each dimension to compute the Hermite series value. The numpy.polynomial.hermite.hermval() function can evaluate Hermite polynomials for single-dimensional inputs, which we can nest to achieve a multidimensional evaluation.

Here’s an example:

import numpy as np
from numpy.polynomial.hermite import hermval

# Coefficients for the Hermite series (example with a 2x2x2 array for clarity)
coefficients = np.array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])

# Cartesian product of x, y, and z values
x_values = np.array([1, 2])
y_values = np.array([0, -1])
z_values = np.array([3, 4])

# Compute the 3D Hermite series
results = np.zeros((len(x_values), len(y_values), len(z_values)))
for i, x in enumerate(x_values):
    for j, y in enumerate(y_values):
        for k, z in enumerate(z_values):
            hx = hermval(x, coefficients[:, j, k])
            hy = hermval(y, coefficients[i, :, k])
            hz = hermval(z, coefficients[i, j, :])
            results[i, j, k] = hx * hy * hz

print(results)

The output would be a 3D array with computed values for each combination of x, y, and z:

[[[123.  234.]
  [ 72.  144.]]

 [[ 72.  144.]
  [288.  576.]]]

This snippet sets up a 3D grid of x, y, and z coordinates and computes the sum of the Hermite series for each point. This is done by iterating over all combinations and using Hermite polynomial evaluation from the NumPy library for each dimension, then combining the results.

Method 2: Using NumPy’s Meshgrid and Vectorized Operations

… (other method descriptions and examples follow) …

Bonus One-Liner Method 5: Using NumPy’s Broadcasting and Advanced Indexing

… (brief method description and one-liner example) …

Summary/Discussion

  • Method 1: Iterative Computation. This approach benefits from the familiar structure of nested loops that are easy to understand. However, it is not the most efficient due to Python’s slower loop execution.
  • … (other methods’ strengths and weaknesses) …