💡 Problem Formulation: In scientific computing and graphical applications, evaluating orthogonal polynomials like the Hermite E series across a three-dimensional space is crucial. Consider you have three separate arrays representing the coordinates x, y, and z. The problem is to evaluate a Hermite E series for each combination within the Cartesian product of these arrays. An expected output is a three-dimensional array, where each element corresponds to the Hermite polynomial evaluated at a specific (x, y, z) point.
Method 1: Using NumPy and Itertools
This method employs the use of the NumPy library for mathematical operations and itertools for generating the Cartesian product. A 3D Hermite E series evaluation is achieved by first creating an array for each of the coordinates x, y, and z, followed by computing their Cartesian product using itertools’ product function. Finally, NumPy’s vectorize method allows you to apply the Hermite E function across all combinations efficiently.
Here’s an example:
import numpy as np
from scipy.special import hermite
from itertools import product
# Define Hermite E polynomial function
def hermite_e(x, n):
Hn = hermite(n)
return Hn(x)
# Example coordinates
x = np.linspace(-1, 1, 5)
y = np.linspace(-1, 1, 5)
z = np.linspace(-1, 1, 5)
# Cartesian product
cartesian_points = list(product(x, y, z))
# Vectorize Hermite E function
hermite_e_vect = np.vectorize(hermite_e)
# Evaluate 3D Hermite E series (example for n=3)
results = hermite_e_vect(cartesian_points, 3)
print(results)The output is a NumPy array containing the evaluated Hermite E series for each (x, y, z) combination.
This snippet creates arrays for the x, y, and z coordinates and computes their Cartesian product. It then vectorizes the Hermite E function to apply it across all coordinate combinations. Note that this method is straightforward but can be memory-intensive for large arrays due to the creation of the Cartesian product list.
Method 2: Using NumPy’s Broadcasting
NumPy’s broadcasting allows us to evaluate the 3D Hermite E series without explicitly forming the Cartesian product. Instead, we broadcast arrays against each other to simulate the same effect, which can be more memory-efficient. The evaluation is then done in a vectorized manner directly on the broadcasted arrays.
Here’s an example:
import numpy as np
from scipy.special import hermite
# Define Hermite E polynomial function
def hermite_e(x, n):
Hn = hermite(n)
return Hn(x)
# Example coordinates
x = np.linspace(-1, 1, 5)[:, None, None]
y = np.linspace(-1, 1, 5)[None, :, None]
z = np.linspace(-1, 1, 5)[None, None, :]
# Broadcasting to simulate Cartesian product
combined = x + y + z
# Evaluate 3D Hermite E series (example for n=3)
results = hermite_e(combined, 3)
print(results.shape)The output is a NumPy array of shape (5, 5, 5), which indicates the 3D structure of the evaluated Hermite E series values.
The code demonstrates how to extend the dimensions of x, y, and z arrays and uses the broadcasting feature of NumPy to simulate the Cartesian product. This avoids memory overhead and increases computational efficiency, especially with larger arrays.
Method 3: Using Outer Products
Another method using NumPy is to evaluate the Hermite E series on the coordinates via outer products. The np.outer function can be applied twice to achieve a 3D grid. It is particularly useful when the function to evaluate is separable in x, y, and z.
Here’s an example:
import numpy as np
from scipy.special import hermite
# Define Hermite E polynomial function
def hermite_e(x, n):
Hn = hermite(n)
return Hn(x)
# Example coordinates
x = np.linspace(-1, 1, 5)
y = np.linspace(-1, 1, 5)
z = np.linspace(-1, 1, 5)
# Using outer products to create a 3D grid
xy = np.outer(hermite_e(x, 3), hermite_e(y, 3))
xyz = np.outer(xy, hermite_e(z, 3)).reshape(5, 5, 5)
print(xyz.shape)The output is a NumPy array with the shape (5, 5, 5) representing the Hermite E series evaluated across the grid.
This example uses the np.outer function to create a 3D grid. It first evaluates the Hermite E series on x and y, and then again with z. The reshape function ensures that the result has a 3D structure. This method is memory efficient for separable functions but might not be suitable for more complex non-separable expressions.
Method 4: SciPy’s Multidimensional Evaluation
SciPy provides multidimensional versions of some special functions that can be used for direct 3D evaluations. If a multidimensional Hermite E polynomial is available in SciPy, you can employ it directly without worrying about manually handling the Cartesian product or outer products.
Here’s an example:
[Note: As of the knowledge cutoff in 2023, SciPy doesn’t offer direct 3D Hermite E polynomial functions. Imagine a future release has included such a feature, the following code would apply.]
import numpy as np from scipy.special import hermite_e_3d # Example coordinates x = np.linspace(-1, 1, 5) y = np.linspace(-1, 1, 5) z = np.linspace(-1, 1, 5) # Multidimensional evaluation (hypothetical function) results = hermite_e_3d(x, y, z, n=3) print(results.shape)
The output would a…