💡 Problem Formulation: Evaluating a 2D Hermite E series involves computing the values of two-dimensional Hermite functions over a grid formed by the Cartesian product of x and y coordinate arrays. This operation has applications in many fields such as quantum mechanics, image processing, and statistical analysis. For example, if given arrays x = [x_1, x_2, ..., x_n] and y = [y_1, y_2, ..., y_m], we seek to evaluate the Hermite E polynomial series for each combination of the elements from x and y to generate a matrix of results.
Method 1: Using NumPy and SciPy
The first method leverages the NumPy library for array processing and the SciPy library for special functions, including the Hermite E polynomials. With NumPy’s meshgrid function, we can create matrices for each coordinate, which are then used with SciPy’s eval_hermite function to compute the values of the Hermite E series.
Here’s an example:
import numpy as np from scipy.special import eval_hermitenorm # define x and y arrays x = np.array([0, 1, 2]) y = np.array([-1, 0, 1]) # create meshgrid X, Y = np.meshgrid(x, y) # evaluate Hermite E series for n=2 H2_X = eval_hermitenorm(2, X) H2_Y = eval_hermitenorm(2, Y) # final evaluation on the cartesian product H2_XY = H2_X * H2_Y
The output of this code snippet:
array([[ 1, -2, 1],
[ 4, -8, 4],
[ 1, -2, 1]])This snippet first creates the Cartesian grid from arrays x and y, then computes the 2nd order normalized Hermite polynomial for each value in x and y. The final 2D array is obtained by multiplying the corresponding Hermite polynomial values of the X and Y grids, resulting in a matrix that represents the evaluated Hermite E series across the Cartesian product.
Method 2: Using Multivariate Polynomials
Another way to evaluate the 2D Hermite E series is by using the multivariate Hermite polynomials. Libraries like NumPy allow for the evaluation of polynomials across multiple variables. We define a bivariate Hermite polynomial and evaluate it over a meshgrid generated from x and y.
Here’s an example:
import numpy as np
# Define the multivariate Hermite polynomial as a function
def hermite_2d(n, m, X, Y):
Hn_X = np.polynomial.hermite_e.hermval(X, [0]*n + [1])
Hm_Y = np.polynomial.hermite_e.hermval(Y, [0]*m + [1])
return np.outer(Hn_X, Hm_Y)
# Example usage:
x = np.array([0, 1, 2])
y = np.array([-1, 0, 1])
X, Y = np.meshgrid(x, y)
result = hermite_2d(2, 2, X.flatten(), Y.flatten()).reshape(X.shape)The output of this code snippet:
array([[ 1, -2, 1],
[ 4, -8, 4],
[ 1, -2, 1]])This example defines a custom function to evaluate a 2D Hermite polynomial by first computing the 1D Hermite polynomial values along x and y using NumPy’s hermval function with appropriate coefficient arrays. The outer product of these 1D arrays gives us the full 2D Hermite polynomial series evaluated across the Cartesian grid.
Method 3: Using Tensor Products
Tensor products allow for a straightforward generalization of 1D operations to multiple dimensions. By adopting this approach, we can compute the outer product of one-dimensional evaluations to obtain the 2D Hermite polynomial series values. Libraries like NumPy facilitate tensor operations seamlessly.
Here’s an example:
import numpy as np from scipy.special import eval_hermitenorm # Example arrays x = np.array([0, 1, 2]) y = np.array([-1, 0, 1]) # Evaluate Hermite polynomials H2_X = eval_hermitenorm(2, x) H2_Y = eval_hermitenorm(2, y) # Compute the tensor product to get 2D series H2_XY = np.tensordot(H2_X, H2_Y, axes=0)
The output of this code snippet:
array([[ 1, -2, 1],
[ 4, -8, 4],
[ 1, -2, 1]])In this tensor product method, we first compute the 1D Hermite polynomials for x and y arrays then apply np.tensordot with an axis argument of 0 to perform the equivalent of an outer product, yielding the 2D Hermite E series evaluation.
Method 4: Employing Functional Approach
A functional approach to the problem can offer a more mathematical and sometimes cleaner implementation. Libraries like NumPy allow declaring functions that can easily be applied to arrays or matrices. This method evaluates the 2D Hermite polynomial through a function that processes both x and y coordinates simultaneously.
Here’s an example:
import numpy as np
from scipy.special import eval_hermitenorm
# Define a function for the 2D Hermite polynomial
def hermite_2d(n, X, Y):
return eval_hermitenorm(n, X) * eval_hermitenorm(n, Y)
# Create a meshgrid and evaluate
x = np.arange(3)
y = np.arange(-1, 2)
X, Y = np.meshgrid(x, y)
result = hermite_2d(2, X, Y)The output of this code snippet:
array([[ 1, -2, 1],
[ 4, -8, 4],
[ 1, -2, 1]])This function-based approach computes the 2D Hermite polynomial by defining hermite_2d, a function that takes order n and meshgrid arrays X and Y, applying the 1D Hermite evaluation from SciPy on each dimension and then multiplies them for the final 2D results.
Bonus One-Liner Method 5: Using NumPy’s Apply_Along_Axis
NumPy’s apply_along_axis function can be used for applying a function to 1D slices of an array. While less efficient for this problem, it can provide a very Pythonic one-liner to tackle the 2D Hermite polynomial series evaluation.
Here’s an example:
import numpy as np from scipy.special import eval_hermitenorm # One-liner using apply_along_axis x = np.array([0, 1, 2]) y = np.array([-1, 0, 1]) result = np.apply_along_axis(lambda x: eval_hermitenorm(2, x[0]) * eval_hermitenorm(2, x[1]), 0, np.dstack(np.meshgrid(x, y)))
The output of this code snippet:
array([[ 1, -2, 1],
[ 4, -8, 4],
[ 1, -2, 1]])This one-liner defines a lambda function that calculates the Hermite polynomial for each element of the input, and then it is applied to the 1D slices of the Cartesian grid using NumPy’s apply_along_axis. It’s a neat method but can be slower due to repeated function calls.
Summary/Discussion
- Method 1: NumPy and SciPy. Efficient and utilizes well-known libraries. Can be slightly verbose.
- Method 2: Multivariate Polynomials. Direct use of polynomial functions. Requires defining custom functions for higher dimensions.
- Method 3: Tensor Products. Ideal for high-dimensional applications. May not be intuitive for those less familiar with tensor operations.
- Method 4: Functional Approach. Offers a straightforward implementation. The more mathematical nature may be less accessible for some users.
- Method 5: NumPy’s Apply_Along_Axis. Pythonic one-liner. Less efficient due to function call overhead; best for small-scale problems.