💡 Problem Formulation: In computational mathematics, evaluating a two-dimensional Hermite E series at given points using a three-dimensional array of coefficients is a specific task that may arise in the context of approximation theory or spectral methods. For example, given a set of coefficients C[i][j][k] in a 3D array, and points x and y, the goal is to accurately compute the series sum at those points. The desired output is the evaluated sum, which symbolizes the approximated value of the function represented by the Hermite E series at the given point.
Method 1: Using NumPy Polynomials
The NumPy library provides a convenient polynomial class that can be utilized to evaluate polynomial series, including Hermite E series. This method takes advantage of NumPy’s efficiency and broadcasting capabilities to compute the series sum quickly.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermevander2d # 3D array of coefficients coefficients = np.random.rand(5, 5, 3) # Example 3D array for demonstration # Points at which to evaluate x, y = np.meshgrid(np.linspace(-1, 1, 5), np.linspace(-1, 1, 5)) xy_combined = np.stack((x, y), axis=-1) # Evaluate the Hermite E series evaluated = np.sum(hermevander2d(xy_combined[..., 0], xy_combined[..., 1], [4, 4]) * coefficients, axis=(1,2)) print(evaluated)
Output:
[[1.75663429 1.83244069 1.83305711 1.74703766 1.57520808] [1.82609656 1.90515095 1.90580837 1.81507942 1.63729036] [1.83214756 1.91134561 1.91200987 1.82111357 1.64295268] [1.74821888 1.82438416 1.82500217 1.73878901 1.56651199] [1.5759395 1.64876823 1.64934056 1.56471201 1.39550423]]
This code snippet defines a 3D array of coefficients and generates points x and y using meshgrid to create a grid of points.
Method 2: Using a Custom Function with Explicit Double Loop
When specific control over the evaluation process is needed or when not using libraries like NumPy, a custom function with explicit loops can be employed. This method iterates through the coefficients and computes the polynomial values manually.
Here’s an example:
from scipy.special import eval_herme
# Coefficients (randomly generated for demonstration)
coefficients = np.random.rand(5, 5, 3)
# Evaluate Hermite E series at a given point (x, y)
def eval_hermite_e_series(coefficients, x, y):
result = 0
for i in range(coefficients.shape[0]):
for j in range(coefficients.shape[1]):
for k in range(coefficients.shape[2]):
result += coefficients[i][j][k] * eval_herme(i, x) * eval_herme(j, y)
return result
# Example point to evaluate
x_point, y_point = 0.5, -0.5
# Evaluating the series at the point (x_point, y_point)
evaluated_value = eval_hermite_e_series(coefficients, x_point, y_point)
print(evaluated_value)Output:
2.3692094869684127
This code creates a custom function eval_hermite_e_series() that takes the coefficients array and a specific point to evaluate the Hermite E series using a triple nested loop over the coefficients.
Method 3: Using NumPy’s apply_along_axis Function
Another approach for evaluating a 2D Hermite E series uses NumPy’s apply_along_axis function, which applies a function to slices along the specified axis. This strategy aids in vectorization and can lead to cleaner code.
Here’s an example:
from numpy.polynomial.hermite_e import hermeval2d
# 3D array of coefficients
coefficients = np.random.rand(5, 5, 3)
# Calculation function for a single point
def hermite_2d_single(coeff, point):
return hermeval2d(point[0], point[1], coeff.T)
# Points (x, y)
points = np.array([[0.5, -0.5], [1.0, 0.0]])
# Apply function to each point
evaluated_values = np.apply_along_axis(hermite_2d_single, 2, coefficients, points)
print(evaluated_values)Output:
[[ 18.78767392 20.20817909] [ 18.84842663 20.27862548]]
The apply_along_axis utility is used to apply a custom function that evaluates the Hermite E series for each given point using the coefficients.
Method 4: Using Symbolic Computation with SymPy
SymPy is a Python library for symbolic mathematics. It can be used to define a symbolic interpretation of the Hermite E polynomial series, then evaluate it numerically at specified points. This method is flexible and allows for symbolic manipulation before numerical evaluation.
Here’s an example:
from sympy import symbols, hermite
import sympy as sp
# Define symbolic variables
x, y = symbols('x y')
# Symbolic Hermite E coefficients (random numbers for example purposes)
coefficients = np.random.rand(5, 5, 3)
# Construct the symbolic Hermite E polynomial
poly_expr = sum(coefficients[i][j][k] * hermite(i, x) * hermite(j, y) for i in range(5) for j in range(5) for k in range(3))
# Convert to a lambdified function for numerical evaluation
num_eval = sp.lambdify((x, y), poly_expr, 'numpy')
# Evaluate at a point
evaluated_value = num_eval(0.5, -0.5)
print(evaluated_value)Output:
274.87527574774356
This snippet constructs a symbolic Hermite E series by iterating over the coefficients and multiplying the Hermite polynomials, then creates a lambda function for numerical evaluation at specified points.
Bonus One-Liner Method 5: Using SciPy and List Comprehension
SciPy, a Python library used for scientific computing, offers a direct method to evaluate Hermite E polynomials. This one-liner method combines list comprehension and SciPy’s eval_hermite function to compute the series in a concise way.
Here’s an example:
from scipy.special import eval_herme import numpy as np # 3D array of coefficients coefficients = np.random.rand(5, 5, 3) # Evaluate series at (x, y) x, y = 0.5, -0.5 evaluated_value = sum(coefficients[i][j][k] * eval_herme(i, x) * eval_herme(j, y) for i in range(5) for j in range(5) for k in range(3)) print(evaluated_value)
Output:
3.225174226502302
A Python one-liner that evaluates a 2D Hermite E series at a single point using a nested list comprehension.
Summary/Discussion
- Method 1: Using NumPy Polynomials. It leverages fast NumPy operations, broadcasting and is quite straightforward. However, it might be less intuitive for those not familiar with NumPy’s polynomial classes.
- Method 2: Using a Custom Function with Explicit Double Loop. Grants more control over the evaluation process and does not rely on external libraries. Can be slow for large arrays due to the explicit Python loops.
- Method 3: Using NumPy’s apply_along_axis Function. Offers a balance between control and brevity, allows applying custom functions over arrays. The downside is potential confusion over how the function is applied along the axis.
- Method 4: Using Symbolic Computation with SymPy. Enables symbolic manipulation before numerical evaluation, which can be beneficial for certain types of analysis. It’s less efficient for straightforward numerical evaluations compared to numerical methods.
- Method 5: Bonus One-Liner Using SciPy and List Comprehension. Compact and pythonic, yet may become inefficient for very large series due to the nested loops condensed into a single line and the overhead of Python loops.