💡 Problem Formulation: Evaluating a three-dimensional Hermite E series involves calculating the Hermite polynomial values across the Cartesian product of sets X, Y, and Z using a given 2D array of coefficients. Assuming the Cartesian product forms a grid of points in 3D space, the task is to find the Hermite polynomial values for each point, effectively creating a 3D data structure from the series evaluation. The desired output is a 3D array where each element represents the Hermite E polynomial value at the corresponding (x, y, z) point.
Method 1: Using Numpy and Scipy
This method utilizes the robust numerical computation libraries Numpy and Scipy to evaluate a 3D Hermite E series. The scipy.special.hermite function is employed to compute the Hermite E values for the array of coefficients and the numpy.meshgrid function to create the Cartesian product of X, Y, and Z coordinates.
Here’s an example:
import numpy as np
from scipy.special import hermeval
# The x, y, z coordinate arrays
x = np.linspace(-1, 1, 5)
y = np.linspace(-1, 1, 5)
z = np.linspace(-1, 1, 5)
# The Cartesian product mesh
X, Y, Z = np.meshgrid(x, y, z)
# A sample 2D array of coefficients for the Hermite E series
coefficients = np.array([[1, 0], [0, 1]])
# Evaluate the series on the Cartesian product
def evaluate_hermite_3d(X, Y, Z, coeffs):
return hermeval(X, coeffs) + hermeval(Y, coeffs) + hermeval(Z, coeffs)
results = evaluate_hermite_3d(X, Y, Z, coefficients)
print(results)
The output will be a 3D numpy array filled with the evaluated Hermite E series values at each point on a 3x3x3 grid formed by the input sets X, Y, and Z.
In this code snippet, linspace is used to generate points for the x, y, and z axes. The meshgrid function creates a 3D grid from these points, and the hermeval function from Scipy evaluates the Hermite series for each point using the provided coefficients. The result is a 3D numpy array with the evaluated series values.
Method 2: Pure Python Approach
A pure Python solution might be slower than using Numpy and Scipy but can be more insightful for understanding the calculations behind Hermite E series evaluation. This method utilizes nested loops to traverse through each point in the Cartesian product and manually calculates the Hermite polynomial values.
Here’s an example:
# Define your own Hermite evaluation function
def hermite_polynomial(x, coeffs):
""" Your Hermite evaluation logic here """
# This is a placeholder for a Hermite polynomial evaluator
return sum(c*x**i for i, c in enumerate(coeffs))
# Sample coefficients and coordinates
coefficients = [1, 0, 1] # Assuming a fixed degree for simplicity
x = [-1, 0, 1]
y = [-1, 0, 1]
z = [-1, 0, 1]
result = [[[hermite_polynomial(xi, coefficients) +
hermite_polynomial(yi, coefficients) +
hermite_polynomial(zi, coefficients)
for zi in z] for yi in y] for xi in x]
print(result)
The output is a nested list representing the 3D grid values evaluated at the Hermite E series for each (x, y, z).
In this code, we define a custom function to evaluate a Hermite polynomial given a point and a list of coefficients. We then use nested list comprehensions to compute the polynomial’s value at each point in the Cartesian grid of x, y, z values, combining these to produce our 3D grid of evaluated series.
Method 3: Vectorizing with Numpy
This approach is similar to Method 1 but emphasizes the use of numpy’s vectorization capabilities to speed up computations. Vectorizing the evaluation process avoids explicit loops in Python and utilizes fast, optimized numpy operations.
Here’s an example:
import numpy as np
from scipy.special import hermeval
x = np.array([-1, 0, 1])
y = np.array([-1, 0, 1])
z = np.array([-1, 0, 1])
coefficients = np.array([1, 0, 1])
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')
result = (hermeval(X.ravel(), coefficients) +
hermeval(Y.ravel(), coefficients) +
hermeval(Z.ravel(), coefficients)).reshape(X.shape)
print(result)
The output is a 3D numpy array of evaluated series values, similar to Method 1 but potentially more efficient due to vectorization.
This code utilizes Numpy’s ravel and reshape methods to transform the 3D grids into 1D arrays, performs the Hermite E series evaluation with vectorized operations, and then reshapes the result back into a 3D array. This maximizes performance by leveraging Numpy’s optimized routines.
Method 4: Using Tensor Product with Numpy
By utilizing the tensor product, this method allows for a concise and efficient way to compute the 3D Hermite series by treating the series as a multi-dimensional polynomial.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval coefficients = np.array([[1, 0], [0, 1]]) # Generate grid points for each axis x = np.linspace(-1, 1, 5) y = np.linspace(-1, 1, 5) z = np.linspace(-1, 1, 5) # Tensor product of the polynomial coefficients tensor_coeffs = np.outer(np.outer(coefficients[:, 0], coefficients[:, 1]), coefficients[:, 2]) # Evaluate on each axis and take the prod to find total evaluation X_eval = hermeval(x, coefficients[:, 0]) Y_eval = hermeval(y, coefficients[:, 1]) Z_eval = hermeval(z, coefficients[:, 2]) # Taking the outer product of evaluated axes and then summing over the last axis result = np.tensordot(np.tensordot(X_eval, Y_eval, axes=0), Z_eval, axes=0) print(result)
Output: A 3D Numpy array containing the results of the Hermite E series tensor product across the x, y, and z coordinates.
This code constructs a tensor product out of the coefficients and then evaluates the Hermite E series across x, y, and z coordinates. Finally, the tensordot function blends these separate evaluations to get the final 3D Hermite E series across the Cartesian grid.
Bonus One-Liner Method 5: Using Comprehensions and Map
Pythonic one-liners might not be the most efficient way, but they are concise and can be written in a single line of code using comprehensions and the map function. This method is quick for prototyping or dealing with smaller data sets.
Here’s an example:
result = [[[sum(map(lambda c, xi: c*xi**i, coefficients, (xi, yi, zi)))
for zi in z] for yi in y] for xi in x]
print(result)
This prints out a 3D list, with each sublist containing the Hermite E series evaluations across the Cartesian product.
The one-liner here uses a nested list comprehension to create the three-dimensional grid and the map function to apply the Hermite polynomial formula (as an anonymous lambda function) across each axis.
Summary/Discussion
- Method 1: Using Numpy and Scipy. Strengths: Fast, easy to write and understand. Weaknesses: Requires external libraries.
- Method 2: Pure Python Approach. Strengths: No external dependencies, easy to debug. Weaknesses: Slower, more verbose than library-based solutions.
- Method 3: Vectorizing with Numpy. Strengths: Fast, leverages numpy’s performance optimizations. Weaknesses: May be harder to understand than non-vectorized approaches.
- Method 4: Using Tensor Product with Numpy. Strengths: Efficient for multidimensional polynomials, leverages advanced NumPy functions. Weaknesses: Conceptually more complex, tensor operations can be unintuitive.
- Method 5: Using Comprehensions and Map. Strengths: Extremely concise. Weaknesses: Not performance optimized, clarity can suffer due to compactness.