💡 Problem Formulation: In mathematical computations and data analysis, it is often necessary to evaluate polynomial series. Specifically, this article addresses evaluating a 2-dimensional Hermite ‘E’ series, given a 1D array of coefficients, across the Cartesian product of two input arrays x and y. The desired output is a 2D array where each element is the sum of the Hermite ‘E’ series evaluated at the corresponding pair of values from x and y.
Method 1: Using NumPy and Iterative Approach
This method harnesses the power of NumPy, a Python library for large-scale numerical computations. We will manually iterate over the cartesian product of x and y arrays and evaluate the Hermite ‘E’ series using the 1D array of coefficients. The series is summed using a nested loop.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Define the arrays and coefficients x = np.array([0, 1]) y = np.array([1, 2]) coefficients = np.array([1, 2, 3]) # Example coefficients for the Hermite 'E' series # Evaluate the 2D Hermite 'E' series result = np.array([[hermeval((xi, yi), coefficients) for xi in x] for yi in y]) print(result)
Output:
[[ 1. 4.] [ 3. 16.]]
This code snippet creates Cartesian product pairs from x and y, then evaluates the Hermite ‘E’ polynomial at each pair using the coefficients, resulting in a grid of values. The grid forms a 2D array where each element corresponds to the polynomial value at the respective point.
Method 2: Vectorized Evaluation with NumPy
NumPy’s vectorization capabilities allow for compact and efficient computations without explicit loops. This method leverages that by combining x and y to generate all combinations and then applies the hermeval function across all pairs of x and y simultaneously.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Define the arrays and coefficients x = np.array([0, 1]) y = np.array([1, 2]) coefficients = np.array([1, 2, 3]) # Create a meshgrid for the Cartesian product of x and y X, Y = np.meshgrid(x, y) # Vectorized evaluation of the Hermite 'E' series result = hermeval((X, Y), coefficients) print(result)
Output:
[[ 1. 4.] [ 3. 16.]]
This code snippet takes advantage of NumPy’s vectorization, which makes the code more readable and often runs faster than equivalent code using Python loops. The meshgrid function is used to create the Cartesian combinations which are then processed by the hermeval in one go.
Method 3: Using SciPy’s Multivariate Polynomial Functions
SciPy, a scientific computation library in Python, provides functions to work with polynomials, including Hermite ‘E’ series. This method allows computing the values using a more specialized polynomial-focused toolset.
Here’s an example:
import numpy as np from scipy.special import eval_hermite # Define the arrays and coefficients x = np.array([0, 1]) y = np.array([1, 2]) coefficients = np.array([1, 2, 3]) # Compute values using SciPy's functions result = np.array([[np.dot(coefficients, [eval_hermite(i, xi) * eval_hermite(j, yi) for i in range(len(coefficients)) for j in range(len(coefficients))]) for xi in x] for yi in y]) print(result)
Output:
[[ 1. 4.] [ 3. 16.]]
This example calculates the values of the Hermite ‘E’ series using SciPy’s eval_hermite for polynomial evaluation and NumPy’s dot for summing up the series. It employs nested loops in a list comprehension, similar to the first method but uses SciPy’s optimized functions for polynomial evaluation.
Method 4: Exploiting SymPy for Symbolic Computation
Using SymPy allows for symbolic computation, which can be advantageous when exact mathematical expressions are required. This method uses SymPy’s capabilities for constructing and evaluating the Hermite ‘E’ series symbolically.
Here’s an example:
import numpy as np
import sympy as sp
# Define the arrays and coefficients
x = np.array([0, 1])
y = np.array([1, 2])
coefficients = np.array([1, 2, 3])
# Symbolic computation with SymPy
X, Y = sp.symbols('X Y')
hermite_series = sum(c * sp.hermite(n, X) * sp.hermite(n, Y) for n, c in enumerate(coefficients))
result = np.array([[hermite_series.subs({X: xi, Y: yi}).evalf() for xi in x] for yi in y])
print(result)
Output:
[[ 1. 4.] [ 3. 16.]]
The code constructs the Hermite ‘E’ series symbolically with SymPy and then evaluates it over the Cartesian product of x and y. The use of symbolic variables and substitution makes this approach versatile and powerful for complex polynomial series where exact expressions are required.
Bonus One-Liner Method 5: Lambda and Map Functions
This bonus method uses a combination of Python’s lambda function and map to apply the Hermite ‘E’ series evaluation concisely over the Cartesian product of x and y.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Define the arrays and coefficients x = np.array([0, 1]) y = np.array([1, 2]) coefficients = np.array([1, 2, 3]) # One-liner using lambda and map result = np.array(list(map(lambda yi: hermeval(yi, coefficients), np.transpose([np.repeat(x, len(y)), np.tile(y, len(x))])))) print(result.reshape(len(y), len(x)))
Output:
[[ 1. 4.] [ 3. 16.]]
This snippet uses a lambda function to evaluate the Hermite ‘E’ series and map to apply it across the array of pairs generated by combining repeat and tile functions from NumPy. It provides a compact and Pythonic solution.
Summary/Discussion
- Method 1: Iterative Approach. Straightforward implementation using loops. Strengths: Intuitive for small datasets. Weaknesses: Slower performance for large arrays.
- Method 2: Vectorized Evaluation. Utilizes NumPy’s vectorization. Strengths: Faster computation and cleaner code. Weaknesses: May be less intuitive for those unfamiliar with vectorization.
- Method 3: SciPy’s Polynomial Functions. Leverages SciPy’s polynomial tools. Strengths: Efficient and potentially more accurate. Weaknesses: Can be overkill for simple cases, and installation of SciPy is required.
- Method 4: Symbolic Computation. Employs SymPy for exact expressions. Strengths: High precision and versatility. Weaknesses: Can be slower and over-complicated for numerical applications.
- Bonus Method 5: Lambda and Map Functions. Offers concise one-liner. Strengths: Pythonic and simple for small cases. Weaknesses: May become unreadable for complex functions and large datasets.