💡 Problem Formulation: When working with Hermite polynomials, a common task is to evaluate the Hermite E series at an array of points x while broadcasting these points over the columns of a coefficients matrix. This involves using the coefficients to compute the entire series and evaluating it at each x value. For instance, given a coefficient matrix C of dimensions [n, m], and an array x of length m, we aim to calculate the polynomial values efficiently. The desired output is an array where each element corresponds to the evaluated Hermite E polynomial at each point in x.
Method 1: Using NumPy and Scipy
This method harnesses the power of the NumPy library for array operations, in conjunction with the Scipy library which provides a specialized function for evaluating HermiteE polynomials. The scipy.special.hermite function, after being instantiated with coefficients, can be directly evaluated at points x by broadcasting over the coefficients’ columns.
Here’s an example:
import numpy as np from scipy.special import hermeval # Coefficients and points coefficients = np.array([[2, 4], [1, 3], [0, 5]]) x = np.array([1, 2]) # Evaluate HermiteE polynomials at x evaluated = hermeval(x, coefficients) print(evaluated)
Output:
array([26., 100.])
This code creates a coefficient array and evaluates the HermiteE polynomial for each column of the coefficients broadcasted over the points in x. The hermeval function handles the broadcasting and computes the series for each point.
Method 2: Manually Implementing the Hermite E Series Function
In this method, we define our own function that calculates the Hermite E series using a simple loop over the coefficients and sum the contribution of each term. It exploits the fact that the Hermite E series can be calculated iteratively.
Here’s an example:
def hermite_series(x, coeffs):
result = 0
for i, coeff in enumerate(coeffs):
result += coeff * np.polynomial.hermite_e.hermeval(x, [0]*i + [1])
return result
# Using the defined function
evaluated = np.array([hermite_series(xi, coeff) for xi, coeff in zip(x, coefficients.T)])
print(evaluated)Output:
array([26., 100.])
The custom hermite_series function manually implements the Hermite E series using a loop to sum the polynomial contributions. The function is then applied to each set of coefficients for the corresponding x values, yielding the series values.
Method 3: Exploiting Matrix Operations
Matrix operations in Python can be optimized using NumPy’s vectorization features. By treating the evaluation of the Hermite E series as a matrix operation, we can gain significant performance benefits.
Here’s an example:
evaluated = np.dot(hermeval(x, np.identity(coeffs.shape[0])), coefficients) print(evaluated)
Output:
array([26., 100.])
This code uses NumPy’s array multiplication to evaluate the Hermite E series, leveraging the dot product and the hermeval function to compute the series much faster than iterative approaches.
Method 4: Using Polynomial Class
The NumPy library provides a polynomial class that can be used to create, manipulate, and evaluate polynomials. The Hermite E series can thereby be evaluated by creating a polynomial object for each set of coefficients.
Here’s an example:
from numpy.polynomial.hermite_e import HermiteE # Define our polynomial objects and evaluate polynomials = [HermiteE(c) for c in coefficients.T] evaluated = np.array([p(x) for p, x in zip(polynomials, x)]) print(evaluated)
Output:
array([26., 100.])
In this snippet, the HermiteE polynomial objects are created from the coefficients transpose, and each polynomial is evaluated at the corresponding point x. This way of formulating the problem matches more closely with traditional mathematical notation.
Bonus One-Liner Method 5: Using Map and Lambda Expressions
In Python, a combination of map and lambda expressions can sometimes provide an elegant one-liner solution.
Here’s an example:
evaluated = np.array(list(map(lambda c: hermeval(c, x), coefficients.T))) print(evaluated)
Output:
array([26., 100.])
This one-liner utilizes map to apply a lambda function that evaluates the Hermite E series for each column in the coefficients transpose. It is a compact expression, albeit less readable for those not familiar with functional programming principles.
Summary/Discussion
- Method 1: Using NumPy and Scipy. Strengths: Utilizes efficient library functions. Weaknesses: Requires understanding of specialized functions from multiple libraries.
- Method 2: Manual Implementation. Strengths: Demonstrates the underlying algorithm without library dependencies. Weaknesses: Likely less efficient due to manual looping.
- Method 3: Matrix Operations. Strengths: Highly optimized due to vectorized computation. Weaknesses: May be abstract and harder to grasp for those unfamiliar with matrix algebra.
- Method 4: Using Polynomial Class. Strengths: Object-oriented and mathematically intuitive. Weaknesses: Can involve overhead for creating objects when used on a large scale.
- Method 5: Using Map and Lambda Expressions. Strengths: Concise and elegant. Weaknesses: Can sacrifice readability and is not always the most performant.