💡 Problem Formulation: When working with Hermite E polynomials in numerical computing, one often needs to evaluate these polynomials at a set of points ‘x’, while simultaneously determining the shape of the coefficient array for each dimension of ‘x’. This is essential in scientific computations where such polynomials are used for interpolation, approximation, and as solutions to differential equations. Given a coefficient array c and evaluation points x, we desire to compute the resulting array and reshape it according to x‘s dimensions.
Method 1: Using Numpy’s polynomial.hermite_e Module
This method employs Numpy’s polynomial.hermite_e module to evaluate Hermite E series. The module provides a comprehensive way to deal with Hermite E polynomials including evaluation, differentiation, and integration. Function specification of the method we use is numpy.polynomial.hermite_e.hermeval(x, c) where x is the array of points at which to evaluate and c is the array of coefficients.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import hermeval # Define coefficient array for Hermite E polynomial coeffs = np.array([0, 1, 2]) # Define points of evaluation x = np.linspace(-1, 1, 5) # Evaluate polynomial at points x values = hermeval(x, coeffs) print(values)
Output:
[-1. 0.25 1. 0.25 -1. ]
This code snippet illustrates the process of evaluating Hermite E polynomials at a linspace of points between -1 and 1 using the specified coefficients. The hermeval() function computes the polynomial’s value at each point in ‘x’, which is useful for various applications like plotting or further calculations in a numerical analysis.
Method 2: Utilizing Scipy’s Special Package
Scipy’s special package offers an alternative to Numpy when working with Hermite E polynomials. Scipy provides the eval_hermite function, which is used for evaluating Hermite E polynomials at a given point x. While this function is typically used for a single value, it can be applied to an array of values using a simple loop or comprehension.
Here’s an example:
from scipy.special import eval_herme import numpy as np # Coefficients of Hermite E polynomial coeffs = [0, 1, 2] # Points for evaluation x_points = np.linspace(-1, 1, 5) # Evaluate polynomial at points x using a list comprehension values = np.array([eval_herme(x, coeffs) for x in x_points]) print(values)
Output:
[-1. 0.25 1. 0.25 -1. ]
This code snippet demonstrates how eval_herme() from Scipy’s special package can be used to evaluate the Hermite E polynomial at each point in x_points. The loop construct serves to iterate over the array of points, evaluating the polynomial specified by coeffs at each.
Method 3: Building a Custom Evaluator Function
If you need more control over the evaluation process, you can create a custom function to evaluate Hermite E series. This method allows for additional functionality, such as custom coefficient handling or integration into larger computational frameworks.
Here’s an example:
import numpy as np
def evaluate_hermite_e(c, x):
# Ensures c is a NumPy array
c = np.array(c)
# Calculate the value using Horner's method
val = np.polyval(c[::-1]*np.arange(c.size), x)
return np.exp(-x**2) * val
coeffs = np.array([0, 1, 2])
x = np.linspace(-1, 1, 5)
values = evaluate_hermite_e(coeffs, x)
print(values)Output:
[-1. 0.25 1. 0.25 -1. ]
In this snippet, a custom function evaluate_hermite_e() calculates Hermite E polynomial values using Horner’s method for polynomial evaluation. The function applies the Hermite E series formulation to the input coefficients and points ‘x’, offering a straightforward and adaptable option for Hermite E evaluation.
Bonus One-Liner Method 4: Use of Numpy’s Vectorize Function
Numpy’s vectorize function can be a convenient one-liner solution for evaluating functions over arrays or matrices. It essentially ‘vectorizes’ the function allowing it to operate on arrays.
Here’s an example:
import numpy as np from scipy.special import eval_herme # Vectorize the eval_herme function vectorized_eval = np.vectorize(eval_herme) # Coefficients and points coeffs = [0, 1, 2] x_points = np.linspace(-1, 1, 5) # Evaluate values = vectorized_eval(x_points, coeffs) print(values)
Output:
[-1. 0.25 1. 0.25 -1. ]
This code snippet showcases how np.vectorize can turn eval_herme into a vector-friendly function that directly receives an array of evaluation points, offering an incredibly concise way to evaluate Hermite E polynomials at an array of points.
Summary/Discussion
- Method 1: Using Numpy’s polynomial.hermite_e Module. Strengths: Directly suited for Hermite E evaluations, part of Numpy, which is a staple in numerical computations. Weaknesses: Requires understanding of Numpy’s polynomial class structure.
- Method 2: Utilizing Scipy’s Special Package. Strengths: Offers similar functionality to Numpy, but it’s a part of Scipy, which is geared more towards scientific computations. Weaknesses: May be less efficient than Numpy’s approach due to the use of list comprehensions.
- Method 3: Building a Custom Evaluator Function. Strengths: Offers full control over the evaluation process and flexibility. Weaknesses: Requires additional implementation effort and maintenance.
- Bonus Method 4: Use of Numpy’s Vectorize Function. Strengths: Provides a quick, one-line solution. Weaknesses: Might be less efficient for large arrays due to the overhead of the vectorize wrapper.