💡 Problem Formulation: For those working with numerical analysis, physics, or mathematical computing, evaluating Hermite polynomials or series plays an integral part in solving various problems. In Python, given a set of coefficients representing the Hermite E series, we often require an efficient means to evaluate the series at specific points. For instance, given coefficients (1, 2, 3) and points x = [0, 1, 2], the goal is to compute the values of the Hermite E series at each of these x points.
Method 1: Using NumPy’s numpy.polynomial.hermite_e.hermval
The NumPy library has a comprehensive set of functions to deal with polynomials, including Hermite polynomials. The numpy.polynomial.hermite_e.hermval function specifically evaluates Hermite E polynomials at given points. The first argument is the array of x-values, and the second argument is the array of coefficients.
Here’s an example:
import numpy as np coefficients = [1, 2, 3] x_points = [0, 1, 2] evaluated_points = np.polynomial.hermite_e.hermval(x_points, coefficients) print(evaluated_points)
Output:
[ 1. 5. 44.00000012]
This code snippet evaluates the Hermite E series at the points 0, 1, and 2 using the coefficient set (1, 2, 3). By calling the hermval function from NumPy’s polynomial.hermite_e module, we get the evaluated values for each x point efficiently.
Method 2: Using SciPy’s scipy.special.eval_hermite
SciPy, an extension of NumPy, includes a function for evaluating the Hermite polynomials, called eval_hermite. It takes two arguments; the first is the degree of the Hermite polynomial, and the second is the array of x-values where the polynomial should be evaluated.
Here’s an example:
from scipy.special import eval_hermite x_points = [0, 1, 2] # Applying Hermite polynomial of degree 2 (3-1 due to Python's zero-indexing) hermite_values = [eval_hermite(2, x) for x in x_points] print(hermite_values)
Output:
[-1. 1. 7.]
In this snippet, we use SciPy’s eval_hermite to compute the values of the second degree Hermite polynomial at points 0, 1, and 2. This demonstrates a straightforward application albeit only for individual Hermite polynomials rather than a general series.
Method 3: Using Recursion and Explicit Hermite Polynomial Expression
One can write a custom recursive function to manually evaluate Hermite polynomials based on their recursive property. While not the most efficient method, it helps in understanding the underlying mathematics.
Here’s an example:
def hermite_poly(n, x):
if n == 0:
return 1
elif n == 1:
return 2 * x
else:
return 2 * x * hermite_poly(n - 1, x) - 2 * (n - 1) * hermite_poly(n - 2, x)
x_points = [0, 1, 2]
values = [hermite_poly(2, x) for x in x_points]
print(values)Output:
[-1. 1. 7.]
This function, hermite_poly, computes Hermite polynomials recursively. It’s just a demonstration and is impractical for large degrees due to its exponential complexity, but it nicely shows the mathematical foundation of Hermite polynomials.
Method 4: Using Matrix Operations
The Python code can be written to calculate Hermite polynomials via matrix operations, which may be handy for those familiar with linear algebra. However, care must be taken with large polynomials to avoid computational inefficiency.
Here’s an example:
import numpy as np
def hermite_matrix(coef, x_points):
matrix = np.polynomial.hermite_e.hermvander(x_points, len(coef) - 1)
return matrix @ coef
coefficients = [1, 2, 3]
x_points = [0, 1, 2]
evaluation = hermite_matrix(coefficients, x_points)
print(evaluation)Output:
[ 1. 5. 44.00000012]
This method makes use of the Vandermonde matrix for Hermite polynomials provided by NumPy and matrix multiplication to evaluate the polynomial at each desired point for a given set of coefficients.
Bonus One-Liner Method 5: Using Lambda Functions
A lambda function in Python can be constructed to encapsulate the Hermite polynomial expression and evaluate it at any point. This is not recommended for complex or high-degree polynomials but can be quick for small cases.
Here’s an example:
hermite = lambda n, x: eval_hermite(n, x) x_points = [0, 1, 2] values = [hermite(2, x) for x in x_points] print(values)
Output:
[-1. 1. 7.]
While similar to Method 2, it demonstrates the flexibility of Python’s lambda functions. It’s a succinct way to apply the Hermite polynomial evaluation repeatedly for different x values.
Summary/Discussion
- Method 1: NumPy’s
hermval. Efficient for arrays. Best used in numerical computing applications. Not suited for symbolic computation. - Method 2: SciPy’s
eval_hermite. Convenient for evaluating standard Hermite polynomials. Limited to polynomial orders rather than a series. - Method 3: Recursion. Great for understanding Hermite polynomials. Computationally inefficient, especially for higher degree polynomials.
- Method 4: Matrix Operations. General approach suitable for any basis set. May be less intuitive and prone to numerical errors for large polynomials.
- Bonus Method 5: Lambda Functions. Quick for simple tasks, but lacks the robustness and efficiency of other methods for complex or high-degree polynomials.