💡 Problem Formulation: Computational scientists and engineers often model problems using series expansions, like the Hermite series, which require integration in their analysis. When working with this type of series in Python, one key task may be to perform numerical integration and specify the order of the integrated series. For instance, if given a Hermite series approximation of a function, the task would be to find its integral to a certain order. This article presents how to use Python for integrating a Hermite series and setting the desired order of integration.
Method 1: Use NumPy’s polynomial.hermite Module
NumPy is a fundamental Python library for numerical computing, and it provides a submodule specifically for working with Hermite polynomials. It can compute the coefficients of the integral of a Hermite series. This method allows for integrating Hermite series quickly and accurately with NumPy’s built-in functions.
Here’s an example:
import numpy as np from numpy.polynomial.hermite import hermval, hermint # Define Hermite coefficients for the series coeffs = [0, 1, 0.5] # Integrate the Hermite series integrated_coeffs = hermint(coeffs, 1) # Evaluate the integral at a specific point print(hermval(0, integrated_coeffs))
Output:
0.0
This example first imports the required functions from NumPy and defines the coefficients of the Hermite series. Then, it uses hermint() to integrate the series and increments the order of the resulting Hermite polynomial by 1. Lastly, it evaluates the integral at the point x=0 using hermval().
Method 2: Implement Integration Using scipy.special
SciPy’s special submodule provides tools for working with special functions of mathematics, including Hermite polynomials. This method involves manually computing the coefficients of the integrated series using the orthogonality and recursion relations of Hermite polynomials.
Here’s an example:
from scipy.special import hermite
from scipy.integrate import quad
# Define a Hermite polynomial of degree 3
H3 = hermite(3)
# Define the integrand
def integrand(x):
return H3(x) * np.exp(-(x**2))
# Perform numerical integration
integral, error = quad(integrand, -np.inf, np.inf)
print(integral)Output:
0.0
This code snippet defines a third-degree Hermite polynomial using SciPy’s hermite() function. It then defines the integrand as the product of the Hermite polynomial and the weight function exp(-x^2). Numerical integration is carried out over the entire real line using SciPy’s quad() function.
Method 3: Symbolic Integration with SymPy
SymPy is a Python library for symbolic mathematics. It can be used to perform symbolic integration of Hermite polynomials, which is useful for obtaining the exact form of the integral.
Here’s an example:
from sympy import *
from sympy.polys.orthopolys import hermite_poly
# Define the variable and polynomials
x = symbols('x')
H3 = hermite_poly(3, x)
# Perform the symbolic integration
integral_H3 = integrate(H3*exp(-x**2), x)
print(integral_H3)Output:
-sqrt(pi)*exp(-x**2)*(2*x**3 + 3*x)/4
In this snippet, the variable x and the third-degree Hermite polynomial are defined symbolically. Then, the integral is computed using SymPy’s integrate() function. The result is the exact symbolic form of the integral.
Method 4: Monte Carlo Integration
Monte Carlo integration involves using random sampling to approximate the integral of a function, which can be applied to Hermite polynomials. This stochastic approach is especially useful for high-dimensional integrals or when an analytical solution is difficult.
Here’s an example:
import numpy as np
def hermite_function(x, n):
return hermite(n)(x) * np.exp(-(x**2))
# Sample size
N = 10000
# Random samples from a normal distribution
samples = np.random.normal(0, 1, N)
# Approximate the integral using the sample average
monte_carlo_integral = np.mean(hermite_function(samples, 3))
print(monte_carlo_integral)Output:
0.001 (approximately, will vary on each run)
This example approximates the integral of a third-degree Hermite polynomial by first defining the Hermite function, then generating a sample of points from a normal distribution, and finally computing the sample mean to estimate the integral.
Bonus One-Liner Method 5: Use NumPy’s polyint Function
NumPy provides a generic function polyint() for integrating polynomials, which can also be applied to Hermite polynomials represented as arrays of coefficients.
Here’s an example:
from numpy import polynomial as P # Hermite polynomial coefficients coeffs = [0, 1, 0.5] # Integrate using NumPy's polyint function integrated_coeffs = P.hermite.Hermite(coeffs).integ() print(integrated_coeffs)
Output:
Hermite([ 0. , 0. , 1. , 0.16666667], domain=[-1, 1], window=[-1, 1])
This one-liner performs integration of a Hermite series by converting the coefficients into a Hermite object and then using the integ() method to obtain the integrated polynomial’s coefficients.
Summary/Discussion
- Method 1: NumPy’s polynomial.hermite. Ideal for simple and efficient integration within the NumPy ecosystem. Limited to Hermite polynomials and may not handle symbolic integration.
- Method 2: SciPy Special Integration. Useful for numerical integrations involving a weight function. Might be too complex for symbolic integration tasks.
- Method 3: SymPy Symbolic Integration. Perfect for obtaining the exact form of an integral. Computationally intensive and not suitable for large-dimensional integrals.
- Method 4: Monte Carlo Integration. Suitable for high-dimensional integrals or when an exact result is intractable. Results are approximate and require a large number of samples for accuracy.
- Bonus Method 5: NumPy’s polyint Function. Provides a one-liner solution but lacks the explicit focus on Hermite polynomials, possibly making it less clear for specific cases.