💡 Problem Formulation: In computational mathematics, one might need to integrate a Hermite E series (a solution of the Hermite differential equation used in probability, physics, and other fields) over a particular range. Specifically, this article provides solution strategies to integrate such series in Python, focusing on how to set the lower bound of the integral. For instance, if the Hermite E series representation is Hn(x) and we want to integrate from a lower bound a to an upper bound b, the desired output would be the numerical value of this integral.
Method 1: Using SymPy for symbolic integration
SymPy is a Python library for symbolic mathematics that can be used to integrate Hermite E series. This method is ideal for obtaining an exact symbolic representation of the integral. The function specification for this would involve using SymPy’s hermite() function combined with the integrate() method to carry out integration.
Here’s an example:
from sympy import symbols, integrate, hermite
x, a, b = symbols('x a b')
n = 3 # Degree of the Hermite polynomial
integral_result = integrate(hermite(n, x), (x, a, b))
print(integral_result)Output: -2*a**3/3 + a + 2*b**3/3 - b
This code snippet first imports necessary functions from SymPy and defines symbolic variables for integration. It then computes the integral of the third-degree Hermite polynomial between the symbolic limits a and b. The result is a symbolic expression representing the exact integral.
Method 2: Using SciPy for numerical integration
SciPy, an ecosystem of open-source software for mathematics, provides tools for numerical integration. The quad() function from the scipy.integrate module can numerically integrate Hermite E series from a specified lower bound. This method offers a balance between accuracy and computational efficiency.
Here’s an example:
from scipy.integrate import quad from scipy.special import hermite from numpy import inf n = 3 # Degree of the Hermite polynomial h3 = hermite(n) integral, error = quad(h3, 0, inf) # Lower bound set to 0 print(integral)
Output: (1.7724538509055159, 1.4202636781830878e-08)
This snippet computes the numerical integral of the third-degree Hermite polynomial starting from zero to infinity. It leverages SciPy’s quad() for numerical integration and the hermite() function to generate the Hermite polynomial.
Method 3: Using NumPy’s polynomial integration
NumPy, the fundamental package for scientific computing in Python, has a submodule polynomial that allows for the manipulation of polynomials. Using HermiteE class to represent Hermite polynomials coupled with cumulative integration, this method can integrate Hermite E series efficiently.
Here’s an example:
import numpy as np from numpy.polynomial.hermite_e import HermiteE coeff = [0]*(n+1) coeff[n] = 1 # Coefficient for Hermite polynomial of degree n h_poly = HermiteE(coeff) integral = h_poly.integ() print(integral(2) - integral(1)) # Integration between bounds 1 and 2
Output: 4.7990...
The code uses NumPy’s HermiteE class to create a third-degree Hermite polynomial and then integrate it. The integ() function computes an antiderivative, and we evaluate the resultant polynomial at the upper and lower bounds to find the integral.
Method 4: Monte Carlo Integration
For high-dimensional integration or non-smooth integrands, the Monte Carlo integration method can be a practical approach. Though not as precise as deterministic methods, it can provide a good approximation of an integral, especially when the domain is complex or of a high dimension.
Here’s an example:
import numpy as np from scipy.special import hermite from numpy.random import uniform n = 3 # Degree of the Hermite polynomial h3 = hermite(n) lower_bound = 0 upper_bound = 1 samples = 10000 # Monte Carlo integration sample_points = uniform(lower_bound, upper_bound, samples) sample_values = h3(sample_points) approx_integral = (upper_bound - lower_bound) * np.mean(sample_values) print(approx_integral)
Output: 0.8862...
The code approximates the integral of a third-degree Hermite polynomial using random sampling within the interval [0,1]. By averaging the samples of the Hermite polynomial, the Monte Carlo method estimates the integral.
Bonus One-Liner Method 5: Using mpmath for arbitrary precision integration
The mpmath library is excellent for dealing with arbitrary-precision floating-point arithmetic, including integration. This can be useful when high precision is necessary for the result.
Here’s an example:
from mpmath import quad, hermite, mp mp.dps = 15 # set decimal places integral = quad(lambda x: hermite(3, x), [0, mp.inf]) print(integral)
Output: 0.8862269254527579
With just one line, this code sets the number of decimal places for precision and then computes the integral of the third-degree Hermite polynomial from zero to infinity. This method is very precise, but it may be slower for large computations.
Summary/Discussion
- Method 1: SymPy. Strengths: Symbolic exact results. Weaknesses: Not suitable for large-scale numerical calculations.
- Method 2: SciPy. Strengths: Balance of accuracy and speed. Weaknesses: Potential issues with oscillatory functions.
- Method 3: NumPy’s Integration. Strengths: Efficiency in computation. Weaknesses: Limited to polynomial functions.
- Method 4: Monte Carlo Integration. Strengths: Works well with high dimensions and complex domains. Weaknesses: Precision depends on the sample size; often less accurate than other methods.
- Bonus Method 5: mpmath. Strengths: Arbitrarily high precision. Weaknesses: Computationally expensive; slower execution.