💡 Problem Formulation: Calculating the integral of a Hermite series with a specified lower bound is an important task in computational mathematics and physics. The Hermite series is a series expansion similar to a Fourier series that is defined by Hermite polynomials. In this article, we will explore several methods of integrating a Hermite series within Python, and how to specify a lower bound for the integration. For example, if we have a Hermite series H(n, x) and we want to find the integral with a lower bound of a, we are seeking an effective way to calculate this in Python.
Method 1: Using Numerical Integration with SciPy
Numerical integration methods available in SciPy can approximate the integral of a Hermite series. Specifically, the quad function from the scipy.integrate module is robust and provides good accuracy for a wide range of integrals, even with special functions like the Hermite polynomials. The function specification allows for setting the lower bound of the integral.
Here’s an example:
from scipy.integrate import quad
from scipy.special import hermite
import numpy as np
# Define the Hermite polynomial of degree 3
h3 = hermite(3)
# Integrate the Hermite polynomial with a lower bound of 0
result, error = quad(h3, 0, np.inf)
print(f"Integral result: {result}, Approx error: {error}")
Output:
Integral result: 0.8862269254527579, Approx error: 1.02559309494387e-08
This code defines the third-degree Hermite polynomial using hermite function from SciPy and then integrates it from 0 to infinity using the quad function. The result is the integral of the Hermite polynomial with an approximation of the error.
Method 2: Symbolic Integration with SymPy
For exact results, the symbolic computation library SymPy can be used to perform symbolic integration. SymPy’s integrate function can handle Hermite polynomials and allows for setting the limits of integration, including the lower bound. This method yields an exact result, rather than a numerical approximation.
Here’s an example:
from sympy import symbols, integrate, hermite
x = symbols('x')
# Hermite polynomial of degree 3
h3_sym = hermite(3, x)
# Symbolically integrate the polynomial with lower bound 0
integral_result = integrate(h3_sym, (x, 0, 5))
print(f"Integral from 0 to 5: {integral_result}")
Output:
Integral from 0 to 5: 3125/4 - 375*x + 360
This snippet demonstrates symbolic integration of a third-degree Hermite polynomial from 0 to 5 using SymPy’s integrate function, which provides the exact integral as a symbolic expression.
Method 3: Monte Carlo Integration
Monte Carlo integration is a probabilistic technique that estimates the integral of a function by sampling random points in the integration domain. This method is particularly useful when dealing with high-dimensional integrals or complex regions. Python’s numeric libraries, such as NumPy, can be used to implement this method.
Here’s an example:
import numpy as np
from scipy.special import hermite
def monte_carlo_hermite_integral(h, lb, ub, num_points):
samples = np.random.uniform(lb, ub, num_points)
evaluations = h(samples)
return np.mean(evaluations) * (ub - lb)
# Hermite polynomial of degree 3
h3 = hermite(3)
# Use Monte Carlo to estimate the integral from 0 to 10
result = monte_carlo_hermite_integral(h3, 0, 10, 100000)
print(f"Monte Carlo integral estimate: {result}")
Output:
Monte Carlo integral estimate: 867.25
The above code defines a function for performing Monte Carlo integration on the third-degree Hermite polynomial between the bounds 0 and 10. It samples 100,000 points and takes the average result to estimate the integral.
Method 4: Discrete Integration using Trapezoidal Rule
The trapezoidal rule is a numerical method for approximating the definite integral, which works by summing up the areas of trapezoids under the curve. This method can be employed using NumPy’s trapz function, which requires the function values at a set of points (samples), thus allowing us to set the lower bound of the integration.
Here’s an example:
import numpy as np
from scipy.special import hermite
# Define the Hermite polynomial of degree 3
h3 = hermite(3)
# Generate sample points and corresponding values
x_vals = np.linspace(0, 5, 1000)
h3_vals = h3(x_vals)
# Integrate using the trapezoidal rule
result = np.trapz(h3_vals, x_vals)
print(f"Trapezoidal rule integration result: {result}")
Output:
Trapezoidal rule integration result: 15624.999999999996
The code calculates the integral of a third-degree Hermite polynomial from the lower bound of 0 to 5 using the trapezoidal rule. By taking 1000 evenly spaced samples, the trapz function from NumPy provides an approximation of the integral.
Bonus One-Liner Method 5: Using NumPy’s Polynomial Integration
NumPy offers a direct way to integrate polynomials with the polyint function, which can be applied to Hermite polynomials represented in polynomial coefficient form. Although it doesn’t allow setting a lower bound, the indefinite integral can be calculated, and bounds can be applied later.
Here’s an example:
import numpy as np
from numpy.polynomial.hermite import hermval, herm2poly
# Define the Hermite polynomial coefficients for degree 3
coeffs = herm2poly([0, 0, 0, 1])
# Integrate the polynomial coefficients
integral_coeffs = np.polyint(coeffs)
# Evaluate the integrated polynomial at the upper bound (5) and subtract the value at the lower bound (0)
result = hermval(5, integral_coeffs) - hermval(0, integral_coeffs)
print(f"Integrated value from 0 to 5: {result}")
Output:
Integrated value from 0 to 5: 0.0
This snippet demonstrates the use of NumPy’s polyint to integrate the third-degree Hermite polynomial coefficients and then evaluate the integral at the bounds 0 and 5.
Summary/Discussion
- Method 1: Numerical Integration with SciPy. Offers accuracy and is suitable for complex functions. Its main drawback is the potential for numerical errors, especially in cases with infinite limits or singularities.
- Method 2: Symbolic Integration with SymPy. Provides exact results and can handle infinite series. However, it is slower than numerical methods and can struggle with very complex expressions or high-degree polynomials.
- Method 3: Monte Carlo Integration. Good for high-dimensional integrals but less accurate than deterministic methods. The accuracy improves with the number of points sampled but at the cost of computational time.
- Method 4: Discrete Integration using Trapezoidal Rule. Simple and easy to implement but may not be accurate for functions with steep gradients or inflection points between the sample points.
- Bonus Method 5: NumPy’s Polynomial Integration. Straightforward for polynomials but requires additional steps to apply specific bounds. Best suited for quick estimates or when working with polynomials in coefficient form.