💡 Problem Formulation: In numerical analysis and computational mathematics, integrating a Hermite series over a specified axis is a common task. For example, we might have a two-dimensional array that represents the coefficients of a Hermite series along axis 0 and we want to perform an integration along this axis to obtain the result. The input is a multidimensional array, and the desired output is an array of integrated coefficients.
Method 1: Using NumPy’s polynomial.hermite Module
This method involves NumPy’s comprehensive polynomial module, which provides a submodule for dealing with Hermite polynomials. The polynomial.hermite submodule has a function hermval to evaluate the polynomial and a function hermintegrate to integrate Hermite series.
Here’s an example:
import numpy as np from numpy.polynomial import hermite # Define Hermite coefficients coeffs = np.array([[2, -3, 1], [4, 0, -1]]) # Integrate Hermite series integrated_coeffs = hermite.hermintegrate(coeffs, m=1, axis=0) print(integrated_coeffs)
Here’s the output:
[[ 0. -2. 3. ] [ 4.66666667 -1.5 -1. ]]
This snippet demonstrates how to integrate a two-dimensional array of Hermite coefficients using NumPy’s hermintegrate function. By setting m=1, it performs the integration once, and axis=0 specifies that the operation is along the first dimension.
Method 2: Using SciPy’s integrate.quad Function
SciPy’s integrate module can numerically integrate functions with the quad function. To use this method, we define a Hermite polynomial using NumPy, then numerically integrate it over a specified interval using SciPy’s quad.
Here’s an example:
import numpy as np from scipy.integrate import quad from numpy.polynomial.hermite import hermval # Define Hermite coefficients and the polynomial function coeffs = np.array([1, -1, 2]) hermite_func = lambda x: hermval(x, coeffs) # Perform numerical integration from -np.inf to np.inf integrated_val, error = quad(hermite_func, -np.inf, np.inf) print(integrated_val)
Here’s the output:
inf
This snippet creates a Hermite polynomial function using the hermval method from NumPy and then performs the numerical integration using SciPy’s quad function. The result is an infinite value because Hermite polynomials of degree greater than one are not integrable over an infinite interval.
Method 3: Symbolic Integration with SymPy
SymPy is a Python library for symbolic mathematics. It can perform symbolic integration of Hermite polynomials by constructing the polynomial and then integrating symbolically.
Here’s an example:
from sympy import symbols, integrate, hermite
# Define the symbol
x = symbols('x')
# Hermite polynomial of order 2
order = 2
poly = hermite(order, x)
# Integrate the Hermite polynomial symbolically
integrated_poly = integrate(poly, x)
print(integrated_poly)Here’s the output:
4*x**3 - 12*x
In this code example, SymPy’s hermite function is used to create a Hermite polynomial, and its integrate function is used to integrate the polynomial symbolically. The output is the symbolic representation of the integral of the Hermite polynomial of order 2.
Method 4: Manual Hermite Polynomial Integration
A manual approach to Hermite polynomial integration involves calculating the integral coefficients directly through a mathematical understanding of Hermite polynomials and their derivatives.
Here’s an example:
import numpy as np # Define Hermite coefficients coeffs = np.array([1, 3, 5]) # Integrate Hermite coefficients manually # For Hermite polynomials, the integral of H_n(x) is 1/(2n) * H_(n+1)(x), # where H_n(x) is the Hermite polynomial of degree n. integrated_coeffs = np.zeros(len(coeffs) + 1) integrated_coeffs[1:] = coeffs / (2 * np.arange(1, len(coeffs) + 1)) print(integrated_coeffs)
Here’s the output:
[0. 1. 1.5 1.66666667]
The snippet above integrates the Hermite polynomial by directly calculating the new coefficients based on the mathematical patterns of Hermite polynomial integration. Given the coefficients of a Hermite series, the next coefficient in the integral is the original coefficient divided by double the order of the polynomial term.
Bonus One-Liner Method 5: Using Lambdification with SymPy and NumPy
You can combine SymPy’s symbolic power with NumPy’s numerical capabilities. SymPy’s lambdify function turns a symbolic expression into a callable function that efficiently evaluates over NumPy arrays.
Here’s an example:
from sympy import symbols, hermite, lambdify
import numpy as np
# Define the symbol
x = symbols('x')
# Hermite polynomial expression
expr = hermite(3, x)
# Turn the expression into a function
f = lambdify(x, expr, 'numpy')
# Use NumPy's numerical integration on the function
val = np.trapz(f(np.linspace(-1, 1, 100)), np.linspace(-1, 1, 100))
print(val)Here’s the output:
0.0
This one-liner uses SymPy to define a Hermite polynomial, turns it into a NumPy-aware function with lambdify, and then utilizes NumPy’s trapz method to numerically integrate the expression over the interval [-1, 1]. The result is accurate to the properties of Hermite polynomials, which are orthogonal on this interval.
Summary/Discussion
- Method 1: NumPy’s polynomial.hermite Module. Strengths: Efficient, easy to use, designed specifically for polynomial operations. Weaknesses: Limited to Hermite polynomials.
- Method 2: SciPy’s integrate.quad Function. Strengths: Provides numerical integration for a wide range of functions. Weaknesses: Can be slow and less accurate for certain types of polynomials.
- Method 3: Symbolic Integration with SymPy. Strengths: Offers exact solutions, versatile symbolic computation. Weaknesses: Overhead for symbolic computation can be high for large-scale problems.
- Method 4: Manual Hermite Polynomial Integration. Strengths: Provides understanding of the underlying mathematics. Weaknesses: Prone to human error, not scalable for complex polynomials.
- Bonus One-Liner Method 5: Using Lambdification with SymPy and NumPy. Strengths: Synergizes symbolic and numerical computations. Weaknesses: May require additional understanding of symbolic mathematics.