Overview, Algorithm and Implementation of Gauss-Hermite Integral

Gaussian-Hermite Integration

Gaussian-Hermite Integration is a method of numerical integration, often used for stochastic problems, especially those in which the probability density function is Gaussian (normally distributed), and for integrals such as the wave function in quantum mechanics, which is approximated by the Gaussian-Hermite polynomial The Gauss-Hermite integral is a method of approximating the integral using the Gauss-Hermite polynomial.

The Gauss-Hermite integral is commonly expressed in the following form.

\[\displaystyle\int_{-\infty}^{\infty}f(x)e^{-x^2}dx\approx \sum_{i=1}^n\omega_if(x_i)\]

Algorithm used for Gauss-Hermite integration

To compute the Gauss-Hermite integral, it is common to use the roots and weights of the Gauss-Hermite polynomial. The following is the general procedure of the algorithm for computing the Gauss-Hermite integral.

1. Calculate the roots and weights of the Gauss-Hermite polynomial: In order to compute the Gauss-Hermite integral, the roots and weights of the Gauss-Hermite polynomial must be computed in advance. Since Gauss-Hermite polynomials have recursive relations, they are computed using a dedicated algorithm.
2. Application of integration points and weights: In the approximate formulation of the Gauss-Hermite integral, the function f(x) is approximately integrated using the computed integration points (roots of the Gauss-Hermite polynomial) and weights. The number of integration points affects the accuracy of the calculation, and in many cases the number of integration points is adjusted within numerically computable limits.
3. Calculation of the approximate value: The approximate value of the Gauss-Hermite integral is calculated by the following equation. \[\displaystyle\int_{-\infty}^{\infty} f(x)e^{-x^2}dx\approx\sum_{i=1}^n \omega_i f(x_i)\] where \(\omega_i\) is the weight and xi is the integration point. The function f(x) is applied to this equation to compute an approximation of the integral.

The algorithm for Gauss-Hermite integration is based on the properties of Gauss-Hermite polynomials and is commonly provided in numerical libraries and mathematical software. The algorithm can be adjusted according to the number of integration points and the accuracy of the calculation. Gauss-Hermite integrals are used especially in areas such as stochastic problems and expectation calculations in quantum mechanics, providing high accuracy and efficiency in numerical calculations.

Libraries and algorithms used for Gauss-Hermite integration

Many numerical libraries and mathematical software are available to compute Gauss-Hermite integrals. They are listed below.

• SciPy (Python library): SciPy is one of the Python scientific computing libraries, and its scipy.integrate module includes functions for computing Gauss-Hermite integrals. scipy.special module also includes functions for computing Gauss-Hermite polynomials and roots. The following is an example of using SciPy to compute Gauss-Hermite integrals.

from scipy import integrate
from scipy.special import hermite
import numpy as np

# Compute roots and weights of Gauss-Hermite polynomials
n = 5  # Number of Integral Points
x, w = np.polynomial.hermite.hermgauss(n)

# Definition of function f(x)
def f(x):
    return x**2

# Calculation of Gauss-Hermite integrals
result = np.sum(w * f(x))
print(result)

• MATLAB (numerical software): MATLAB is also a powerful tool for computing Gauss-Hermite integrals: it can compute integrals using integral functions and also provides functions for computing Gauss-Hermite polynomials and roots.

% Compute roots and weights of Gauss-Hermite polynomials
n = 5;  % Number of Integral Points
[x, w] = gaussHermite(n);

% Definition of function f(x)
f = @(x) x.^2;

% Calculation of Gauss-Hermite integrals
result = sum(w .* f(x));
disp(result);

• Manual Calculation: It is also possible to manually calculate the roots and weights of the Gauss-Hermite integral. In this case, the roots and weights must be derived based on the definition of the Gauss-Hermite polynomial. This is not a common method and requires mathematical expertise, but can be useful when customized calculations are needed for a specific problem.

The Gauss-Hermite integral is used especially in fields such as probability theory and quantum mechanics and is supported by many numerical tools. Depending on the chosen programming language and numerical computing environment, the Gauss-Hermite integral can be computed using appropriate libraries and algorithms.

Application of Gauss-Hermite integrals

The Gauss-Hermite integral has a wide range of applications in scientific and technical fields and areas of mathematics, primarily in the following areas

• Probability and Statistics: The Gauss-Hermite integral is used to compute statistical properties such as expectation and variance in probability distributions. If the probability density function is Gaussian, these statistical properties can be calculated analytically using the Gauss-Hermite integral.
• Quantum Mechanics: In quantum mechanics, the Gauss-Hermite integral is used to compute the expectation value and normalization constant of a wave function. In particular, Hermite polynomials for potentials such as harmonic oscillators play an important role in Gauss-Hermite integrals.
• Quantum Chemistry: Gauss-Hermite integrals are used to calculate molecular orbital integrals and electronic interactions when calculating the electronic structure of molecules. This allows the prediction of molecular energies and reactivity.
• Statistical Mechanics: In statistical mechanics, the Gauss-Hermite integral is applied to calculate physical quantities (e.g. partition function, average energy, heat capacity) for many-particle systems. This allows for the theoretical study of phase transitions and properties of matter.
• Quantum field theory: The Gauss-Hermite integral is also applied in field theory to integrals related to fermionic fields (fields describing particles with spin 1/2). It is used to calculate expectation values and correlation functions for fermionic fields.
• Information Theory: The Gauss-Hermite integral is also applied to entropy calculations in information theory. They are used to compute the entropy of continuous probability distributions and are relevant to the study of data compression and information theory.

Although these are common applications of the Gauss-Hermite integral, it is used in practice in a variety of scientific and technical fields. In particular, the Gauss-Hermite integral is of great importance in stochastic problems, quantum mechanics, statistics, physics, chemistry, and information theory.

Example implementation of probability statistics using the Gauss-Hermite integral

This section describes an example implementation of probability statistics using the Gauss-Hermite integral. In the following example, the Gauss-Hermite integral is used to compute the mean and variance of a normal distribution. Python and the SciPy library are used here.

import numpy as np
from scipy import integrate
from scipy.special import hermite
import math

# Mean and variance of normal distribution
mu = 2.0
sigma = 1.0

# Calculation of integration points and weights for Gauss-Hermite integrals
n = 50  # Number of Integral Points
x, w = np.polynomial.hermite.hermgauss(n)

# Calculation of Gauss-Hermite integrals
def gaussian_hermite_integration(f):
    result = 0.0
    for i in range(n):
        result += w[i] * f(x[i])
    return result

# Probability density function of normal distribution
def normal_pdf(x):
    return (1.0 / (math.sqrt(2 * math.pi) * sigma)) * math.exp(-((x - mu) ** 2) / (2 * sigma ** 2))

# Calculation of averages
mean = gaussian_hermite_integration(lambda x: x * normal_pdf(x))

# Calculation of Variance
variance = gaussian_hermite_integration(lambda x: (x - mean) ** 2 * normal_pdf(x))

print("average:", mean)
print("dispersion:", variance)

The code integrates the probability density function of the normal distribution using Gauss-Hermite integration to compute the mean and variance. The integration points and weights of the Gauss-Hermite integral are computed with np.polynomial.hermite.hermgauss(n).

Reference Information and Reference Books

Various examples of mathematical and probabilistic approaches to machine learning are described in “On Mathematics in Machine Learning“. Also see “Stochastic Optimization” “Statistical Learning Theory” and “Continuous Optimization for Machine Learning” especially for optimization using machine learning.

For reference books, see “Riemann and Algebraic Function Theory: The Nodes of Modern Western Mathematics.

“Metric Spaces of Non-Positive Curvature”