The Gauss-Legendre Quadrature method is a numerical integration technique used to approximate the definite integral of a function. It involves using a weighted sum of function values at specific points within the integration interval, known as Gauss-Legendre Quadrature coefficients, to estimate the integral.
The Gauss-Legendre Quadrature coefficients are calculated using a mathematical formula that involves solving a system of equations. The number of coefficients used depends on the desired level of accuracy, with higher numbers of coefficients resulting in a more precise estimate of the integral.
The use of Gauss-Legendre Quadrature coefficients allows for a more accurate estimation of definite integrals compared to other numerical integration methods. It also has the advantage of being able to handle integrands with singularities or discontinuities with relative ease.
The Gauss-Legendre Quadrature coefficients are closely related to orthogonal polynomials, specifically the Legendre polynomials. These polynomials are used to generate the coefficients and are also used as the basis functions for the integration.
The Gauss-Legendre Quadrature method is commonly used in various fields of engineering and science to numerically solve integrals that arise in physical and mathematical models. It also has applications in numerical analysis, signal processing, and numerical weather prediction.