The Gaussian Quadrature is a numerical method used for approximating definite integrals. It involves the use of weighted sum of function values at specific points within the interval of integration.
The Gaussian Quadrature works by choosing specific points (called nodes) within the interval of integration and calculating the corresponding weights. These weights are then multiplied with the function values at the nodes and added together to approximate the integral.
The Gaussian Quadrature has several advantages over other numerical methods for approximating integrals. It is very accurate, especially when compared to simpler methods such as the trapezoidal rule. It also requires fewer function evaluations, making it more efficient.
The Gaussian Quadrature is limited by the number of nodes chosen and the degree of the polynomial it can accurately integrate. If the function being integrated is highly oscillatory or has singularities, the accuracy of the approximation may be affected.
The Gaussian Quadrature differs from other methods in that it chooses the nodes and weights in a way that minimizes the error in the approximation. This allows for more accurate results with fewer function evaluations. It is also more versatile, as it can be applied to a wider range of functions.