Numerical Integration: Gaussian Quadrature | Theory and Python Implementation #masteringpython

Gaussian quadrature is a numerical integration technique that aims to approximate the definite integral of a function over a specified interval. Unlike traditional methods such as the trapezoidal rule or Simpson's rule, Gaussian quadrature achieves higher accuracy by strategically selecting the integration points and weights based on the properties of orthogonal polynomials. The weights in Gaussian quadrature are determined through a mathematical process that involves the properties of orthogonal polynomials and the desired accuracy of the quadrature rule. The weights are chosen to ensure that the quadrature rule is exact for polynomials up to a certain degree. #cuphysics #physics #engineering #numerical #numericalanalysis #gaussianQuadrature #numpy #pythonprogramming #1/3 simpsons #simpson's #cu #sem iii#mastering t1/3 Simpsons Method #datascience #integration #numerical physics #computationalthinking #numerical analysis#quadrature #Mastering Numerical Integration: Simpson's 1/3 Rule in Python #bscphysicscontent #bscphysicsclass #bsc physics numerical ------------------------------------------------------------------------------------------------------------- ''np.polynomial.legendre.leggauss(n)": This function is part of NumPy and generates the Legendre-Gauss quadrature points (x) and weights (w) for a specified number of points n. The points are by default in the interval [-1, 1]. Translation of x to the interval [a, b]: The line x = (b - a) * 0.5 * x + (b + a) * 0.5 scales and translates the quadrature points from the default interval [-1, 1] to the desired interval [a, b]. This is necessary because the Legendre-Gauss points are typically defined on the interval [-1, 1], and the function adjusts them to the specific interval of integration. Return values: The function returns the translated quadrature points (x) and weights (w). -------------------------------------------------------------------------------------------------------------- #datascience #computationalthinking #bscmaths #pythontutorial #numpy #scipy