How accurate is Gauss quadrature?

How accurate is Gauss quadrature?

The important property of Gauss quadrature is that it yields exact values of integrals for polynomials of degree up to 2n – 1. A Gauss quadrature rule with 3 points will yield exact value of integral for a polynomial of degree 2 × 3 – 1 = 5. Simpson’s rule also uses 3 points, but the order of accuracy is 3.

What is the maximum no of degree of polynomial that could be solved using Gauss quadrature method?

It is accurate for polynomials up to degree 2n – 3, where n is the number of integration points (Quarteroni, Sacco & Saleri 2000).

How do you calculate Gaussian quadrature?

The Gaussian quadrature method is an approximate method of calculation of a certain integral . By replacing the variables x = (b – a)t/2 + (a + b)t/2, f(t) = (b – a)y(x)/2 the desired integral is reduced to the form .

What is the degree of a quadrature rule?

Recall that each Newton–Cotes quadrature rule came from integrating the Lagrange polynomial that interpolates the integrand f at n equally spaced nodes in the interval [a, b]. Therefore, the highest degree of exactness we can expect to achieve with any quadrature rule is 2n − 1.

What is a quadrature point?

For a function of one independent variable, the basic idea of a quadrature rule is to replace the definite integral by a sum of the integrand evaluated at certain points (called quadrature points ) multiplied by a number (called quadrature weights ).

How do you calculate quadrature?

xj=a+jh, j=0…n, h=(b−a)n, where n is a positive integer, N=n+1, is called the Newton–Cotes quadrature formula; this quadrature formula has algebraic degree of accuracy d=n when n is odd and d=n+1 when n is even.

Why is Gauss quadrature used in FEM?

1 Gaussian Quadrature. The finite element method requires the calculation of integrals over individual elements, for example, Gaussian quadrature approximates an integral as the weighted sum of the values of its integrand. Consider integrating the general function , over the domain − 1 ≤ ξ ≤ 1 .

What is a quadrature formula?

An approximate formula for the calculation of a definite integral: b∫ap(x)f(x)dx≅N∑j=1Cjf(xj). The sum on the right-hand side of (1) is called the quadrature sum, the numbers xj are called the nodes of the quadrature formula, while the numbers Cj are called its weights.

What is general quadrature formula?

What means quadrature?

noun. the act of squaring. Mathematics. the act or process of finding a square equal in area to a given surface, especially a surface bounded by a curve. the act or process of finding an area or calculating an integral, especially by numerical methods.