We have seen how quadrature rules can be derived based on interpolating a set of function values by a polynomial or piecewise polynomial of a given degree and integrating the interpolant exactly.

The word “quadrature” reminds us of an elementary technique for finding this area—plot the function on graph paper and count the number of little squares that lie underneath the curve.

Then any quadrature formula ˆI is exact for polynomials up to order n if and only if it is exact up to order 2n + 1. Corresponding quadrature rules are usually prefixed with “Gauss-”, i.e., “Gauss-Legendre …

We note that I(L2(x)q(x)) = 0 due to orthogonality of L2(x) to all of the poly-nomials of smaller degrees, I(r) = Q(r) due to exactness of the quadrature for all polynomials of degree less than 2 by …

Gauss Quadrature Derivation of Gauss-3 point quadrature formula. b f(x) dx ≈ PN i=1 [w1hf(xi−1 + r1h) + w2hf(xi−1 + r2h) + w3hf(xi−1 + r3h)] To make the algebra simpler, let’s take a = −1, b = 1, N = 1, h …

— Gaussian quadrature uses good choices of xi nodes and ωi weights. — Gaussian quadrature formulas use n points and are exact of degree 2n − 1 Theorem 1 Suppose that {φ k(x)}∞ is an …

First we discuss the general non-composite Newton-Cotes (NC) quadrature rule. This rule is based on polynomial interpolation. To define it, we choose n + 1 points in [a, b] as our nodes xj, and then …