On Generating Gaussian Quadrature Rules

Abstract

Given a mass distribution dσ(x) on the (finite or infinite) interval (a,b), where σ(x) has at least n+1 points of increase, and assuming the existence of the first 2n moments of dσ(x),

$${\mu _k} = \int_a^b {{x^k}} d\sigma (x),\;\;\;\;k = 0,1,2,...,2n - 1$$

((1.1))

it is well known that the n-point Gaussian quadrature rule associated with the distribution da(x) exists and is unique. That is, there exist unique nodes ⁽ⁿ⁾_v ∊ (a,b) and weights λ ⁽ⁿ⁾_v > 0 such that

$$\int_a^b {f(x)} d\sigma (x) = \sum\limits_{v = 1}^n {\lambda _v^{(n)}f(\xi _v^{(n)})} + {R_n}(f)$$

((1.2))

with

$$ {R_n}(f) = 0\,for\,all\,f \in {P_{2n - 1}} $$

((1.3))

.