How Does Gaussian-Legendre Quadrature Approximate Non-Polynomial Functions?

[FranzS]

TL;DR Summary

What is the specific polynomial associated with the Gaussian-Legendre quadrature?

The n-point Gaussian-Legendre quadrature gives an exact value for the numerical integration of polynomials with degree up to 2n-1.
For the integration of non-polynomial functions, the n-point Gaussian-Legendre quadrature gives a good approximation as long as the function is well approximated by a polynomial with degree 2n-1.

My question is: given a non-polynomial function to be integrated, is its n-point Gaussian-Legendre quadrature associated with a specific polynomial with degree 2n-1?
In that case, how do you find it?
Will that polynomial be the best approximation (with degree 2n-1) of the original function between the limits of integration? In other words, will that polynomial be the hypothetical result of applying a multilinear ("polynomial") regression (2n-1 degree) to "all" the points of the original function between the limits of integration?

Thanks for your attention.

 

[pasmith]

What Gauss-Legendre quadrature does is to interpolate a function $f$ by a polynomial $p_f$ of degree $n-1$ defined by $$p_f(x_i) = f(x_i),\qquad 1 \leq i \leq n,$$ and use the approximation $$\int_{a}^{b} f(x)\,dx \approx \int_a^b p_f(x)\,dx = \sum_{i=1}^n f(x_i)w_i.$$ The method is an $n$-point method since it uses $n$ sample points; the idea is that the specific choice of the $x_i$ and $w_i$ guarantees that the integral of a product of two degree $n - 1$ polynomials (ie. a polynomial of degree $2n-2$) will also be exact, which assists in approximating inner products in applications where that is relevant. If that is not relevant to you, then there are other $n$-point methods which may be more accurate.

 

[FranzS]

Thanks for your reply. I apparently got fooled by the fact that the $n$-point G-L quadrature gives an exact result for the integral of a polynomial of degree $2n-1$ (I do not understand why you write $2n-2$, would you mind to explain that to me?), but I did not consider it does so by interpolating the function with a polynomial with degree $n-1$, which is uniquely defined by the $n$ fixed points.

EDIT: is it extended to $2n-1$ because the monomial of such degree (the highest in the polynomial) does not contribute to the integral, being an odd power which is integrated over a symmetrical interval?