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

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• FranzS
In summary, the n-point Gaussian-Legendre quadrature method is used to numerically integrate polynomials of degree up to 2n-1 with exact results. For non-polynomial functions, it provides a good approximation as long as the function can be well approximated by a polynomial of degree 2n-1. The method uses n sample points and interpolates the function with a polynomial of degree n-1. The specific choice of sample points guarantees that the integral of a product of two degree n-1 polynomials will also be exact. This method may not be relevant in all cases, as there are other n-point methods that may be more accurate.
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?

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.

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?

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

Gaussian-Legendre quadrature is a numerical integration technique used to approximate the definite integral of a function. It involves dividing the interval of integration into smaller subintervals and using a weighted sum of function values at specific points within each subinterval to estimate the integral.

Gaussian-Legendre quadrature works by using a set of predetermined points and weights to approximate the integral of a function. These points and weights are chosen in a way that minimizes the error in the approximation, making Gaussian-Legendre quadrature a highly accurate numerical integration method.

One of the main advantages of Gaussian-Legendre quadrature is its high accuracy. It is also a relatively simple and efficient method, making it a popular choice for numerical integration. Additionally, Gaussian-Legendre quadrature can be used for a wide range of functions and intervals, making it a versatile technique.

What are the limitations of Gaussian-Legendre quadrature?

One limitation of Gaussian-Legendre quadrature is that it can only be used for smooth functions. It also requires the function to be integrated to be known at the predetermined points, which may not always be feasible. Additionally, Gaussian-Legendre quadrature may not be the most efficient method for functions with highly oscillatory behavior.

How is Gaussian-Legendre quadrature different from other numerical integration methods?

Gaussian-Legendre quadrature differs from other numerical integration methods in the way it chooses the points and weights for approximating the integral. Other methods, such as the trapezoidal rule or Simpson's rule, use equally spaced points, while Gaussian-Legendre quadrature uses a specific set of points and weights that are optimized for accuracy.

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