Lagmental Vicfred

Gaussian Quadrature Is Exact through Degree Two n Minus One by Vicfred

tags: mathematics, numerical analysis, root finding, interpolation

Choosing nodes as roots of an orthogonal polynomial doubles the polynomial exactness available from n samples. I want the notation, the mechanism, and the failure mode visible at the same time.

Statement

Numerical analysis replaces an exact object \(x^\ast\) by approximations \(x_n\) with controlled error. Stability asks how rounding or data perturbations affect the answer.

$$ \int_a^bf(x)w(x)\,dx\approx\sum_{i=1}^{n}\omega_if(x_i) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \deg f\le2n-1\Longrightarrow\int_a^bf(x)w(x)\,dx=\sum_{i=1}^{n}\omega_if(x_i) $$

Worked algebra

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ n=2,\ [-1,1]:\qquad\int_{-1}^{1}f(x)\,dx\approx f(-1/\sqrt3)+f(1/\sqrt3) $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \int_a^bf(x)w(x)\,dx\approx\sum_{i=1}^{n}\omega_if(x_i),\\[5pt] \mathsf{C}\;&:\quad \deg f\le2n-1\Longrightarrow\int_a^bf(x)w(x)\,dx=\sum_{i=1}^{n}\omega_if(x_i). \end{aligned} $$

Caveat

Fast local convergence is not a global guarantee. Newton's method can diverge, interpolation can oscillate, and a small residual can coexist with a large forward error.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \deg f\le2n-1\Longrightarrow\int_a^bf(x)w(x)\,dx=\sum_{i=1}^{n}\omega_if(x_i) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Sun 02 August 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.