Vandermonde with Arnoldi

Vandermonde matrices are exponentially ill-conditioned, rendering the familiar polyval(polyfit)algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem. This amounts to on-the-fly construction of discrete orthogonal polynomials by Stieltjes orthogonalization.

Automated summary

Vandermonde matrices become exponentially ill-conditioned at higher degrees, causing the familiar polyval(polyfit) algorithm for polynomial interpolation and least-squares fitting to be ineffective. However, this issue can be fixed by using Arnoldi orthogonalization, which involves constructing discrete orthogonal polynomials on-the-fly.