Vandermonde with Arnoldi - Brubeck, PD; Nakatsukasa, Y; Trefethen, LN (Society for Industrial and Applied Mathematics, 2021)
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.