Vandermonde with Arnoldi
Pablo Brubeck, Yuji Nakatsukasa, Lloyd N. Trefethen · SIAM Review · 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.