Confluent Vandermonde with Arnoldi

TL;DR

This paper extends the Vandermonde with Arnoldi method to handle confluent Vandermonde matrices, enabling high-precision computation of polynomial derivatives and broadening applications involving derivatives.

Contribution

It establishes a theorem for Krylov subspaces generating the confluent Vandermonde matrix, facilitating high-precision derivative calculations and comparing two orthogonalization approaches.

Findings

The method allows high-precision derivatives of high-degree polynomials.

Two orthogonalization approaches are compared, with problem-dependent preferences.

Numerical examples demonstrate the method's effectiveness.

Abstract

In this note, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by P. D. Brubeck and L. N. Trefethen. The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.