On polynomial interpolation in the monomial basis

TL;DR

This paper demonstrates that the monomial basis can be effectively used for polynomial interpolation with proper condition number bounds, leading to practical algorithms and new bounds for Vandermonde matrices.

Contribution

It establishes conditions under which the monomial basis is as effective as well-conditioned bases and introduces a new upper bound for Vandermonde matrix condition numbers.

Findings

Monomial basis is viable for interpolation under certain condition number bounds.

Provides a practical algorithm for piecewise polynomial interpolation in the complex plane.

Derives a new upper bound for Vandermonde matrix condition numbers.

Abstract

In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.