Faber polynomials in a deltoid region and power iteration momentum methods

TL;DR

This paper introduces a family of polynomials related to Faber polynomials in a deltoid region, providing new approximation results and applying them to develop a momentum-based acceleration method for power iteration in linear algebra.

Contribution

It constructs a polynomial family for the deltoid region, proves approximation properties, and applies these results to design an accelerated power iteration method with momentum.

Findings

Polynomial family approximates powers within the deltoid region

Bounded polynomial magnitude inside the deltoid

Accelerated power iteration method with momentum

Abstract

We consider a region in the complex plane enclosed by a deltoid curve inscribed in the unit circle, and define a family of polynomials $P_n$ that satisfy the same recurrence relation as the Faber polynomials for this region. We use this family of polynomials to give a constructive proof that $z^n$ is approximately a polynomial of degree $\sim\sqrt{n}$ within the deltoid region. Moreover, we show that $|P_n| \le 1$ in this deltoid region, and that, if $|z| = 1+\varepsilon$, then the magnitude $|P_n(z)|$ is at least $\frac{1}{3}(1+\sqrt{\varepsilon})^n$, for all $\varepsilon > 0$. We illustrate our polynomial approximation theory with an application to iterative linear algebra. In particular, we construct a higher-order momentum-based method that accelerates the power iteration for certain matrices with complex eigenvalues. We show how the method can be run dynamically when the two dominant eigenvalues are real and positive.