Peter Cowal | Searcharxiv

arXiv2025

Random Walks, Faber Polynomials and Accelerated Power Methods

In this paper, we construct families of polynomials defined by recurrence relations related to mean-zero random walks. We show these families of polynomials can be used to approximate $z^n$ by a polynomial of degree $\sim \sqrt{n}$ in associated radially convex domains in the complex plane. Moreover, we show that the constructed families of polynomials have a useful rapid growth property and a connection to Faber polynomials. Applications to iterative linear algebra are presented, including the development of arbitrary-order dynamic momentum power iteration methods suitable for classes of non-symmetric matrices.

math.NA↗

arXiv2025

Faber polynomials in a deltoid region and power iteration momentum methods

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.

math.NA↗