Faber polynomials in a deltoid region and power iteration momentum methods

TL;DR

This work generalizes a Chebyshev-like acceleration to a deltoid region by introducing a family of polynomials $P_n$ satisfying the same recurrence as Faber polynomials for the deltoid domain $Γ$. It proves $|P_n(z)| ≤ 1$ on $Γ$ and a superpolynomial growth outside the unit disk, and shows that $z^n$ can be approximated by a low-degree $P_k$ basis with coefficients given by random-walk probabilities, yielding a square-root degree efficiency. These analytic results underpin a higher-order momentum approach for the power iteration: a Deltoid Momentum Power Method and a dynamic-parameter variant that accelerates convergence for matrices with complex eigenvalues, with theoretical guarantees and numerical demonstrations on toy, circulant, and Markov-chain examples. The combination of polynomial approximation theory and momentum-enabled iterations offers practical acceleration for eigenvalue computations and related matrix functions in spectral regions containing a deltoid shape, with potential extensions to broader domains and higher-order recurrences.

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.

Figures (3)

• Figure 1: The curve $\gamma$ defined by \ref{['gamma']}, called a deltoid, intersects the circle $|z|=1$ at three points and circle $|z|=1/3$ at three points. The closure of the region enclosed by $\gamma$ is denoted by $\Gamma$.
• Figure 2: Plots of $|P_n(z)|$ on $\{z \in \mathbb{C}: |\mathop{\mathrm{Re}}\nolimits(z)| \leq 1, |\mathop{\mathrm{Im}}\nolimits(z)| \leq 1 \}$, where points such that $|P_n(z)| \leq 10^{-3}$ are shown in black, and points such that $|P_n(z)| \geq 1$ are shown in white. The contours indicate $|P_n(z)| \in \{0.2, 0.4, 0.6, 0.8, 1\}$.
• Figure 3: Results for the numerical example of Section \ref{['toyexample']} (top left), Section \ref{['circulantexample']} (top right) and Section \ref{['stationarydist']} (bottom), performed with Algorithm \ref{['alg1']} (power method), the algorithm of xu2018accelerated (order 1), Algorithm \ref{['algnew']} (order 2), and Algorithm \ref{['alg:dymo2']} (order 2 dyn). The dotted line is the asymptotic rate given by Theorem \ref{['algoworks']}.

Theorems & Definitions (30)

• Lemma 1
• Theorem 1: Sachdeva, Vishnoi, sachdeva2014faster
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1: Optimizing the error bound
• Lemma 2
• proof : Proof of Lemma \ref{['lem:contraction']}
• Remark 2: Stability of dynamic algorithm
• Lemma 3