Linear convergence of the Collatz method for computing the Perron eigenpair of primitive dual number matrix

TL;DR

This paper confirms two conjectures about dual number matrices, proving the Collatz method's linear convergence and providing explicit convergence rates for computing the Perron eigenpair.

Contribution

It offers a theoretical proof that the Collatz method linearly converges for dual number matrices, confirming prior conjectures and extending Perron-Frobenius theory.

Findings

The spectral radius of the standard part determines the decay of the matrix power.

The Collatz method converges linearly with an explicit rate.

Conjectures by Qi and Cui are validated through rigorous proof.

Abstract

Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method was also extended to find Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix tends to zero if and only if the spectral radius of its standard part less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.