Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

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