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.Comment: arXiv admin note: text overlap with arXiv:2306.16140 by other author