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

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