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

doi:10.1007/s42967-024-00426-6

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 232-247.doi: 10.1007/s42967-024-00426-6

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

Yongjun Chen, Liping Zhang

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

Received:2024-02-11 Revised:2024-05-01 Online:2026-02-20 Published:2026-02-11
Contact: Liping Zhang,E-mail:lipingzhang@tsinghua.edu.cn E-mail:lipingzhang@tsinghua.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China (Grant no. 12171271).

Abstract

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 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.

Key words: Dual numbers, Dual primitive matrices, Eigenvalues, Collatz method, Linear convergence

CLC Number:

Yongjun Chen, Liping Zhang. Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 232-247.

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

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