Relaxed Fixed Point Iterations for Matrix Equations Arising in Markov Chains Modeling

TL;DR

This paper introduces accelerated fixed point iteration methods for efficiently computing solutions to matrix equations in Markov chain models, demonstrating improved convergence through theoretical analysis and numerical experiments.

Contribution

It proposes new staircase regular splittings and two-step fixed point iterations with relaxation for faster convergence in Markov chain matrix equations.

Findings

Proposed methods outperform classical iterations in numerical tests.

Theoretical convergence of the new iterations is rigorously proved.

Accelerated variants reduce computation time in practical applications.

Abstract

We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration and of its relaxed modification is proved. Our theoretical analysis, along with several numerical experiments show that the proposed variants generally outperform the classical iterations.