Accurate Numerical Solution for Shifted M-Matrix Algebraic Riccati Equations

Abstract

An algebraic Riccati equation (are) is called a shifted M-matrix algebraic Riccati equation (mare) if it can be turned into an mare after its matrix variable is partially shifted by a diagonal matrix. Such an are can arise from computing the invariant density of a Markov modulated Brownian motion. Sufficient and necessary conditions for an are to be a shifted mare are obtained. Based on the conditions, a highly accurate implementation of the alternating directional doubling algorithm (adda) is established to compute the extremal solution of a shifted mare, as well as a quantity needed for computing the invariant density in the application, with high entrywise relative accuracy. Numerical examples are presented to demonstrate the theory and algorithms.