Highly accurate decoupled doubling algorithm for large-scale M-matrix algebraic Riccati equations

TL;DR

This paper introduces a highly accurate, decoupled doubling algorithm for solving large-scale M-matrix algebraic Riccati equations, improving efficiency and precision through novel kernel analysis.

Contribution

The paper develops a new decoupled doubling iteration with small M-matrix kernels and constructs highly accurate triplet representations for improved numerical solutions.

Findings

The decoupled algorithm is efficient for large-scale problems.

Small M-matrix kernels enable high-accuracy solutions.

Numerical examples demonstrate the algorithm's effectiveness.

Abstract

We consider the numerical solution of large-scale M-matrix algebraic Riccati equations with low-rank structures. We derive a new doubling iteration, decoupling the four original iteration formulae in the alternating-directional doubling algorithm. We prove that the kernels in the decoupled algorithm are small M-matrices. Illumined by the highly accurate algorithm proposed by Xue and Li in 2017, we construct the triplet representations for the small M-matrix kernels in a highly accurate doubling algorithm. Illustrative numerical examples will be presented on the efficiency of our algorithm.