The Shifting Technique for Solving a Nonsymmetric Algebraic Riccati Equation

TL;DR

This paper introduces a shifted structure-preserving doubling algorithm for solving a specific nonsymmetric algebraic Riccati equation, achieving guaranteed quadratic convergence and improved efficiency through a novel shifting technique.

Contribution

It develops a shifted SDA that guarantees quadratic convergence without breakdown and enhances simple iteration speed in critical cases.

Findings

The shifted SDA converges quadratically with no breakdown.

Numerical experiments show reduced computational steps.

The modified simple iteration significantly speeds up convergence.

Abstract

This paper analyzes a special instance of nonsymmetric algebraic matrix Riccati equations arising from transport theory. Traditional approaches for finding the minimal nonnegative solution of the matrix Riccati equations are based on the fixed point iteration and the speed of the convergence is linear. Relying on simultaneously matrix computation, a structure-preserving doubling algorithm (SDA) with quadratic convergence is designed for improving the speed of convergence. The difficulty is that the double algorithm with quadratic convergence cannot guarantee to work all the time. Our main trust in this work is to show that applied with a suitable shifted technique, the SDA is guaranteed to converge quadratically with no breakdown. Also, we modify the conventional simple iteration algorithm in the critical case to dramatically improve the speed of convergence. Numerical experiments strongly suggest that the total number of computational steps can be significantly reduced via the shifting procedure.