Palindromic linearization and numerical solution of nonsymmetric algebraic T-Riccati equations

TL;DR

This paper explores the relationship between nonsymmetric algebraic T-Riccati equations and palindromic matrix pencils, introducing new theoretical insights and numerical methods for solving these equations efficiently.

Contribution

It establishes a connection between T-Riccati solutions and palindromic pencils, and develops novel numerical algorithms based on the QZ and doubling methods.

Findings

Theoretical properties of T-Riccati solutions are derived.

New numerical algorithms demonstrate effectiveness in tests.

Palindromic QZ and doubling algorithms outperform existing methods.

Abstract

We identify a relationship between the solutions of a nonsymmetric algebraic T-Riccati equation (T-NARE) and the deflating subspaces of a palindromic matrix pencil, obtained by arranging the coefficients of the T-NARE. The interplay between T-NARE and palindromic pencils allows one to derive both theoretical properties of the solutions of the equation, and new methods for its numerical solution. In particular, we propose methods based on the (palindromic) QZ algorithm and the doubling algorithm, whose effectiveness is demonstrated by several numerical tests