RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations

TL;DR

This paper presents RADI, a low-rank ADI-type algorithm for efficiently solving large-scale algebraic Riccati equations, unifying several existing methods into a single iterative framework.

Contribution

The paper introduces RADI, a novel low-rank ADI-type algorithm for CARE that generalizes and unifies existing methods, with improved implementation strategies.

Findings

RADI produces the same iterates as three other known algorithms when using identical parameters.

The algorithm reduces complex arithmetic and optimizes shift strategies.

It offers an efficient solution approach for large-scale CARE problems.

Abstract

This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature -- all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.