A Low-rank ADI Algorithm for Solving Large-scale Non-symmetric Algebraic Riccati Equations

TL;DR

This paper introduces a novel low-rank ADI algorithm specifically designed for large-scale non-symmetric algebraic Riccati equations, enhancing efficiency and autonomy in solving these complex matrix equations.

Contribution

It develops the first low-rank ADI algorithm tailored for large-scale NAREs, unifying and extending existing methods for related matrix equations.

Findings

Successfully solves a benchmark problem of order 10^6

Demonstrates high computational efficiency and accuracy

Automatically generates shifts without user intervention

Abstract

This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving several matrix equations, including Lyapunov equations, Sylvester equations, and symmetric Riccati equations. Although a low-rank algorithm for the Sylvester equation has been used as an inner loop in computing low-rank solutions of NAREs, no low-rank ADI algorithm currently exists for NAREs themselves. This paper fills this gap by developing a low-rank ADI algorithm for large-scale NAREs that admit a low-rank solution. Since Lyapunov equations, Sylvester equations, and symmetric Riccati equations are special cases of the NARE, the existing low-rank ADI methods in the literature are special cases of the more general low-rank ADI method proposed here. An automatic and computationally efficient method for shift generation is also discussed, and a subspace-accelerated projection approach is presented to generate shifts for subsequent iterations without user intervention. Once initialized with arbitrary shifts, the proposed algorithm solves large-scale NAREs autonomously, generating its own shifts. Numerical results are presented using benchmark example of order $10^6$, demonstrating the computational efficiency and accuracy of the proposed algorithm.