An inexact low-rank Newton–ADI method for large-scale algebraic Riccati equations

Abstract

This paper improves the inexact Kleinman–Newton method for solving algebraic Riccati equations by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman–Newton step. A convergence result is presented that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the Lyapunov equation residual, which in general do not hold for low-rank approaches. In the convergence proof of this paper, the line search is needed to ensure that the Riccati residuals decrease monotonically in norm. In the numerical experiments, the line search can lead to substantial reduction in the overall number of ADI iterations and, therefore, overall computational cost.