A New Subspace Iteration method for the Algebraic Riccati Equation

TL;DR

This paper introduces a novel subspace iteration method for efficiently approximating solutions to large, sparse algebraic Riccati equations by leveraging invariant subspace techniques and low rank updates.

Contribution

The paper presents a new subspace iteration algorithm for the algebraic Riccati equation that combines low rank updates with invariant subspace methods, enhancing solution efficiency and understanding.

Findings

Effective low rank approximation of X achieved

Method inherits properties from ADI iteration

New matrix relations improve understanding of solution strategies

Abstract

We consider the numerical solution of the continuous algebraic Riccati equation $A^*X+XA-XFX+G=0$, with $F=F^*, G=G^*$ of low rank and $A$ large and sparse. We develop an algorithm for the low rank approximation of $X$ by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought after approximation can be obtained by a low rank update, in the style of the well known ADI iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection-type methods, which will help increase our understanding of this latter class of solution strategies.