A New Subspace Iteration method for the Algebraic Riccati Equation

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.Comment: 25page

Numerical computation and new output bounds for time-limited balanced truncation of discrete-time systems

In this paper, balancing based model order reduction (MOR) for large-scale linear discrete-time time-invariant systems in prescribed finite time intervals is studied. The first main topic is the development of error bounds regarding the approximated output vector within the time limits. The influence of different components in the established bounds will be highlighted. After that, the second part of the article proposes strategies that enable an efficient numerical execution of time-limited balanced truncation for large-scale systems. Numerical experiments illustrate the performance of the proposed techniques.Comment: 23 pages, 4 figure