A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null recurrent. While in this case the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well-behaved one. Ill-conditioning and slow convergence appear also in close-to-critical problems, but when none of the eigenvalues is exactly zero the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present a theoretical analysis and several numerical experiments which confirm the efficiency of the new method

Order reduction methods for solving large-scale differential matrix Riccati equations

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming low-rank approximations to the sought after solution at selected timesteps. We show that great computational and memory savings are obtained by a reduction process onto rational Krylov subspaces, as opposed to current approaches. By specifically addressing the solution of the reduced differential equation and reliable stopping criteria, we are able to obtain accurate final approximations at low memory and computational requirements. This is obtained by employing a two-phase strategy that separately enhances the accuracy of the algebraic approximation and the time integration. The new method allows us to numerically solve much larger problems than in the current literature. Numerical experiments on benchmark problems illustrate the effectiveness of the procedure with respect to existing solvers

Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms

We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton\u27s iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms

The problem of reducing an algebraic Riccati equation $XCX-AX-XD+B=0$ to a unilateral quadratic matrix equation (UQME) of the kind $PX^2+QX+R$ is analyzed. New reductions are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the cyclic reduction algorithm applied to a suitable UQME. A new algorithm obtained by complementing our reductions with the shrink-and-shift tech- nique of Ramaswami is presented. Finally, faster algorithms which require some non-singularity conditions, are designed. The non-singularity re- striction is relaxed by introducing a suitable similarity transformation of the Hamiltonian

Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring. The equations we consider are Stein equations $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, and nonlinear matrix equations $X+A^*X^{-1}A=Q$. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge

A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process

We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems