45 research outputs found
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
On a family of low-rank algorithms for large-scale algebraic Riccati equations
In [3] it was shown that four seemingly different algorithms for computing
low-rank approximate solutions to the solution of large-scale
continuous-time algebraic Riccati equations (CAREs) generate the same sequence when used with the same
parameters. The Hermitian low-rank approximations are of the form where is a matrix with only few columns and is a
small square Hermitian matrix. Each generates a low-rank Riccati residual
such that the norm of the residual can be evaluated easily
allowing for an efficient termination criterion. Here a new family of methods
to generate such low-rank approximate solutions of CAREs is proposed.
Each member of this family of algorithms proposed here generates the same
sequence of as the four previously known algorithms. The approach is
based on a block rational Arnoldi decomposition and an associated block
rational Krylov subspace spanned by and Two specific versions of
the general algorithm will be considered; one will turn out to be a rediscovery
of the RADI algorithm, the other one allows for a slightly more efficient
implementation compared to the RADI algorithm. Moreover, our approach allows
for adding more than one shift at a time
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 of another naturally-arising fixed-point
iteration via a sort of repeated squaring.
The equations we consider are Stein equations , Lyapunov
equations , discrete-time algebraic Riccati equations
, continuous-time algebraic Riccati equations
, palindromic quadratic matrix equations , and
nonlinear matrix equations . 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