Balancing a matrix by a simple and accurate similarity transformation can improve
the speed and accuracy of numerical methods for computing eigenvalues. We describe
balancing strategies for a large and sparse Hamiltonian matrix H. It is first shown how
to permute H to irreducible form while retaining its structure. This form can be used to
decompose the Hamiltonian eigenproblem into smaller-sized problems. Next, we discuss
the computation of a symplectic scaling matrix D so that the norm of D 1 HD is reduced.
The considered scaling algorithm is solely based on matrix-vector products and thus particularly
suitable if the elements of H are not explicitly given. The merits of balancing
for eigenvalue computations are illustrated by several practically relevant examples.
In this paper we investigate the use of parallel computing to deal with the high computational cost of numerical algorithms for model reduction of large linear descriptor systems. The state-space truncation methods considered here are composed of iterative schemes which can be efficiently implemented on parallel architectures using existing parallel linear algebra libraries. Our experimental results on a cluster of Intel Pentium processors show the performance of the parallel algorithms.
We discuss a parallel library of efficient algorithms for model reduction of largescale
systems with state-space dimension up to O(104). We survey the numerical
algorithms underlying the implementation of the chosen model reduction methods.
The approach considered here is based on state-space truncation of the system
matrices and includes absolute and relative error methods for both stable and unstable
systems. In contrast to serial implementations of these methods, we employ
Newton-type iterative algorithms for the solution of the major computational tasks.
Experimental results report the numerical accuracy and the parallel performance of
our approach on a cluster of Intel Pentium II processors.
We describe a prototype web service for model reduction of very large-scale linear systems, with
dimension in the order of millions of states, that includes a user-friendly interface designed so that the computation
can be easily performed via the HTTP protocol. Access via a web browser isolates the user of the service from the
complexities of installing and using the parallel model reduction codes and the maintenance of the hardware. In case
the routines are found to be appropriate for the problem the user needs to solve, the library can be then downloaded
and installed on the user’s own computing resources.
This paper illustrates the major issues of the access procedure by means of graphical examples, and describes
the structure and implementation of the remote model reduction service. The service is offered in a cluster of Linux
machines.
A Structure-Preserving Method for Generalized Algebraic RiccatiEquations Based on Pencil Arithmetic
(2004)
This paper describes a numerical method for extracting the stable
right deflating subspace of a matrix pencil Z Y using
a spectral projection method. It has several advantages compared
to other spectral projection methods like the sign function
method. In particular it avoids the rounding error induced
loss of accuracy associated with matrix inversions. The new algorithm
is particularly well adapted to solving continuous-time
algebraic Riccati equations. In numerical examples, it solves
Riccati equations to high accuracy.
We discuss solvers for Sylvester, Lyapunov, and Stein equations
that are available in the SLICOT Library (Subroutine
Library In COntrol Theory). These solvers offer improved
efficiency, reliability, and functionality compared to corresponding
solvers in other computer-aided control system design
packages. The performance of the SLICOT solvers is
compared with the corresponding MATLAB solvers.
We investigate numerical methods for passive
model reduction of linear dynamical systems. This
is an important task in circuit simulation when
modeling parasitic effects of interconnect. We will
show how positive real balancing, based on balancing
the solutions of two algebraic Riccati equations,
can be used for passive model reduction
of large-scale systems on parallel computers. Numerical
experiments demonstrate the performance
of the parallel algorithms using several examples
from circuit simulation.
This article describes Fortran 77 subroutines for computing eigenvalues and invariant subspaces
of Hamiltonian and skew-Hamiltonian matrices. The implemented algorithms are based on orthogonal
symplectic decompositions, implying numerical backward stability as well as symmetry
preservation for the computed eigenvalues. These algorithms are supplemented with balancing and
block algorithms, which can lead to considerable accuracy and performance improvements. As a
by-product, an efficient implementation for computing symplectic QR decompositions is provided.
We demonstrate the usefulness of the subroutines for several, practically relevant examples.
We discuss solvers for Sylvester, Lyapunov, and Stein equations that are available in the SLICOT
Library (Subroutine Library In COntrol Theory). These solvers offer improved efficiency, reliability,
and functionality compared to corresponding solvers in other computer-aided control system
design packages. The performance of the SLICOT solvers is compared with the corresponding
Matlab solvers. This note can also serve as a guide to the SLICOT and SLICOT-based Matlab
solvers for Linear Matrix Equations.