## On BLAS Level-3 Implementations of Common Solvers for (Quasi-) Triangular Generalized Lyapunov Equations

### Martin Köhler, Jens Saak

SLICOT Working Note 2014-1: September 2014.

The solutions of Lyapunov and generalized Lyapunov equations are a key player in many applications in systems and control theory. Their stable numerical computation, when the full solution is sought, is considered solved since the seminal work of Bartels and Stewart [1]. A number of variants of their algorithm have been proposed, but none of them goes beyond BLAS level-2 style implementation. On modern computers, however, the formulation of BLAS level-3 type implementations is crucial to enable optimal usage of cache hierarchies and modern block scheduling methods based on directed acyclic graphs describing the interdependence of single block computations. Our contribution closes this gap by a transformation of the aforementioned level-2 variants to level-3 versions and a comparison on a standard multicore machine.

## MB04BV - A FORTRAN 77 Subroutine to Compute the Eigenvectors Associated to the Purely Imaginary Eigenvalues of Skew-Hamiltonian/Hamiltonian Matrix Pencils 1

### Peihong Jiang, Matthias Voigt

SLICOT Working Note 2013-3: September 2013.

We implement a structure-preserving numerical algorithm for extracting the eigenvectors asso ciated to the purely imaginary eigenvalues of skew-Hamiltonian/Hamiltonian matrix p encils. We compare the new algorithm with the QZ algorithm using random examples with different difficulty. The results show that the new algorithm is signficantly faster, more robust, and more accurate, esp ecially for hard examples.

## FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems - Part II: Implementation and Numerical Results

### Peter Benner, Vasile Sima, and Matthias Voigt

SLICOT Working Note 2013-2: August 2013.

Skew-Hamiltonian/Hamiltonian matrix pencils S - lambda H appear in many applications, including linear quadratic optimal control problems, H1-optimization, certain multi-body systems and many other areas in applied mathematics, physics, and chemistry. In these applications it is necessary to compute certain eigenvalues and/or corresponding deflating subspaces of these matrix pencils. Recently developed methods exploit and preserve the skew-Hamiltonian/Hamiltonian structure and hence increase reliability, accuracy and performance of the computations. In this paper we describe the implementation in SLICOT of the algorithms described in Part I of this work (see SLICOT Working Note 2013-1) and address various details. Furthermore, we perform numerical tests using real-world examples to demonstrate the superiority of the new algorithms compared to standard methods.

## FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems - Part I: Algorithms and Applications

### Peter Benner, Vasile Sima, and Matthias Voigt

SLICOT Working Note 2013-1: August 2013.

Skew-Hamiltonian/Hamiltonian matrix pencils S - lambda H appear in many applications, including linear quadratic optimal control problems, H1-optimization, certain multi-body systems and many other areas in applied mathematics, physics, and chemistry. In these applications it is necessary to compute certain eigenvalues and/or corresponding deflating subspaces of these matrix pencils. Recently developed methods exploit and preserve the skew-Hamiltonian/Hamiltonian structure and hence increase reliability, accuracy and performance of the computations. In this paper we describe the corresponding algorithms whose implementations have been included in the Subroutine Library in Control Theory (SLICOT). Furthermore we address some of their applications. We describe variants for real and complex problems with versions for factored and unfactored matrices S.

## How a Numerical Rank Revealing Instability Affects Computer Aided Control System Design

### Zvonimir Bujanovic, and Zlatko Drmac

SLICOT Working Note 2010-1: January 2010.

Since numerical libraries are used in engineering design in a variety of industrial applications, it is important that their numerical reliability is the top priority of both the developers of numerical algorithms and users from industry. Following that principle, we have examined a state of the art control library (case study: SLICOT) with respect to use of rank revealing subroutines in computing various canonical decompositions of linear time invariant systems. This issue seems to be critical, with potential for causing numerical catastrophes, because the deployed rank revealing code is prone to severe instabilities, causing completely wrongly computed parameters of systems under analysis. We analyze the SLICOT library in detail and propose modifications of critical parts of the code, based on our recent work published in the ACM Trans. Math. Softw. 35, 2008, where we analyze and solve the problem. The proposed modifications increase numerical reliability of all of the sixty affected subroutines. We recommend that the developers of other control theory numerical libraries examine their codes with respect to the issue discussed in this paper.