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.

Balancing a matrix by a simple and accurate similarity transformation can improve the performance 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.

Abstract — The numerical solution of algebraic Riccati equations is a central issue in computer-aided control systems design. It is the key step in many computational methods for model reduction, filtering, and controller design for linear control systems. We discuss recent advances in the solvers for continuous-time and discrete-time algebraic Riccati equations available in the SLICOT Library (Subroutine Library In COntrol Theory) and compare their performance with the corresponding solvers in the Matlab toolboxes.

A set of utilities for generating web computing environments related to mathematical and engineering library software is presented.

We investigate the numerical solution of algebraic Bernoulli equations via the Newton iteration for the matrix sign function. Bernoulli equations are nonlinear matrix equations arising in control and systems theory in the context of stabilization of linear systems, coprime factorization of rational matrix-valued functions, as well as model reduction. The algorithm proposed here is easily parallelizable and thus provides an efficient tool to solve large-scale problems. We report the parallel performance and scalability of our parallel implementations on a cluster of Intel Xeon processors. Key words. Bernoulli equation; linear and nonlinear matrix equations; matrix sign function; control and systems theory; parallel computers.

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The numerical treatment of linear-quadratic regulator problems on finite time horizons for parabolic partial differential equations requires the solution of large-scale differential Riccati equations (DREs). Typically the coefficient matrices of the resulting DRE have a given structure (e.g. sparse, symmetric or low rank). Here we discuss numerical methods for solving DREs capable of exploiting this structure. These methods are based on a matrix-valued implementation of the BDF methods. The crucial question of suitable stepsize and order selection strategies is also addressed. 1

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This paper describes a numerical method for extracting the stable right deflating subspace of a matrix pencil Z − λY 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. 1

SLICOT is a comprehensive numerical software package for control systems analysis and design. While based on highly performant Fortran routines, Matlab and Scilab interfaces provide convenient access for users. In this survey, we summarize the functionality contained in the three SLICOT toolboxes for (i) basic tasks in systems and control, (ii) system identification, and (iii) model reduction. Several examples illustrate the use of these toolboxes for addressing frequent computational tasks.