Future Generation Computer Systems 19 (2003) 1123–1124
Guest editorial
Structural dynamical systems in linear algebra and control: computational aspects
This special issue collects 12 recent and original research papers on theoretical and computational aspects of structural dynamical systems in linear algebra and control. It has been motived by the workshop “Structural Dynamical Systems in Linear Algebra and Control”, taken place at Monopoli, Bari, Italy, July 1–4, 2001, although some papers not presented at that workshop have been added. The aim of this issue is to show how problems arising in different fields have a common denominator: structural dynamical system. By structure of the dynamical system we mean the belonging of its solution to the orthogonal, the symplectic or to a general linear matrix group (manifold), or to different set of matrices such as that of symmetric positive definite matrices, etc. Structural dynamical systems describe the dynamics of several physical models (see [6]), they are useful for solving inverse and control problems (see [5,9]), they also appear in linear algebra problems as decomposition problems [4]. A fundamental aspect of structural dynamical systems study is their integration by suitable numerical methods being able to preserve the underlying structure (see for instances [1–3,7,8,10]). This volume tries to summarize all recent theoretical and numerical developments on structural dynamical systems and to stimulate the reader to future studies. The first five papers consider mathematical problems in linear algebra or control theory by a theoretical point of view, suggesting continuous dynamical system approaches for their solution. In their work, Dieci and Papini describe continuation methods of block eigendecomposition of matrix valued functions. They provide new theoretical results on the reduction to Hessenberg and bidiagonal forms,
and introduce and implement algorithms to continue eigendecompositions. Chu, Diele and Sgura consider the inverse eigenvalue problem of matrix completion with prescribed eigenvalues. They investigate some continuation techniques by recasting the problem as an optimization one using a cost function measuring the distance between the manifold of the isospectral matrices to a given matrix with prescribed eigenvalues and the set of the affine matrices with prescribed entries. Przybylska shows in her paper a general connection between matrix differential equations of Lax type and FG factorization algorithm for a matrix, generalizing some results known for the QR algorithm. Trendafilov considers the Procrustes problem replacing the usual least square objective function by one based on a smooth approximation of the 1 matrix norm and solve it by the continuous projected gradient method. In their article, Del Buono and Lopez propose a dynamical system approach for computing Euclidean diagonal norm balanced realizations in control theory, showing how the limiting solution of a differential flow is a possible way to solve such a linear algebra problem. As observed before, once obtained a continuous dynamical systems an important issue is their numerical solution by suitable structure preserving algorithms. The following three papers presents some results on structural numerical procedures. McLachlan, Permutter and Quispel propose numerical integrators, which preserve the foliation of the considered system. Foliation systems are particular systems which preserve some (possibly singular)
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Guest editorial / Future Generation Computer Systems 19 (2003) 1123–1124
foliation of phase space, such as systems with integrals, systems with continuous symmetries and skew product systems. In the context of numerical methods for solving ODEs evolving on Lie groups, Munthe-Kaas and Krogstad present a characterization of the algebraic structure underlying non-commutative Lie–Butcher series in terms of balanced Lyndon words over a binary alphabet. In their paper, Elia and Lopez study numerical methods preserving the exponential monotonicity of the solutions of ordinary differential equations. They show that Gauss Legendre Runge–Kutta methods do not preserve this qualitative feature while certain Lie group methods, some projection and splitting techniques do it. Finally, the four papers of the issue presents results and overviews on dynamical systems in control theory. In Varga’s paper a general, numerically reliable approach to solve the pole and eigenstructure assignment problem for descriptor systems is proposed. For this purpose, a minimum norm robust pole assignment problem is formulated and solved as an unconstrained minimization problem for suitably chosen cost function. By using a generalized Sylvester equation based parameterization, an explicit expression of the gradient of the cost function is derived to allow the efficient solution of the minimization problem by using gradient techniques. Benner, Kressner and Mehrmann present a review paper on current and future directions in the developments of numerical methods and numerical software for control problems where higher accuracy, robustness of the methods and the need for methods to solve large scale problems are frequent demands to satisfy. They show that, to address these demands it is essential to preserve any underlying physical structure of the problem. Datta presents in his paper a brief but state-of-the-art survey of some existing Krylov subspace methods for large-scale matrix problems in control. Some research problems are also suggested.
Hüper and Van Dooren present new methods for refining estimates of invariant subspaces of a non-symmetric matrix. Global analysis is used to show local quadratic convergence of the proposed methods under mild conditions on the spectrum of the matrix. Before concluding, I wish to thank the authors for their contributions and cooperation in preparing this special issue, and all the referees for their help in the review process. I am also grateful to the publisher of Future Generation Computer Systems Journal and to Prof. P.M.A. Sloot, editor-in-chief on Computational Science, for having offered me the opportunity to edit this issue. References [1] P.E. Crouch, R. Grossman, Numerical integration of ordinary differential equation on manifolds, J. Nonlinear Sci. 3 (1993) 1–33. [2] F. Diele, L. Lopez, R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math. 8 (1998) 317–334. [3] L. Dieci, D. Russell, E.S. Van Vleck, Unitary integration and applications to continuous orthonormalization techniques, SIAM J. Numer. Anal. 31 (1994) 261–281. [4] L. Dieci, T. Eirola, On smooth decomposition of matrices, SIAM J. Matrix Anal. Appl. 20 (3) (1999) 800–819. [5] M.T. Chu, G. Golub, Structural inverse eigenvalue problems, Acta Numer. (2002) 1–71. [6] H. Flaschka, The Toda lattice I, Phys. Rev. B 9 (1974) 1924– 1925. [7] H. Munthe-Kaas, Runge–Kutta methods on Lie groups, BIT 38 (1) (1998) 92–111. [8] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, 2001. [9] U. Helmke, J.B. Moore, Optimization and Dynamical Systems, Springer, Berlin, 1994. [10] A. Iserles, H. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie group methods, Acta Numer. 9 (2000) 215–365.
Luciano Lopez Dipartimento di Matematica Università degli Studi di Bari Via Orabona 4, I-70125 Bari, Italy E-mail address:
[email protected] (L. Lopez)
