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STRUCTURE-PRESERVING METHODS FOR COMPUTING EIGENPAIRS OF LARGE SPARSE SKEW-HAMILTONIAN/HAMILTONIAN PENCILS VOLKER MEHRMANN AND DAVID WATKINSy

Abstract. We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linear-quadratic control problem for partial dierential equations. We develop a structure-preserving skew-Hamiltonian, isotropic, implicitly-restarted shift-and-invert Arnoldi algorithm (SHIRA). Several numerical examples demonstrate the superiority of SHIRA over a competing unstructured method.

Keywords: Skew-Hamiltonian/Hamiltonian pencil, generalized eigenvalue problem, quadratic eigenvalue problem, implicitly restarted Arnoldi method, Lamé equations, classical mechanics, linear quadratic control, algebraic Riccati equation 1. Introduction. In this paper we study the numerical computation of a small number of eigenvalues (and the associated eigenvectors) of large-scale generalized eigenvalue problems having a certain structure that arises frequently in applications. A 2n  2n real matrix pencil (1.1)









F1 G1 ? F2 G2 ; N ? H =  H H2 ?F2T 1 F1T

where G1 = ?GT1 , H1 = ?H1T , G2 = GT2 , and H2 = H2T , is called a skewHamiltonian/Hamiltonian pencil or, more briey, an SHH pencil. Throughout this paper we will assume that N and H are large and sparse. Several examples of applications that give rise to large, sparse generalized eigenvalue problems with SHH pencils are given in Section 2. The reason for the terminology is simply this: Matrices the form of N and H in (1.1) are called skew Hamiltonian and Hamiltonian, respectively. If (1.2)



0

J = ?I n



In ; 0

where In is the nn identity matrix, then skew-Hamiltonian matrices satisfy (N J )T = ?(N J ) and Hamiltonian matrices satisfy (HJ )T = HJ . An SHH pencil and in particular its spectrum have considerable structure. The  ?; ?  ), or in real or purely imaginary pairs eigenvalues occur in quadruplets (; ; (; ?) [24, 25, 26]. The objective of this paper is to develop algorithms that preserve  Fakultät für Mathematik, Technische Universität Chemnitz, D09107 Chemnitz, Fed. Rep. Germany. This work has been supported by Deutsche Forschungsgemeinschaft within SFB393 Numerische Simulation auf massiv parallelen Rechnern. y Dept. of Mathematics, Washington State Univerity, Pullman, WA, 99164-3113, USA currently: Fakultät für Mathematik, Technische Universität Chemnitz, D09107 Chemnitz, Fed. Rep. Germany. This work has been supported by Deutsche Forschungsgemeinschaft within project A8 of SFB393 Numerische Simulation auf massiv parallelen Rechnern and by Deutscher Akademischer Austauschdienst within the program HSP III, Förderung ausländischer Gastdozenten. 1

and exploit this structure. The payos are more ecient and more accurate algorithms. In some cases preservation of the structure is crucial to the stable solution of a problem. Typical applications require the few eigenvalues that are smallest in magnitude or closest to the imaginary axis. To achieve this we must apply transformations that have the eect of shifting the desired eigenvalues to the periphery of the spectrum, so that they can be computed eciently by Krylov subspace methods, e.g. Arnoldi or Lanczos, possibly with implicit restarts. This is a standard procedure in methods for large sparse eigenvalue problems, [33, 39]. What is special to this paper is that our transformations and our Krylov subspace methods respect the structure of the problem. Our approach requires that the skew-Hamiltonian matrix N be presented as a product in the special form (1.3) N = Z1 Z2 where Z2T J = J Z1 ; with Z1 and Z2 sparse. This allows us to transform the pencil N ? H to a standard eigenvalue problem I ? W = I ? Z1?1 HZ2?1 , in which the matrix W = Z1?1 HZ2?1 turns out to be Hamiltonian. This procedure is analogous to the technique by which a symmetric pencil A ? B , with B positive denite, is transformed to a standard eigenvalue problem using a Cholesky decomposition B = LLT . Because of the symmetry of the decomposition, the matrix L?1 AL?T inherits the symmetry of A. In the current context, if we introduce the skew-symmetric inner product hx; yiJ = yT Jx, we nd that a matrix is Hamiltonian if and only if it is skew symmetric with respect to this J -inner product, i.e. hHx; yiJ = ?hx; HyiJ for all x; y 2 R2n . The relationship between Z1 and Z2 given in (1.3) implies that Z2 is (plus or minus) the adjoint of Z1 with respect to the J inner product, i.e. hZ1 x; yiJ = hx; Z2 yiJ for all x; y 2 R2n . Thus the decomposition N?=1 Z1?Z12 is a symmetric, Cholesky-like decomposition of N . Consequently W = Z1 HZ2 inherits the J -skew symmetry of H; that is, W is Hamiltonian. We discuss two approaches that make dierent transformations of the Hamiltonian operator W . The rst maps W to a skew-Hamiltonian operator, from which the eigenvalues can be extracted by an implicitly restarted Arnoldi method that has been modied to preserve the structure. For this approach we provide numerical results demonstrating its eectiveness. The second approach maps W to a symplectic operator by a generalized Cayley transform. The desired eigenvalues can then be extracted by an implicitly restarted symplectic Lanczos method. We only outline this approach and discuss its advantages and disadvantages. 2. Applications. The need to solve SHH generalized eigenvalue problems arises in many applications. The best-known example is the linear quadratic optimal control problem for descriptor systems, where the pencil typically has the particular form 







T  E0 E0T ? CAT C ??BB ; T A with B of size n  m, C of size p  n and m