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 differential 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.

We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n x 2n symplectic matrix can be reduced to this condensed form which contains 8n x 4 nonzero entries and is determined by 4n x 1 parameters. The symplectic eigenvalue problem can be solved using the SR algorithm based on this condensed form. The SR algorithm preserves this form and can be modified to work only with the 4n x 1 parameters instead of the 4n² matrix elements. The reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to Hessenberg form. A Lanczos-like algorithm for reducing a symplectic matrix to butterfly form is also presented.

Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (log p)&sup2:. The paper consists...

SR and SZ algorithms for the symplectic (generalized) eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed. A 2n x 2n symplectic butterfly matrix has 8n - 4 (generically) nonzero entries, which are determined by 4n - 1 parameters. While the SR algorithm operates directly on the matrix entries, the SZ algorithm works with the 4n - 1 parameters. The algorithms are made more compact and efficient by using Laurent polynomials, instead of standard polynomials, to drive the iterations.

Large sparse Hamiltonian eigenvalue problems arise in a variety of contexts. These problems can be attacked directly, or they can rst be transformed to problems having some related structure, such as symplectic or skew Hamiltonian. In the interest of eciency, stability, and accuracy, such problems should be solved by methods that preserve the structure, whether it be Hamiltonian, skew Hamiltonian, or symplectic.

This paper deals with the numerical solution of large scale polynomial or rational eigenvalue problems with Hamiltonian or symplectic symmetry in the spectrum. Applications where such problems arise are introduced briey. It is shown how these problems may be formulated as linear generalized eigenvalue problems that have either symmetric/skew symmetric, skew Hamiltonian/Hamiltonian or symplectic pencils. The presented numerical methods are designed to preserve these structures.

Abstract. We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a factor √ 2 of the unstructured backward error. This paper collects, unifies, and extends existing work on this subject.

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.

Introduction Quadratic matrix equations in their general form 0 = Q+A T PB + C T PD E T PGPF (1) where P 2 IR nm is unknown and A; B; C; D;E;F; G; Q are of appropriate dimensions, arise in systems and control theory, order reduction, structural mechanics and vibration problems, economical dynamics, ltering, decoupling of parabolic systems, etc.. Here we will consider only a special case of (1)|the symmetric algebraic Riccati equation (ARE) 0 = R(P ) := Q+A T P + PA PGP (2) resulting from (1) by setting B = C = E = F = I n , D = A and assumi