This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.

. In this paper we derive a unitary eigendecomposition for a sequence of matrices which we call the periodic Schur decomposition. We prove its existence and discuss its application to the solution of periodic difference equations arising in control. We show how the classical QR algorithm can be extended to provide a stable algorithm for computing this generalized decomposition. We apply the decomposition also to cyclic matrices and two point boundary value problems. Key words. Numerical algorithms, linear algebra, periodic systems, K-cyclic matrices, two-point boundary value problems 1 Introduction In the study of time-varying control systems in (generalized) state space form : ( E k \Delta z k+1 = F k \Delta z k +G k \Delta u k y k = H k \Delta z k + J k \Delta u k (1) the periodic coefficients case has always been considered the simplest extension of the time-invariant case. Here the coefficients satisfy, for some K ? 0 the periodicity conditions E k = E k+K , F k = F k+K , G k...

We discuss the numerical solution of the Lyapunov equation

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order p ", where " is the machine precision, the new method computes the eigenvalues to full possible accuracy. Keywords. eigenvalue problem, Hamiltonian pencil (matrix), symplectic pencil (matrix), skew-Hamiltonian matrix AMS subject classification. 65F15 1 Introduction The eigenproblem for Hamiltonian and symplectic matrices has received a lot of attention in the last 25 years, since the landmark papers of Laub [13] and Paige/Van Loan [20]. The reason for this is the importance of this problem in many applications in c...

Let U \Gamma V be an n \Theta n pencil with unitary matrices U and V . An algorithm is presented which reduces U and V simultaneously to unitary block diagonal matrices Go = Q H UP and Ge = Q H V P with block size at most two. It is an O(n 3 ) process using Householder eliminations and it is backward stable. In the special case V = I the block diagonal matrices Go ; G H e can be normalized such that their entries are just the Schur parameters of the Hessenberg condensed form of U . We call Go \Gamma Ge a Schur parameter pencil. It can also be derived from U; V by a Lanczos--like process. For the solution of the eigenvalue problem for Go \Gamma Ge a QR--type algorithm can be developed based on this unitary reduction of a pencil U \Gamma V to a Schur parameter pencil. The condensed form is preserved throughout the process. Each iteration step needs only O(n) operations. This method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the QR meth...

We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem (GEP). For the special case of the quadratic eigenvalue problem (QEP), we show that solving the QEP by applying the QZ algorithm to a corresponding GEP can be backward unstable. The QEP can be reformulated as a GEP in many ways. We investigate the sensitivity of a given eigenvalue to perturbations in each of the GEP formulations and identify which formulations are to be preferred for large and small eigenvalues, respectively. Key words. Polynomial eigenvalue problem, quadratic eigenvalue problem, generalized eigenvalue problem, backward error, condition number. 1 Introduction We are concerned with backward error analysis and conditioning for the nonlinear eigenvalue problem P ()x = 0; (1.1) where P () is a matrix whose elements are polynomials in a scalar . We write P in the form P () =...

Abstract. We derive a stable technique, based upon matrix pencils, for the reconstruction of (or approximation by) polygonal shapes from moments. We point out that this problem can be considered the dual of 2 − D numerical quadrature over polygonal domains. An analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points. Finally, an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field. Key words. gravimetry

. In the year 2000 the dominant method for solving matrix eigenvalue problems is still the QR algorithm. This paper discusses the family of GR algorithms, with emphasis on the QR algorithm. Included are historical remarks, an outline of what GR algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the QR algorithm. Now that we know how to parallelize it, the QR algorithm seems likely to retain its dominance for many years to come. 1. Introduction Since the early 1960's the standard algorithms for calculating the eigenvalues and (optionally) eigenvectors of "small" matrices have been the QR algorithm [29] and its variants. This is still the case in the year 2000 and is likely to remain so for many years to come. For us a small matrix is one that can be stored in the conventional way in a computer's main memory and whose complete eigenstructure can be calculated in a matter of minutes without exploiting whatever sparsity the matrix m...

Abstract. Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear structure in the coefficient matrices (for example, Hermitian, Toeplitz, Hamiltonian, or band structure) is preserved by the perturbations.

Kronecker-like forms of a system pencil are useful in solving many computational problems encountered in the analysis and synthesis of linear systems. The reduction of system pencils to various Kronecker-like forms can be performed by structure preserving 0(n complexity numerically stable algorithms. The presented algorithms form the basis of a modular collection of LAPACK compatible FORTRAN 77 subroutines to perform the reduction of a system pencil to several Kronecker-like forms.