On the Solution of Block Hessenberg Systems

This paper describes a divide-and-conquer strategy for solving block Hessenberg systems. For dense matrices the method is a little more efficient than Gaussian elimination; however, because it works almost entirely with the original blocks, it is be much more efficient for sparse matrices or matrices whose blocks can be generated on the fly. For Toeplitz matrices, the algorithm can be combined with the fast Fourier transform to give a new superfast algorithm. (Also cross-referenced as UMIACS-TR-92-109

Numerical Methods for M/G/1 Type Queues

Queues of M/G/1 type give rise to infinite embedded Markov chains whose transition matrices are upper block Hessenberg. The traditional algorithms for solving these queues have involved the computation of an intermediate matrix G. Recently a recursive descent method for solving block Hessenberg systems has been proposed. In this paper we explore the interrelations of the two methods. (Also cross-referenced as UMIACS-TR-95-37

Rounding Errors in Solving Block Hessenberg Systems

A rounding error analysis is presented for a divide-and-conquer algorithm to solve linear systems with block Hessenberg matrices. Conditions are derived under which the algorithm computes a backward stable solution. The algorithm is shown to be stable for diagonally dominant matrices and for M-matrices. (Also cross-referenced as UMIACS-TR-94-105

Implementing an Algorithm for Solving Block Hessenberg Systems

This paper describes the implementation of a recursive descent method for solving block Hessenberg systems. Although the algorithm is conceptually simple, its implementation in C (a natural choice of language given the recursive nature of the algorithm and its data) is nontrivial. Particularly important is the balance between ease of use, computational efficiency, and flexibility. (Also cross-referenced as UMIACS-TR-94-70

A survey on recursive algorithms for unbalanced banded Toeplitz systems: computational issues

Several direct recursive algorithms for the solution of band Toeplitz systems are considered. All the methods exploit the displacement rank properties, which allow a large reduction of computational efforts and storage requirements. Some algorithms make use of the Sherman-Morrison- Woodbury formula and result to be particularly suitable for the case of unbalanced bandwidths. The computational costs of the algorithms under consideration are compared both in a theoretical and practical setting. Some stability issues are discussed as well

Preconditioning for standard and two-sided Krylov subspace methods

This thesis is concerned with the solution of large nonsymmetric sparse linear systems. The main focus is on iterative solution methods and preconditioning. Assuming the linear system has a special structure, a minimal residual method called TSMRES, based on a generalization of a Krylov subspace, is presented and its convergence properties studied. In numerical experiments it is shown that there are cases where the convergence speed of TSMRES is faster than that of GMRES and vice versa. The numerical implementation of TSMRES is studied and a new numerically stable formulation is presented. In addition it is shown that preconditioning general linear systems for TSMRES by splittings is feasible in some cases. The direct solution of sparse linear systems of the Hessenberg type is also studied. Finally, a new approach to compute a factorized approximate inverse of a matrix suitable for preconditioning is presented