65 research outputs found
A Thesis Submitted in Partial Fulfilmen of the Requirements for the Degree of Doctor of Philosophy in Mathematics
The Cayley transform method is a Newton-like method for solving in-
verse eigenvalue problems. If the problem is large, one can solve the Ja-
cobian equation by iterative methods. However, iterative methods usually
oversolve the problem in the sense that they require far more (inner) it-
erations than is required for the convergence of the Newton (outer) itera-
tions. In this paper, we develop an inexact version of the Cayley transform
method. Our method can reduce the oversolving problem and improves the
e±ciency with respect to the exact version. We show that the convergence
rate of our method is superlinear and that a good tradeo® between the
required inner and outer iterations can be obtained
Structured Eigenvalue Problems
Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
- …