New Structured Matrix Methods for Real and Complex Polynomial Root-finding

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular numerical approximation of the real roots of a polynomial. Our analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page

Spectra of indefinite linear operator pencils

In recent years, there has been a rapid growth of interest in spectral properties of non-self-adjoint operators and operator pencils. This thesis is concerned with indefinite self-adjoint linear pencils which lead to a special class of non-self-adjoint spectral problems. These problems are not well understood, and, in general, many sign-indefinite problems which are trivial to state require some highly non-trivial analysis. We look at indefinite linear pencil problems from the perspective of a two parameter eigenvalue problem. We derive localisation results for real eigenvalues and present several examples. We also use different approaches to obtain estimates of non-real eigenvalues, supported by a large number of numerical experiments. Additionally, these experiments lead to various open questions and conjectures

Well-Posed Boundary Element Formulations in Electromagnetics.

New well-conditioned frequency and time domain integral equations providing rapidly convergent solutions to electromagnetic radiation and scattering problems have been obtained. By leveraging (i) novel integral identities, (ii) frequency and time domain Calderon formulas, and (iii) novel wavelet bases, the proposed constructs permit the resonant-free and accurate analysis of ultrabroadband and multiscale electromagnetic phenomena that hitherto resisted analysis by all known simulation methodologies. The new formulations have been applied successfully to the analysis of real-life problems, including the characterization of integrated circuit interconnects, the design of broadband antennas, and the simulation of electromagnetic interactions with spacecraft; effciency gains ranging between twenty and fifty with respect to previously available solvers, were obtained. This, together with the high degree of integrability of the presented techniques into the existing technology, implies they could rapidly impact the state of the art in boundary element solvers in use in academia and industry. The ideas presented here have applications that go beyond the scope of this thesis; future studies will analyze their extension to volume/surface integral equations, to high order and singular basis functions, and to their hybridization with finite element technology.Ph.D.Electrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58409/1/fandri_1.pd

Numerical Methods I.

A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült

New bounds for roots of polynomials based on Fiedler companion matrices

Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots lambda of p(z). Recently, M. Fiedler (2003) [9] has introduced a new family of companion matrices of p(z) that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on vertical bar lambda vertical bar. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1- and infinity-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on vertical bar lambda vertical bar is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials.This work has been supported by the Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542