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springer-1989-cover

SPRINGER-VERLAG, Lecture Notes in Mathematics 1387, 1989

ITERATIVE METHODS FOR SIMULTANEOUS INCLUSION OF POLYNOMIAL ZEROS

Miodrag S. Petković, University of Niš, Serbia

ISBN: 3-540-51486-6

263 pages

The simultaneous inclusion of polynomial complex zeros is a crucial problem in numerical analysis. Rapidly converging algorithms are presented in these monograph, including convergence analysis in terms of circular regions, and in complex arithmetic. Parallel circular iterations, where the approximations to the zeros have the form of circular regions containing these zeros, are efficient because they also provide error estimates. There are at present no book publications on this topic and one of the aims of this book is to collect most of the algorithms produced in the last 15 years.
To decrease the high computational cost of interval methods, several effective iterative processes for the simultaneous inclusion of polynomial zeros which combine the efficiency of ordinary floating-point arithmetic with the accuracy control that may be obtained by the interval methods, are set down, and their computational efficiency is described. The rate of these methods is of interest in designing a package for the simultaneous approximation of polynomial zeros, where automatic procedure selection is desired.
The book is both a text and a reference source for mathematicians, engineers, physicists and a computer scientists who are interested in new developments and applications, but the material is also accessible to anyone with graduate level mathematical background and some knowledge of basic computational complex analysis and programming.

springer-2008-cover

SPRINGER, Lecture Notes in Mathematics 1933, 2008

POINT ESTIMATION OF ROOT FINDING METHODS
Miodrag S. Petković, University of Niš, Serbia

ISBN: 978-3-540-77850-9

210 pages

This book sets out to state computationally verifiable initial conditions for predicting the immediate appearance of the guaranteed and fast convergence of iterative root finding methods. Attention is paid to iterative methods for simultaneous determination of polynomial zeros in the spirit of Smale's point estimation theory, introduced in 1986. Some basic concept and Smale's theory for Newton's method, together with its modifications and higher-order methods, are presented in the first two chapters. The remaining chapters contain the recent author's results on initial conditions guaranteing convergence of a wide class of iterative methods for solving algebraic equations. These conditions are of practical interest since they depend only on available data, the information of a function whose zeros are sought and initial approximations. The convergence approach presented can be applied in designing a package for the simultaneous approximations of polynomial zeros.

wiley-cover

WILEY-VCH, Mathematical Research, Vol. 105, 1998

COMPLEX INTERVAL ARITHMETIC AND ITS APPLICATIONS

Miodrag S. Petković, University of Niš, Serbia
Ljiljana D. Petković, University of Niš, Serbia

ISBN: 3-527-40134-2

284 pages

The aim of this monograph is to present formulas and methods developed using complex interval arithmetic. While most of numerical methods described in the literature deal with real intervals and real vectors, there is no systematic study of methods in complex interval arithmetic. The book fills this gap. Several main subjects are considered: outer estimates for the range of complex functions, especially complex centered forms, the best approximations of elementary complex functions by disks, iterative methods for the inclusion of polynomial zeros including their implementation on parallel computers, the analysis of numerical stability of iterative methods by using complex interval arithmetic and numerical computation of curvilinear integrals with error bounds. Mainly new methods are presented developed over the last years, including a lot of very recent results by the authors some of which have not been published before.
The book is both a text and a reference source for mathematicians, engineers, physicists, and computer scientists who are interested in new developments and applications, but material is also accessible to anyone with graduate level mathematical background and some knowledge of basic computational complex analysis and programming.