Iterative Methods for Simultaneous Inclusion of Polynomial Zeros
The simultaneous inclusion of polynomial complex zeros is a crucial problem in numerical analysis. Rapidly converging algorithms are presented in these notes, including convergence analysis in terms of circular regions, and in complex arithmetic. Parallel
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1387 Miodrag Petkovi6
Iterative Methods for Simultaneous Inclusion of Polynomial Zeros
Springer-Verlag Berlin Heidelberg NewYork LondonParis Tokyo Hong Kong
Author Miodrag Petkovi6 University of Nis Faculty of Electronic Engineering P.O. Box 73, 18000 Nis, Yugoslavia
Mathematics Subject Classification (1980): Primary: 65H05 Secondary: 65G05, 65G 10, 30C 15 ISBN 3-540-51485-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51485-6 Springer-Verlag New York Berlin Heidelberg
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To my sons, Ivan and Vladimir
PREFACE Galois' famous theorem states that a general direct method in terms of explicit formulas exists only for the polynomial equations degree of which is less than five. Because of that, to find the zeros of polynomials of higher degree one must apply some of numerical methods; moreover, such methods are already used for polynomials of degree three or four since the corresponding explicit formulas are remarkably complicated. A great importance of the problem of determining polynomial zeros in the theory and practice (e.g., in the theory of control systems, stability of systems, nonlinear circuits, analysis of transfer functions, various mathematical models, differential and difference equations, eigenvalues problems and other disciplines) has led to the development of a great number of numerical methods in this field. The list of contributors contains many of the most famous names of mathematical history. These numerical methods, which generally take the form of an iterative procedure, have become practically applicable together with the rapid development of digital computers some thirty years ago. In connection with any implementation of the numerical methods on a computer, it is important to note that the selection of zero-finding routine may depend heavily on extrarnathematical considerations such as speed and memory of the computing equipment and trustworthiness of the result. Anyone using a computing has surely inquired about the effect of rounding error and, eventually, propagated 'error due to uncertain values of polynomial coefficients. The computed solution of a polynomial equation is only an approximation of the true solution, since there are errors originating from discretization or truncation and from rounding. In connection with this effect we quote Henrici's arg
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