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The problem of solving nonlinear equations and systems of equations
ranks among the most signi?cant in the theory and practice, not
only of applied
mathematicsbutalsoofmanybranchesofengineeringsciences, physics, c-
puter science, astronomy, ?nance, and so on. A glance at the
bibliography and the list of great mathematicians who have worked
on this topic points to a high level of contemporary interest.
Although the rapid development of digital computers led to the
e?ective implementation of many numerical methods, in practical
realization, it is necessary to solve various problems such as
computational e?ciency based on the total central processor unit
time, the construction of iterative methods which possess a fast
convergence in the presence of multiplicity (or clusters) of a
desired solution, the control of rounding errors, information about
error bounds of obtained approximate solution, stating
computationally veri?able initial conditions that ensure a safe
convergence, etc. It is the solution of these challenging problems
that was the principal motivation for the present study. In this
book, we are mainly concerned with the statement and study of
initial conditions that provide the guaranteed convergence of an
iterative method for solving equations of the form f(z) = 0. The
traditional approach to this problem is mainly based on asymptotic
convergence analysis using some strong hypotheses on
di?erentiability and derivative bounds in a rather wide domai
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 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 mathematicans, engineers, physicists and 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.
Splendid collection of 99 puzzles, all built around the chessboard. Arithmetical and probability problems, chessboard recreations, geometrical puzzles, mathematical amusements and games, more. Many standard problems with new, interesting twists. You don't need to be a chess player, just know the rules of the game. Solutions.
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