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Suitable as supplemental reading in courses in differential and integral calculus, numerical analysis, approximation theory and computer-aided geometric design. Relations of mathematical objects to each other are expressed by transformations. The repeated application of a transformation over and over again, i.e., its iteration leads to solution of equations, as in Newton's method for finding roots, or Picard's method for solving differential equations. This book studies a treasure trove of iterations, in number theory, analysis and geometry, and applied them to various problems, many of them taken from international and national Mathematical Olympiad competitions. Among topics treated are classical and not so classical inequalities, Sharkovskii's theorem, interpolation, Bernstein polynomials, BŽzier curves and surfaces, and splines. Most of the book requires only high school mathematics; the last part requires elementary calculus. This book would be an excellent supplement to courses in calculus, differential equations,, numerical analysis, approximation theory and computer-aided geometric design.

Iterative methods (Mathematics) --- Transformations (Mathematics) --- Algorithms --- Differential invariants --- Geometry, Differential --- Iteration (Mathematics) --- Numerical analysis --- Iterative methods (Mathematics). --- Transformations (Mathematics).

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With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

Iterative methods (Mathematics) --- Combinatorial optimization --- Itération (Mathématiques) --- Optimisation combinatoire --- Combinatorial optimization. --- Optimization, Combinatorial --- Combinatorial analysis --- Mathematical optimization --- Iteration (Mathematics) --- Numerical analysis --- Iterative methods (Mathematics).

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Iterative methods (Mathematics) --- Algorithms. --- Numerical analysis. --- Mathematical analysis --- Algorism --- Algebra --- Arithmetic --- Iteration (Mathematics) --- Numerical analysis --- Foundations

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In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Gröbner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugère (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.

Equations --- Polynomials. --- Iterative methods (Mathematics) --- Numerical solutions. --- Iteration (Mathematics) --- Numerical analysis --- Algebra --- Graphic methods

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Polynomial equations have been long studied, both theoretically and with a view to solving them. Until recently, manual computation was the only solution method and the theory was developed to accommodate it. With the advent of computers, the situation changed dramatically. Many classical results can be more usefully recast within a different framework which in turn lends itself to further theoretical development tuned to computation. This first book in a trilogy is devoted to the new approach. It is a handbook covering the classical theory of finding roots of a univariate polynomial, emphasising computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model and the Thom Codification. Mora aims to show that solving a polynomial equation really means finding algorithms that help one manipulate roots rather than simply computing them; to that end he also surveys algorithms for factorizing univariate polynomials.

Equations --- Polynomials. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Algebra --- Numerical solutions. --- Graphic methods --- Polynomials --- Equations - numerical solutions --- 512.62 --- -Iterative methods (Mathematics) --- Mathematics --- 512.62 Fields. Polynomials --- Fields. Polynomials --- Numerical solutions