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

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Control theory. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Dynamics --- Machine theory

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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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The second volume of this comprehensive treatise focusses on Buchberger theory and its application to the algorithmic view of commutative algebra. In distinction to other works, the presentation here is based on the intrinsic linear algebra structure of Groebner bases, and thus elementary considerations lead easily to the state-of-the-art in issues of implementation. The same language describes the applications of Groebner technology to the central problems of commutative algebra. The book can be also used as a reference on elementary ideal theory and a source for the state-of-the-art in its algorithmization. Aiming to provide a complete survey on Groebner bases and their applications, the author also includes advanced aspects of Buchberger theory, such as the complexity of the algorithm, Galligo's theorem, the optimality of degrevlex, the Gianni-Kalkbrener theorem, the FGLM algorithm, and so on. Thus it will be essential for all workers in commutative algebra, computational algebra and algebraic geometry.

Equations --- Polynomials. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Algebra --- Numerical solutions. --- Graphic methods --- Gröbner bases. --- Gröbner basis theory --- Commutative algebra