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This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance ---regarding both stability and complexity--- of numerical algorithms. While the role of condition was shaped in the last half-century, so far there has not been a monograph treating this subject in a uniform and systematic way.   The book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition.   The exposition's level increases along the book, starting in the context of linear algebra at an undergraduate level and reaching in its third part the recent developments and partial solutions for Smale's 17th problem which can be explained within a graduate course. Its middle part contains a condition-based course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming.

Geometry, Differential -- Data processing. --- Engineering & Applied Sciences --- Computer Science --- Numerical analysis. --- Geometry. --- Mathematics. --- Computer science --- Computer mathematics. --- Algorithms. --- Mathematical optimization. --- Probabilities. --- Mathematics of Computing. --- Probability Theory and Stochastic Processes. --- Computational Mathematics and Numerical Analysis. --- Computational Science and Engineering. --- Optimization. --- Mathematics --- Euclid's Elements --- Mathematical analysis --- Computer science. --- Distribution (Probability theory. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Foundations --- Computer science—Mathematics. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Computer science --- Numerical analysis --- Congresses

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The goal of learning theory is to approximate a function from sample values. To attain this goal learning theory draws on a variety of diverse subjects, specifically statistics, approximation theory, and algorithmics. Ideas from all these areas blended to form a subject whose many successful applications have triggered a rapid growth during the last two decades. This is the first book to give a general overview of the theoretical foundations of the subject emphasizing the approximation theory, while still giving a balanced overview. It is based on courses taught by the authors, and is reasonably self-contained so will appeal to a broad spectrum of researchers in learning theory and adjacent fields. It will also serve as an introduction for graduate students and others entering the field, who wish to see how the problems raised in learning theory relate to other disciplines.

Computational learning theory. --- Approximation theory.

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Mathematics

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Stephen Smale is one of the great mathematicians of the 20th century. His work encompasses a wide variety of subjects: differential topology, dynamical systems, calculus of variations, theory of computation, mechanics and mathematical economy. In all these subjects he has left the imprint of a collection of fundamental results. He has obtained several distinctions, including the Fields Medal, the Veblen Prize, the Chauvenet Prize, the von Neumann Award and the National Medal of Science.This invaluable book contains the collected papers of Stephen Smale. These are divided into eight groups: top

Mathematics. --- Computer science. --- Economics.