Overview

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.

Full Product Details

Author:   Peter Bürgisser ,  Felipe Cucker
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2013 ed.
Volume:   349
Dimensions:   Width: 15.50cm , Height: 3.20cm , Length: 23.50cm
Weight:   9.989kg
ISBN:  

9783642388958

ISBN 10:   3642388957
Pages:   554
Publication Date:   23 August 2013
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

Preface.- Overture: On the Condition of Numerical Problems and the Numbers that Measure It.- I Condition in Linear Algebra (Adagio): 1 Normwise Condition of Linear Equation Solving.- 2 Probabilistic Analysis.- 3 Error Analysis of Triangular Linear Systems.- 4 Probabilistic Analysis of Rectangular Matrices.- 5 Condition Numbers and Iterative Algorithms.- Intermezzo I: Condition of Structured Data.- II Condition in Linear Optimization (Andante): 6 A Condition Number for Polyhedral Conic Systems.- 7 The Ellipsoid Method.- 8 Linear Programs and their Solution Sets.- 9 Interior-point Methods.- 10 The Linear Programming Feasibility Problem.- 11 Condition and Linear Programming Optimization.- 12 Average Analysis of the RCC Condition Number.- 13 Probabilistic Analyses of the GCC Condition Number.- Intermezzo II: The Condition of the Condition.- III Condition in Polynomial Equation Solving (Allegro con brio): 14 A Geometric Framework for Condition Numbers.- 15 Homotopy Continuation and Newton's Method.- 16 Homogeneous Polynomial Systems.- 17 Smale's 17th Problem: I.- 18 Smale's 17th Problem: II.- 19 Real Polynomial Systems.- 20 Probabilistic Analysis of Conic Condition Numbers: I. The Complex Case 4.- 21 Probabilistic Analysis of Conic Condition Numbers: II. The Real Case.- Appendix​.

Reviews

From the reviews: The authors of this book discuss the ways that such errors are produced in a computer, and consider the use of condition numbers to understand the performance of numerical algorithms. ... this monograph not only offers a well-organized and systematic introduction to the subject, but also works as a useful reference for advanced researchers. (Tanbir Ahmed, Computing Reviews, November, 2013)

Author Information

Peter Bürgisser is an internationally recognized expert in complexity theory. He is associate editor of the journal Computational Complexity and he was invited speaker at the 2010 International Congress Mathematicians. Felipe Cucker is well known for his work on complexity over the real numbers, jointly with L. Blum, S. Smale and M. Shub. He also worked in learning theory and made seminal contributions to condition numbers in optimization and their probabilistic analyses. F.C. is former chair of the Society for the Foundations of Computational Mathematics and the current managing editor of the society's journal.

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