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Grundlehren der mathematischen Wissenschaften

Condition

The Geometry of Numerical Algorithms

Authors: Bürgisser, Peter, Cucker, Felipe

  • Unique book combining methods from numerical computation and complexity
  • Excellent pedagogical presentation
  • Explanation of Smale's 17th problem
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eBook $109.00
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  • ISBN 978-3-642-38896-5
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Hardcover $149.00
price for USA
  • ISBN 978-3-642-38895-8
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About this book

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.

About the authors

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.

Reviews

“The book under review is divided into three parts, ‘which approximately correspond to themes of conditioning in linear algebra, linear programming, and polynomial equation solving’. … Given its detailed covering of a wide range of topics and its geometric approach, I think this book may well become a must-have for all who are seriously interested in numerical algorithms.” (S. C. Coutinho, The Mathematical Gazette, Vol. 99 (546), November, 2015)

“This book published in 2013 is the first book devoted entirely on this subject. It must be said that this book is a full success since it realizes a synthesis of ideas and works on the mathematical foundations on conditioning. … The book is self contained and easy to read … . The book ends with the statement of eighteen open problems that shows that Mr. Condition has a bright future ahead of him.” (Jean-Claude Yakoubsohn, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 117, 2015)

“The authors intend with this book to fill the gap by addressing the probabilistic analysis of the algorithms related to condition numbers. … Under the vigilant eyes of so many famous scientists, it is sure that this book is a milestone in this area of research. … The monograph under review is without any doubt a very carefully prepared one, and researchers interested in numerical analysis (and related topics) should become familiar with this book.” (Elena Pelican, Mathematical Reviews, August, 2014)

“This book studies a type of numerical imprecision that arises universally. … Bürgisser (Technical Univ. of Berlin, Germany) and Cucker (City Univ. of Hong Kong) provide the first book-length treatment of the concept. … Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (D. V. Feldman, Choice, Vol. 51 (11), July, 2014)

“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)

Table of contents (21 chapters)

  • Normwise Condition of Linear Equation Solving

    Bürgisser, Peter (et al.)

    Pages 3-19

  • Probabilistic Analysis

    Bürgisser, Peter (et al.)

    Pages 21-58

  • Error Analysis of Triangular Linear Systems

    Bürgisser, Peter (et al.)

    Pages 59-75

  • Probabilistic Analysis of Rectangular Matrices

    Bürgisser, Peter (et al.)

    Pages 77-100

  • Condition Numbers and Iterative Algorithms

    Bürgisser, Peter (et al.)

    Pages 101-117

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-642-38896-5
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-642-38895-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Condition
Book Subtitle
The Geometry of Numerical Algorithms
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
349
Copyright
2013
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-38896-5
DOI
10.1007/978-3-642-38896-5
Hardcover ISBN
978-3-642-38895-8
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XXXI, 554
Topics