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.
From the book reviews: "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. ... Burgisser (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)
作者简介
Peter Burgisser 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.
出版社: Springer-Verlag Berlin and Heidelberg GmbH & Co. K (2013年8月13日)
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