Condition, 1st Edition

  • Published By:
  • ISBN-10: 3642388965
  • ISBN-13: 9783642388965
  • DDC: 518
  • Grade Level Range: College Freshman - College Senior
  • 554 Pages | eBook
  • Original Copyright 2013 | Published/Released June 2014
  • This publication's content originally published in print form: 2013

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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.

Table of Contents

Front Cover.
Editorial Board.
Other Frontmatter.
Title Page.
Copyright Page.
Overture: On the Condition of Numerical Problems.
1: Condition in Linear Algebra (Adagio).
2: Normwise Condition of Linear Equation Solving.
3: Probabilistic Analysis.
4: Error Analysis of Triangular Linear Systems.
5: Probabilistic Analysis of Rectangular Matrices.
6: Condition Numbers and Iterative Algorithms.
7: Intermezzo I: Condition of Structured Data.
8: Condition in Linear Optimization (Andante).
9: A Condition Number for Polyhedral Conic Systems.
10: The Ellipsoid Method.
11: Linear Programs and Their Solution Sets.
12: Interior-Point Methods.
13: The Linear Programming Feasibility Problem.
14: Condition and Linear Programming Optimization.
15: Average Analysis of the RCC Condition Number.
16: Probabilistic Analyses of the GCC Condition Number.
17: Intermezzo II: The Condition of the Condition.
18: Condition in Polynomial Equation Solving (Allegro con brio).
19: A Geometric Framework for Condition Numbers.
20: Homotopy Continuation and Newton's Method.
21: Homogeneous Polynomial Systems.
22: Smale's 17th Problem: I.
23: Smale's 17th Problem: II.
24: Real Polynomial Systems.
25: Probabilistic Analysis of Conic Condition Numbers: I. The Complex Case.
26: Probabilistic Analysis of Conic Condition Numbers: II. The Real Case.
Coda: Open Problems.
Notation ….
… Concepts ….
… and the People Who Crafted Them.