Bürgisser P., Cucker F. Condition: The Geometry of Numerical Algorithms

Springer, 2013. — 567 p. — (Grundlehren der mathematischen Wissenschaften 349). — ISBN: 978-3-642-38895-8.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.Condition in Linear Algebra (Adagio)
Normwise Condition of Linear Equation Solving
Probabilistic Analysis
Error Analysis of Triangular Linear Systems
Probabilistic Analysis of Rectangular Matrices
Condition Numbers and Iterative Algorithms
Condition in Linear Optimization (Andante)
A Condition Number for Polyhedral Conic Systems
The Ellipsoid Method
Linear Programs and Their Solution Sets
Interior-Point Methods
The Linear Programming Feasibility Problem
Condition and Linear Programming Optimization
Average Analysis of the RCC Condition Number
Probabilistic Analyses of the GCC Condition Number
Condition in Polynomial Equation Solving (Allegro con brio)
A Geometric Framework for Condition Numbers
Homotopy Continuation and Newton’s Method
Homogeneous Polynomial Systems
Smale’s 17th Problem: I
Smale’s 17th Problem: II
Real Polynomial Systems
Probabilistic Analysis of Conic Condition Numbers: I. The Complex Case
Probabilistic Analysis of Conic Condition Numbers: II. The Real Case