Condition (eBook)

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.

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​.