Condition : The Geometry of Numerical Algorithms 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 Newtons Method 16 Homogeneous Polynomial Systems 17 Smales 17th Problem: I 18 Smales 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 . 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 expositions 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 Smales 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.