Condition - The Geometry of Numerical Algorithms

Abstract. The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways. A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. This is the theme of my recent monograph Condition, written with Felipe Cucker, that appeared in 2013 in Springer’s Grundlehren series. The monograph is divided into three parts. Its first part deals with the solution of linear systems of equations, where many of the concepts can be explained in an elementary way. The second part is devoted to linear programming, i.e., the solution of systems of linear inequalities (there exist natural extensions to convex programming). The third part is devoted to the solution of systems of polynomial equations, focusing on Smale’s 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in n + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time. Recently, Pierre Lairez succeeded in providing a complete solution of Smale’s 17th problem (“A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time,” to appear in J. FoCM). The enclosed course material in the form of slides follows the three part structure of the monograph and attempts to illustrate the main unifying concepts and key ideas. The framework seems quite generally applicable. For instance, a numerical algorithm for computing eigenpairs of matrices, that is numerically stable and provably runs in average polynomial time, was recently developed along these lines (Armentano, Beltrán, Bürgisser, Cucker, and Shub, “A stable, polynomial-time algorithm for the eigenpair problem,” accepted for J. EMS).