Smoothed Analysis of Moore-Penrose Inversion

We perform a smoothed analysis of the condition number of rectangular matrices. We prove that, asymptotically, the expected value of this condition number depends only of the elongation of the matrix, and not on the center and variance of the underlying probability distribution.Comment: 19 pages. Version 2 contains a new section on application

Computing the homology of basic semialgebraic sets in weak exponential time

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data)