Advancing matrix computations with randomized preprocessing

Abstract

The known algorithms for linear systems of equations perform significantly slower where the input matrix is ill conditioned, that is lies near a matrix of a smaller rank. The known methods counter this problem only for some important but special input classes, but our novel randomized augmentation techniques serve as a remedy for a typical ill conditioned input and similarly facilitates computations with rank deficient input matrices. The resulting acceleration is dramatic, both in terms of the proved bit-operation cost bounds and the actual CPU time observed in our tests. Our methods can be effectively applied to various other fundamental matrix and polynomial computations as well. © 2010 Springer-Verlag.