Null space and eigenspace computations with additive preprocessing

Abstract

We propose and analyze additive preprocessing for computing a vector in the null space of a matrix and a basis for this space. Due to our preprocessing, instead of singular linear systems we solve nonsingular ones, which preserve the conditioning properties and the structure of the input matrices. We extend our preprocessing to decrease the size and the condition number of an ill conditioned input matrix. We also cover applications to the eigenspace computations and to generating effective preconditioners for a linear system of equations.