Randomized Preprocessing of Homogeneous Linear Systems

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations. In the case of Toeplitz inputs, we decrease the solution time from quadratic to nearly linear, and our tests show dramatic decrease of the CPU time as well. We prove numerical stability of our randomized algorithms and extend our approach to solving nonsingular linear systems, inversion and generalized (Moore–Penrose) inversion of general and structured matrices by means of Newton’s iteration, approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank, matrix eigen-solving, and root-finding for polynomial and secular equations. Some by-products and extensions of our study can be of independent technical intersest, e.g., our extensions of the Sherman–Morrison– Woodbury formula for matrix inversion, our estimates for the condition number of randomized matrix products, preprocessing via augmentation, and the link of preprocessing to aggregation.