Toeplitz and Hankel Meet Hensel and Newton Modulo a Power of Two

We extend Hensel lifting for solving general and structured linear systems of equations to the rings of integers modulo nonprimes, e.g. modulo a power of two. This enables significant saving of word operations. We elaborate upon this approach in the case of Toeplitz linear systems. In this case, we initialize lifting with the MBA superfast algorithm, estimate that the overall bit operation (Boolean) cost of the solution is optimal up to roughly a logarithmic factor, and prove that the degeneration is unlikely even where the basic prime is fixed but the input matrix is random. We also comment on the extension of our algorithm to some other fundamental computations with (possibly singular) general and structured matrices and univariate polynomials as well as to the computation of the sign and the value of the determinant of an integer matrix. 2000 Math. Subject Classification: 68W30, 68W20, 65F05, 68Q25 ∗Some results of this paper have been presented at the Annual International Conference on Application of Computer Algebra, Volos, Greece, June 2002; ACM International Symposium on Sympolic and Algebraic Computation, Lille, France, July 2002; and the 5th Annual Conference on Computer Algebra in Scientific Computing, Yalta, Crimea, Ukraine, September 2002. †Supported by NSF Grant CCR 9732206 and PSC CUNY Awards 65393–0034 and 66437– 0035