TR-2008003: Unified Nearly Optimal Algorithms for Structured Integer Matrices and Polynomials

We seek the solution of banded, Toeplitz, Hankel, Vandermonde, Cauchy and other structured linear systems of equations with integer coefficients. By combining Hensel’s symbolic lifting with either divide-and-conquer algorithms or numerical iterative refinement, we unify the solution for all these structures. We yield the solution in nearly optimal randomized Boolean time, which covers both solution and its correctness verification. Our algorithms and nearly optimal time bounds are extended to the computation of the determinant of a structured integer matrix, its rank and a basis for its null space as well as to some fundamental computations with univariate polynomials that have integer coefficients. Furthermore, we allow to perform lifting modulo a properly bounded power of two t

TR-2008007: Degeneration of Structured Integer Matrices Modulo an Integer

AbstractHensel’s lifting modulo a prime q is a customary means of the solution of an integer or rational linear system of equations. In combination with some effective numerical algorithms this technique enables solution in nearly optimal time in the case of most popular structured inputs. Practically one can further benefit from choosing q=2v for a proper positive integer v and performing binary computations within the computer precision. If the input matrix becomes singular because of the reduction modulo q, then the approach fails. For larger integers q and random integer input matrices, however, such degeneration occurs rarely according to the analysis by Brent and McKay 1987. Based on distinct techniques we show that degeneration also occurs rarely for random integer matrices with all most popular structures such as the Toeplitz, Hankel, band and rank (quasiseparable) structures. Furthermore with random small-rank modifications of an input matrix we have good chances to overcome degeneration, safely solve the new linear system, and recover the solution of the original one. The results of our extensive tests support our formal analysis