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 to implement our algorithms in binary within a fixed computer precision. 2000 Math. Subject Classification: 68W30, 68W20, 65F05, 68Q25