TR-2003004: Superfast Algorithms for Singular Toeplitz/Hankel-like Matrices

Applying the superfast divide-and-conquer MBA algorithm for generally singular n × n Toeplitz-like or Hankel-like integer input matrices, we perform computations in the ring of integers modulo a power of a fixed prime, especially power of 2. This is practically faster than computing modulo a random prime but requires additional care to avoid degeneration, particularly at the stages of compression of auxiliary matrices. We supply the necessary techniques. The resulting algorithm combined with Hensel’s lifting and fast rational number reconstruction supports nearly optimal bit cost estimates for the solution of (possibly singular but) consistent Toeplitz/Hankel-like linear systems with integer coefficients (as well as for other fundamental problems of matrix computation). We arrive at nearly optimal bit cost estimates also for computing the univariate polynomial gcd and resultant, Padé approximation, rational interpolation, and Berlekamp–Massey’s problem.