Superfast Algorithms for Singular Toeplitz-like Matrices

We apply the superfast divide-and-conquer MBA algorithm to possibly singular n × n Toeplitz-like integer input matrices and extend it to computations in the ring of integers modulo a power of a random prime. We choose the power which barely fits the size of a computer word; this saves word operations in the subsequent lifting steps. We extend our early techniques for avoiding degeneration while preserving the Toeplitz structure. Our resulting algorithm supports nearly optimal randomized bit cost estimates for the solution of possibly singular Toeplitz and Toeplitz-like linear systems of equations, various related fundamental matrix computations (rank, null space) as well as computing the univariate polynomial gcd and resultant, Padé approximation, and rational interpolation where all input values are integers.