Toeplitz and Hankel Meet Hensel and Newton: Nearly Optimal Algorithms and Their Practical Acceleration with Saturated Initialization

The classical and intensively studied problem of solving a Toeplitz/ Hankel linear system of equations is omnipresent in computations in sciences, engineering and signal processing. By assuming a nonsingular integer input matrix and relying on Hensel’s lifting, we compute the solution faster than with the divide-and-conquer algorithm by Morf 1974/1980 and Bitmead and Anderson 1980 and nearly reach the information lower bound on the bit operation complexity of the solution. Furthermore, we extend lifting to the rings of integers modulo nonprimes, e.g., modulo 2. This allows significant saving of the word operations. We also extend our algorithms and complexity estimates to computations with singular ∗The 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