A Divide and Conquer Algorithm for the Superfast Solution of Toeplitz-like Systems

Abstract

In this paper a new $O(N \,\log^3N)$ solver for $N\times N$ Toeplitz-like systems, based on a divide and conquer technique, is presented. Similarly to the superfast algorithm MBA for the inversion of a Toeplitz-like matrix [R. R. Bitmead and B. D. O. Anderson, Linear Algebra Appl., 34 (1980), pp. 103--116; M. Morf, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1980, pp. 954--959], it exploits the displacement properties. In order to avoid the well-known numerical instability of the explicit inversion, the new algorithm relies on the triangular factorization and back-substitution formula for the system seen as a $2\times 2$ block system with blocks of half size. This idea is the one used in [M. Stewart, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 669--693] to improve the numerical stability of superfast methods based on the generalized Schur algorithm for positive definite Toeplitz matrices, but the algorithm we propose can be applied also to nonsymmetric Toeplitz-like systems. The stability of the algorithm is examined through numerical experiments.