Doubling algorithms for Toeplitz and related equations

A new class of doubling or halving algorithms for solving Toeplitz and related equations is presented. For scalar n by n Toeplitz matrices, they require computations, similarly to the HGCD (half-greatest-common-divisor) based algorithm of Gustavson and Yun. However, these new algorithms are based on the notions of "shift" or displacement rank , an index of how close a matrix is to being Toeplitz, requiring operations, ( ). A basic version of a doubling algorithm for such "α-Toeplitz matrices" is presented, and the applications of these results to related problems are mentioned, such as the inversion of banded-, block- and Hankel matrices.