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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 O(n \log^{2}n) 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 1 \leq \alpha \leq n , an index of how close a matrix is to being Toeplitz, requiring O(\alpha^{d} n \log^{2}n) operations, ( d \leq 2 ). 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.