Asymptotically fast solution of toeplitz and related systems of linear equations

We present an inversion algorithm for the solution of a generic N X N Toeplitz system of linear equations with computational complexity O(Nlog²N) and storage requirements O(N). The algorithm relies upon the known structure of Toeplitz matrices and their inverses and achieves speed through a doubling method. All the results are derived and stated in terms of the recent concept of displacement rank, and this is used to extend the scope of the algorithm to include a wider class of matrices than just Toeplitz and also to include block Toeplitz matrices.