Fast algorithms for rational Hermite approximation and solution of Toeplitz systems

This is a short paper announcing several new results. The best known algorithm for solving a Toeplitz system of linear equations requires
O(n^2)
operations. We present a new algorithm that reduces this computation to
O(n\log^{2}n)
. Furthermore the new algorithm handles all degenerate cases. We show that the solution process of a Toeplitz system is intimately related to the computation of certain Padé approximants. Actually the Padé computation is a special case of rational Hermite interpolation. Our primary result shows that all rational Hermite interpolants along an antidiagonal of the Rational Interpolation Table can be computed by the extended Euclidean algorithm. We present new fast algorithms for such computations and indicate extensions and improvements. Several application areas can make use of these results. We include a brief discussion on coding but leave digital filters and circuit synthesis to the full paper.