Numer. Math. 64, 295-32t (1993) Numerische Mathematik q~ Springer-Verlag1993 A look-ahead algorithm for the solution of general Hankel systems Roland W. Freund 1'* and Hongyuan Zha 2'** 1AT&T Bell Laboratories, Room 2C-420, 600 Mountain Avenue, Murray Hill, NJ 07974-0636, USA 2 Scientific Computing and Computational Mathematics, Stanford Uni~cr~ity. Stanford, CA 94305, USA Received January 21, 1992 Summary. The solution of systems of linear equations with Hankel coefficient matrices can be computed with only O(n 2) arithmetic operations, as compared to O (n 3) operations for the general case. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general non- singular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look- ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices. Mathematics Subject Classification (1991): 65F05, 15A57, 42C05 1. Introduction Many important applications lead to systems of linear equations (1.1) n.x. = b. * The research of the first author was supported by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA). **The research of the second author was supported in part by NSF grant DRC-8412314 and Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA). Correspondence to: R.W. Freund