A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Asymptotic Behavior of Solutions of Emden-Fowler Difference Equations with Oscillating Coefficients

AbstractIt is known that if ∑∞j |pj| < ∞ then the Emden-Fowler difference equation (A) Δ2yn−1 = pnyγn (γ > 0) has a positive solution {yn}, defined for n sufficiently large, such that limn→∞yn = c > 0, while if ∑∞jγ |pj| < ∞ then (A) has a positive solution }yn}, defined for n sufficiently large, such that limn→∞Δyn = c > 0. Here it is shown that these conclusions hold if the series converge (perhaps conditionally) and satisfy secondary conditions which do not imply absolute convergence. Estimates of }yn} and }Δyn} as n → ∞ are also given. Moreover, γ can be any real number other than 0 or 1