Some results concerning the fundamental nature of Wynn's vector epsilon algorithm

Abstract

In this thesis, Wynn's Vector Epsilon Algorithm (VEA) is examined. Although the usefulness of this sequence-to-sequence transformation for inducing and enhancing convergence in vector sequences has been amply demonstrated by others, it is still not well understood. After reviewing some known important theoretical results for the VEA and its kernel (the full set of vector sequences which the VEA transforms to give a constant vector sequence), the author provides a sufficient and necessary condition for membership of a vector sequence in the real part of the kernel of the 1st order VEA. This kernel is shown to be the set of all real vector sequences {xn} converging toward, orbiting, or diverging away from some vector x where each term of the error sequence {xn-x} is a scaled and/or rotated version of the previous term of the error sequence, called λR sequences. This result is contrasted with one by McLeod and Graves-Morris. It is then shown that λR sequences may also be described as those sequences {xn} whose terms satisfy xn = x + znw + zn w where z ≠ 0, z ≠ 1, ||w|| > 0, and < w, w >= 0.

Numerical experiments by the author on vector sequences generated by the formula xn=Axn-1+b are reported. Circumstances are found under which the VEA order of such sequences is lower than the upper bound given by Brezinski. The reduction is triggered by the presence of certain orthogonal relationships between eigenvector and generalised eigenvector components whose corresponding Jordan blocks in the Jordan canonical form of A have complex conjugate eigenvalues. This empirical result anticipates the complex kernel of the 1st order VEA which is shown to be every sequence {xn} whose terms satisfy xn = x + znw₁ + znw₂ with z ≠ 0, z ≠ 1, ||w₁|| + ||w₂|| > 0, and < w₁,w₂ > = 0 and no others. Some remaining open questions are noted in the final chapter.