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

 dc.contributor.committeeMember Dolovich, Allan T. en_US dc.creator Steele, John Arthur en_US dc.date.accessioned 2008-12-12T13:14:20Z en_US dc.date.accessioned 2013-01-04T05:10:19Z dc.date.available 2009-12-16T08:00:00Z en_US dc.date.available 2013-01-04T05:10:19Z dc.date.created 2001-01 en_US dc.date.issued 2001-01-01 en_US dc.date.submitted January 2001 en_US dc.description.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. en_US dc.identifier.uri http://hdl.handle.net/10388/etd-12122008-131420 en_US dc.language.iso en_US en_US dc.title Some results concerning the fundamental nature of Wynn's vector epsilon algorithm en_US dc.type.genre Thesis en_US dc.type.material text en_US thesis.degree.department Mechanical Engineering en_US thesis.degree.discipline Mechanical Engineering en_US thesis.degree.grantor University of Saskatchewan en_US thesis.degree.level Doctoral en_US thesis.degree.name Doctor of Philosophy (Ph.D.) en_US

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