Date of Completion

8-10-2015

Embargo Period

8-10-2015

Major Advisor

Vadim Olshevsky

Associate Advisor

Alexander Teplyaev

Associate Advisor

Patrick McKenna

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In 1971, Householder and Fox [26] introduced a method for computing an orthonormal basis for the range of a projection. Using a Cholesky decompo- sition on a symmetric idempotent matrix A produced A = LL T , where the columns of the lower triangular matrix L form said basis. Moler and Stewart [32] performed an error analysis on the Householder-Fox algorithm in 1978. It was shown that in most cases reasonable results can be expected, however a recent paper by Parlett and Barszcz [36] included a numerical experiment by Kahan in which the Householder-Fox method performed poorly. Parlett proposed an alternate method which focused on exploiting the structure of the n×n projection I−qq T . In the case where the Householder-Fox algorithm produced an error of 1, this new method produced full accuracy. In what fol- lows, additional algorithms will be introduced, exploiting the decomposable, Green’s (H,1)-quasiseparable and Green’s (H,1)-semiseparable structure of unitary Hessenberg matrices. It will be shown that the more general and newly defined unitary k-Hessenberg matrices also have a great deal of struc- ture, and further, that the structure exploiting algorithms mentioned above are readily generalizable to this new unitary k-Hessenberg case.

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