Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.715948 
Title:  Isospectral algorithms, Toeplitz matrices and orthogonal polynomials  
Author:  Webb, Marcus David 
ORCID:
0000000294405361
ISNI:
0000 0004 6348 9851


Awarding Body:  University of Cambridge  
Current Institution:  University of Cambridge  
Date of Award:  2017  
Availability of Full Text: 


Abstract:  
An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pertToeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pertToeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pertToeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pertToeplitz Jacobi operators.


Supervisor:  Not available  Sponsor:  Engineering and Physical Sciences Research Council (EPSRC)  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.715948  DOI:  
Keywords:  Isospectral flows ; Toeplitz matrices ; Orthogonal polynomials ; eigenvalues ; matrices ; spectrum ; spectra ; Jacobi operators ; QR algorithm ; QL algorithm  
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