Ritz values and Arnoldi convergence for non-Hermitian matrices

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dc.contributor.advisorEmbree, Marken_US
dc.creatorCarden, Russellen_US
dc.date.accessioned2013-03-08T00:33:01Zen_US
dc.date.available2013-03-08T00:33:01Zen_US
dc.date.issued2012en_US
dc.description.abstractThis thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.en_US
dc.format.extent142 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH.SCI. 2012 CARDENen_US
dc.identifier.citationCarden, Russell. "Ritz values and Arnoldi convergence for non-Hermitian matrices." (2012) Diss., Rice University. <a href="https://hdl.handle.net/1911/70215">https://hdl.handle.net/1911/70215</a>.en_US
dc.identifier.digitalCardenRen_US
dc.identifier.urihttps://hdl.handle.net/1911/70215en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectApplied sciencesen_US
dc.subjectPure sciencesen_US
dc.subjectRitz valuesen_US
dc.subjectArnoldi convergenceen_US
dc.subjectNon-Hermitian matricesen_US
dc.subjectEigenvaluesen_US
dc.subjectField of valuesen_US
dc.subjectArnoldien_US
dc.subjectMajorizationen_US
dc.subjectApplied mathematicsen_US
dc.subjectMathematicsen_US
dc.titleRitz values and Arnoldi convergence for non-Hermitian matricesen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematical Sciencesen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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