Nonnormality in Lyapunov Equations

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dc.contributor.advisorSorensen, Dannyen_US
dc.contributor.committeeMemberEmbree, Marken_US
dc.creatorBaker, Jonathanen_US
dc.date.accessioned2017-08-02T16:29:26Zen_US
dc.date.available2017-08-02T16:29:26Zen_US
dc.date.created2016-05en_US
dc.date.issued2016-04-22en_US
dc.date.submittedMay 2016en_US
dc.date.updated2017-08-02T16:29:26Zen_US
dc.description.abstractThe singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low- rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. We will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically as the numerical range of the coefficient matrix extends farther into the right half-plane. We also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationBaker, Jonathan. "Nonnormality in Lyapunov Equations." (2016) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/96192">https://hdl.handle.net/1911/96192</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/96192en_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.subjectLyapunov equationen_US
dc.subjectsingular valuesen_US
dc.subjectnonnormalityen_US
dc.subjectHermitian parten_US
dc.subjectnumerical rangeen_US
dc.titleNonnormality in Lyapunov Equationsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Artsen_US
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