<mods:mods xmlns:mods="http://www.loc.gov/mods/v3" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:METS="http://www.loc.gov/METS/" ID="etd617" xsi:schemaLocation="http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd">
          <mods:titleInfo>
            <mods:title>Diagonalizing Random Matrices with Integrable Systems</mods:title>
          </mods:titleInfo>
          <mods:name type="personal">
            <mods:namePart>Pfrang, Christian Werner</mods:namePart>
            <mods:role>
              <mods:roleTerm type="text">creator</mods:roleTerm>
            </mods:role>
          </mods:name>
          <mods:originInfo>
            <mods:copyrightDate>2011</mods:copyrightDate>
          </mods:originInfo>
          <mods:physicalDescription>
            <mods:extent>xviii, 293 p.</mods:extent>
            <mods:digitalOrigin>born digital</mods:digitalOrigin>
          </mods:physicalDescription>
          <mods:note>Thesis (Ph.D. -- Brown University (2011)</mods:note>
          <mods:name type="personal">
            <mods:namePart>Menon, Govind</mods:namePart>
            <mods:role>
              <mods:roleTerm type="text">Director</mods:roleTerm>
            </mods:role>
          </mods:name>
          <mods:name type="personal">
            <mods:namePart>Deift, Percy</mods:namePart>
            <mods:role>
              <mods:roleTerm type="text">Reader</mods:roleTerm>
            </mods:role>
          </mods:name>
          <mods:name type="personal">
            <mods:namePart>Harrison, Matthew</mods:namePart>
            <mods:role>
              <mods:roleTerm type="text">Reader</mods:roleTerm>
            </mods:role>
          </mods:name>
          <mods:name type="corporate">
            <mods:namePart>Brown University. Applied Mathematics</mods:namePart>
            <mods:role>
              <mods:roleTerm type="text">sponsor</mods:roleTerm>
            </mods:role>
          </mods:name>
          <mods:genre authority="aat">theses</mods:genre>
          <mods:subject>
            <mods:topic>Completely Integrable</mods:topic>
          </mods:subject>
          <mods:subject>
            <mods:topic>Eigenvalue Algorithms</mods:topic>
          </mods:subject>
          <mods:subject authority="FAST" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/1089803"><mods:topic>Random matrices</mods:topic></mods:subject><mods:recordInfo>
            <mods:recordContentSource authority="marcorg">RPB</mods:recordContentSource>
            <mods:recordCreationDate encoding="iso8601">20111003</mods:recordCreationDate>
          </mods:recordInfo>
        <mods:language><mods:languageTerm type="code" authority="iso639-2b">eng</mods:languageTerm><mods:languageTerm type="text">English</mods:languageTerm></mods:language><mods:abstract>This thesis is concerned with the applicationof a certain class of eigenvalue algorithmsto random matrix initial data. In particularwe study numerically the time to first deflation ('runtime') as wellas the subdiagonal index at which deflationoccurs as random variables driven by randominitial data.For the QR and the Toda algorithm we show numericallythat initial data with a Wigner--semicircle spectrallimit universally leads to exponential (Gaussian)right tails of the corresponding runtime distributions.Moreover, if the runtime distributions are centeredand scaled to mean zero and variance one, theshapes of the resulting distributions are universalfor both algorithms. We demonstrate that fornon Wigner--type initial data the same scalingbehavior does not necessarily exist.We investigate the dependence of the meanand standard deviations of the runtime distributionfor the QR and the Toda on matrix size &lt;em&gt;n&lt;/em&gt;and deflation tolerance algorithm and find thatthe QR runtimes show little dependence on &lt;em&gt;n&lt;/em&gt; butan approximately linear dependence on log ε.In addition to these well known algorithms we describean algorithm that reliably deflates Hermite--1 initialmatrices in the middle. This is interesting in the lightof recent divide and conquer algorithms for the eigenvalueproblem.We also present theoreticalresults on the scale of the minimal subdiagonal entryof Hermite--1 distributed Jacobi matrices and derive the Liouvilleequation for the evolution of this density under the Toda flow.The very last section mentions a result concerning the behavior of the inverse spectralmap for Jacobi matrices near the boundary of the positive orthant of the sphere.</mods:abstract><mods:identifier type="doi">10.7301/Z00C4T26</mods:identifier><mods:accessCondition type="rights statement" xlink:href="http://rightsstatements.org/vocab/InC/1.0/">In Copyright</mods:accessCondition><mods:accessCondition type="restriction on access">Collection is open for research.</mods:accessCondition><mods:typeOfResource authority="primo">dissertations</mods:typeOfResource></mods:mods>