Description
- 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 <em>n</em>and deflation tolerance algorithm and find thatthe QR runtimes show little dependence on <em>n</em> 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.
- Notes:
- Thesis (Ph.D. -- Brown University (2011)
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Citation
Pfrang, Christian Werner,
"Diagonalizing Random Matrices with Integrable Systems"
(2011).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z00C4T26
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....