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Part I: Multiple bifurcations. Part II: Parallel homotopy method for the real nonsymmetric eigenvalue problem

Citation

Lui, Shiu-Hong (1992) Part I: Multiple bifurcations. Part II: Parallel homotopy method for the real nonsymmetric eigenvalue problem. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06152005-084230

Abstract

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PART I

Consider an analytic operator equation [...] = 0 where [...] is a real parameter. Suppose 0 is a "simple" eigenvalue of the Frechet derivative [...] at [...]. We give a hierarchy of conditions which completely determine the solution structure of the operator equation. It will be shown that multiple bifurcation as well as simple bifurcation can occur. This extends the standard bifurcation theory from a "simple" eigenvalue in which only one branch bifurcates. When 0 is a multiple eigenvalue, we give some sufficient conditions for multiple bifurcations with a lower bound on the multiplicity of the bifurcation. This theory is applied to some semilinear elliptic partial differential equations on a cylinder with a constant cross-section.

PART II

We present a homotopy method to compute the eigenvectors and eigenvalues, i.e., the eigenpairs of a given real matrix [...]. From the eigenpairs of some real matrix [...], we follow the eigenpairs of [...] at successive times from t = 0 to t = 1 using continuation. At t = 1, we have the eigenpairs of the desired matrix [...]. The following phenomena are present for a general nonsymmetric matrix:

- complex eigenpairs

- ill-conditioned problems due to non-orthogonal eigenvectors

- bifurcation (i.e., crossing of eigenpaths)

These can present computational difficulties if not handled properly. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. We will see that the homotopy method is extremely slow for full matrices but has the potential to compete with other algorithms for sparse matrices as well as matrices with defective eigenvalues.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Keller, Herbert Bishop
Thesis Committee:
  • Unknown, Unknown
Defense Date:13 September 1991
Record Number:CaltechETD:etd-06152005-084230
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-06152005-084230
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2602
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:15 Jun 2005
Last Modified:26 Dec 2012 02:52

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