Citation
Lui, ShiuHong (1992) Part I: Multiple bifurcations. Part II: Parallel homotopy method for the real nonsymmetric eigenvalue problem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/b79yvb23. https://resolver.caltech.edu/CaltechETD:etd06152005084230
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 crosssection.
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
 illconditioned problems due to nonorthogonal 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:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  13 September 1991 
Record Number:  CaltechETD:etd06152005084230 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd06152005084230 
DOI:  10.7907/b79yvb23 
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 ETDdb 
Deposited On:  15 Jun 2005 
Last Modified:  16 Apr 2021 23:03 
Thesis Files

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