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. doi:10.7907/b79y-vb23. https://resolver.caltech.edu/CaltechETD:etd-06152005-084230

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

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.

Thesis Files