Title:

The improvement of approximate solutions of the integral equation eigenvalue problem

This thesis is mainly concerned with the improvement of approximate solutions of the eigenvalue problem for Fredholm integral equations with continuous kernels. It is assumed that an approximate solution has been obtained by the Nystrom (quadrature) method and more accurate solutions are required. Particular attention is paid to the application to the integral equation eigenvalue problem of the well known deferred correction technique for the improvement of numerical solutions. To illustrate the ideas the trapezoidal rule is chosen as the basic quadrature rule with the Gregory formula being used to estimate the local truncation error. In chapter 1, the main ideas of this thesis are introduced. The approximation of simple eigenvalues is discussed in chapter 2. An equation for the corrections of eigenvalues and eigenvectors is derived and gives rise to a stable iteration process for the computation of the corrections. For the case of multiple eigenvalues, the equations of chapter 2 are generalised and a more complicated correction process is obtained. Chapter 3 gives some necessary results for the matrix perturbation problem. The results for the approximations of multiple eigenvalues (defective and nondefective) of integral equations are presented in chapter 4. In chapter 5 the ideas are extended to deal with the approximation of eigenvalues for the SturmLiouville ordinary differential equation eigenvalue problem. A suite of programs for the numerical solution of the integral equation eigenvalue problem with continuous kernel has been written using the ideas described in chapters 2 to 4. These are described in chapter 6 and listings of the main programs are given in Appendix VI. Numerical results which illustrate the theoretical results of chapters 2, 4 and 5 are also presented.
