The location of roots of equations with particular reference to the generalized eigenvalue problem

A survey is presented of algorithms which are in current use for the solution of a single algebraic or transcendental equation in one unknown, together with an appraisal of their practical performance.
The first part of the thesis consists of an account of the theoretical basis of a number of iterative methods and an examination of the problems to be overcome in order to achieve a successful computer implementation.
In the selection of specific programs for testing, the emphasis has been placed on methods which are suitable for use, in conjunction with determinant evaluation, for the solution of standard eigenvalue problems and generalized problems of the form A(λ)x = O, where the elements of A are linear or non-linear functions of λ.
The principal requirements for such purposes are that:
1. the algorithm should not be restricted to polynomial
equations
2. derivative evaluation should not be required.
Examples of eigenvalue problems arising from engineering applications illustrate the potential difficulties of determining roots. Particular attention is given to the problem of calculating a number of roots in cases where a priori estimates for each root are not available. The discussion is extended to give a brief account of possible approaches to the problem of locating complex roots.
Interpolation methods are found to be particularly versatile and can be recommended for their accuracy and efficiency. It is also suggested that such algorithms may often be employed as search strategies in the absence of good initial estimates of the roots. Mention is also made of those features of practical implementation which were found to be particularly useful, together with a list of some outstanding difficulties, associated principally with the automatic computation of several roots of an equation.

Submitted to the Council for National Academic Awards in partial fulfilment of the requirements for the degree of Master of Philosophy.