The comparison of algorithms

As we have introduced and discussed the various possible schemes for solving Volterra and Fredholm equations of the second kind, we have made a number of general qualitative and quantitative comparisons of the methods, in terms of their cost (for a given discretisation size N), their likely accuracy (for a given N) and rate of convergence (as N increases), their stability and the ease with which usable error estimates can be provided for the method. It is unfortunately a truism that the performance of a method in practice is also affected by apparently minor details of the implementation (that is, of the program). It is also true that comparisons based on one or two numerical examples can be misleading; an overall picture of the performance of a method depends on its behaviour over a wide class of problems. In this chapter we consider in some detail how objective comparisons between methods can be made and we illustrate the problems by looking at the performance of several specific routines for solving second kind Fredholm equations. These detailed comparisons are very interesting but, when two different algorithms give fairly close results, the reader is warned of the dangers of mentally ranking the algorithms: different implementations of the algorithms might well rank in the reverse order, so that only quite large differences in performance should be taken seriously.

Implementations of a given algorithm can be of two main types, namely automatic or non-automatic routines.

This book considers the practical solution of one-dimensional integral equations. Both integral equations, and methods for solving them, come in many forms and we could not try, and have not tried, to be exhaustive. For the problem classes covered, we have used the ‘classical’ Fredholm/Volterra/first kind/second kind/third kind categorisation. Not all problems fit neatly into such categories; then the methods used to solve standard classes of problems must be modified and tailored to suit the needs of nonstandard ‘real life’ problems. It is hoped that the nature of any such modifications will be obvious to the intelligent reader. Not all categories of problems seem equally important (i.e. frequent) in practice; we have tried to spend most time on the most important classes of problems.

We have also been selective in the choice of methods covered. Here, personal likes and dislikes have helped the selection process, but we have also taken particular note of the fact that the cost of solving even a one-dimensional integral equation of Fredholm type can be unexpectedly high. Methods which converge slowly but steadily are therefore not very attractive in practice and particular emphasis is placed on the ability of a given method to obtain rapid convergence, to provide computable error estimates and to produce reliable results at relatively low cost.

It is hoped that the book will serve as a reference text for the practising numerical mathematician, scientist or engineer, who finds integral problems arising in his work.

Introduction

So far we have been mainly concerned with methods for the solution of Fredholm and Volterra equations under the assumption that the kernel and driving terms are smooth functions of their arguments. When this assumption breaks down, the methods we have discussed will either fail completely (if they try to evaluate the kernel at a point where it is singular, for example) or at best converge only slowly. Singular integral equations are very common in practice, and we discuss in this chapter methods for dealing with them; the aim is, or should be, to produce not merely a method which converges, but one which takes sufficient account of the singularity that it converges as fast as methods for smooth problems.

This aim cannot always be met; when it can, it is clear that the resulting method must be tailored in some way to the form of the singularity. Fortunately a few standard types of singularity appear very common in integral equations. We shall use the term ‘singularity’ rather widely to refer to any lack of analyticity in the problem. Thus, for example, the following features make a problem singular:

1. (i) An infinite or semi-infinite range [a, b] for the integral operator,

2. (ii) A discontinuous derivative in the kernel or driving term. Hence, a Green's function kernel (see for example (0.3.16)) is singular in this sense.

3. (iii) An infinite or non-existing derivative of some finite order. Thus, y(s) = (1 − s2)½ is singular on the range [−, 1] because y′(s) is unbounded at s = α 1. It is nonsingular on the range [−±, ±].

This book

The theory of differential equations is an essential ingredient of any undergraduate course in Mathematics and the majority of Numerical Analysis undergraduate courses introduce the student, at an early stage, to the numerical solution of differential equations. For some reason the theory, and, perhaps more so, the numerical solution, of integral equations are deferred to a later stage: in some sense integral equations must be felt to be either more advanced or of less practical interest than differential equations. This reflects the situation in practical calculations and probably in turn helps to perpetuate it; we turn more readily to a differential formulation of a problem than to an integral formalism. Yet the theory of linear nonsingular integral equations is at least as well developed as that of differential equations and it is in many respects rather simpler. The corresponding operators are bounded rather than unbounded, leading to a very straightforward existence theory (the Fredholm theory); perhaps, as one consequence of this, there is a much tighter link between the theory and practice of integral equations than is the case with differential equations. Most of the convergence proofs are constructive in nature and all or nearly all of the constructions have been used as the basis of algorithms for the numerical solution of the underlying equations (although not always with any great success!).