An Introduction to Numerical Analysis

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viii Preface
required by calculating sufficiently many members of the sequence, or
just one member, sufficiently far along. A ‘pure mathematician’ would
prefer the exact answer, ξ, but the sorts of guaranteed accurate approximations which will be discussed here are entirely satisfactory in real-life
applications.
Numerical analysis brings two new ideas to the usual discussion of
convergence of sequences. First, we need, not just the existence of N0,
but a good estimate of how large it is; and it may be too large for
practical calculations. Second, rather than being asked for the limit of
a given sequence, we are usually given the existence of the limit ξ (or
its approximate location on the real line) and then have to construct a
sequence which converges to it. If the rate of convergence is slow, so
that the value of N0 is large, we must then try to construct a better
sequence, one that converges to ξ more rapidly. These ideas have direct
applications in the solution of a single nonlinear equation in Chapter
1, the solution of systems of nonlinear equations in Chapter 4 and the
calculation of the eigenvalues and eigenvectors of a matrix in Chapter 5.
The next six chapters are concerned with polynomial approximation,
and show how, in various ways, we can construct a polynomial which
approximates, as accurately as required, a given continuous function.
These ideas have an obvious application in the evaluation of integrals,
where we calculate the integral of the approximating polynomial instead
of the integral of the given function.
Finally, Chapters 12 to 14 deal with the numerical solution of ordinary
differential equations, with Chapter 14 presenting the fundamentals of
the finite element method. The results of Chapter 14 can be readily
extended to linear second-order partial differential equations.
We have tried to make the coverage as complete as is consistent with
remaining quite elementary. The limitations of size are most obvious
in Chapter 12 on the solution of initial value problems for ordinary
differential equations. This is an area where a number of excellent books
are available, at least one of which is published in two weighty volumes.
Chapter 12 does not describe or analyse anything approaching all the
available methods, but we hope we have included some of those in most
common use.
There is a selection of Exercises at the end of each chapter. All these
exercises are theoretical; students are urged to apply all the methods
described to some simple examples to see what happens. A few of the
exercises will be found to require some heavy algebraic manipulation;
these have been included because we assume that readers will have ac-

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