Page 11 - Compact Numerical Methods For Computers

Chapter 1

A STARTING POINT

1.1. PURPOSE AND SCOPE
This monograph is written for the person who has to solve problems with (small)
computers. It is a handbook to help him or her obtain reliable answers to specific
questions, posed in a mathematical way, using limited computational resources.
To this end the solution methods proposed are presented not only as formulae but
also as algorithms, those recipes for solving problems which are more than merely
a list of the mathematical ingredients.
There has been an attempt throughout to give examples of each type of
calculation and in particular to give examples of cases which are prone to upset
the execution of algorithms. No doubt there are many gaps in the treatment
where the experience which is condensed into these pages has not been adequate
to guard against all the pitfalls that confront the problem solver. The process of
learning is continuous, as much for the teacher as the taught. Therefore, the user
of this work is advised to think for him/herself and to use his/her own knowledge and
familiarity of particular problems as much as possible. There is, after all, barely a
working career of experience with automatic computation and it should not seem
surprising that satisfactory methods do not exist as yet for many problems. Through-
out the sections which follow, this underlying novelty of the art of solving numerical
problems by automatic algorithms finds expression in a conservative design policy.
Reliability is given priority over speed and, from the title of the work, space
requirements for both the programs and the data are kept low.
Despite this policy, it must be mentioned immediately and with some
emphasis that the algorithms may prove to be surprisingly efficient from a
cost-of-running point of view. In two separate cases where explicit comparisons
were made, programs using the algorithms presented in this book cost less to
run than their large-machine counterparts. Other tests of execution times for
algebraic eigenvalue problems, roots of a function of one variable and function
minimisation showed that the eigenvalue algorithms were by and large ‘slower’
than those recommended for use on large machines, while the other test problems
were solved with notable efficiency by the compact algorithms. That ‘small’
programs may be more frugal than larger, supposedly more efficient, ones based
on different algorithms to do the same job has at least some foundation in the way
today’s computers work.
Since the first edition of this work appeared, a large number and variety of
inexpensive computing machines have appeared. Often termed the ‘microcomputer
revolution’, the widespread availability of computing power in forms as diverse as
programmable calculators to desktop workstations has increased the need for
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