27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
errors introduced as a result of approximations made in setting up the numerical
procedures (truncation errors). For this scale of application, books specifically
devoted to numerical analysis, data analysis and computer programming should
be consulted.
So far as is possible, the method of presentation here is that of indicatingand discussing in a qualitative way the main steps in the procedure, and then
of following this with an elementary worked example. The examples have been
restricted in complexity to a level at which they can be carried out with a pocket
calculator. Naturally it will not be possible for the student to check all the
numerical values presented, unless he or she has a programmable calculator or
computer readily available, and even then it might be tedious to do so. However,
it is advisable to check the initial step and at least one step in the middle of
each repetitive calculation given in the text, so that how the symbolic equations
are used with actual numbers is understood. Clearly the intermediate step should
be chosen to be at a point in the calculation at which the changes are still
sufficiently large that they can be detected by whatever calculating device is
used.
Where alternative methods for solving the same type of problem are discussed,for example in finding the roots of a polynomial equation, we have usually
taken the same example to illustrate each method. This could give the mistaken
impression that the methods are very restricted in applicability, but it is felt by
the authors that using the same examples repeatedly has sufficient advantages, in
terms of illustrating therelativecharacteristics of competing methods, to justify
doing so. Once the principles are clear, little is to be gained by using new
examples each time, and, in fact, having some prior knowledge of the ‘correct
answer’ should allow the reader to judge the efficiency and dangers of particular
methods as the successive steps are followed through.
One other point remains to be mentioned. Here, in contrast with every otherchapter of this book, the value of a large selection of exercises is not clear cut.
The reader with sufficient computing resources to tackle them can easily devise
algebraic or differential equations to be solved, or functions to be integrated
(which perhaps have arisen in other contexts). Further, the solutions of these
problems will be self-checking, for the most part. Consequently, although a
number of exercises are included, no attempt has been made to test the full range
of ideas treated in this chapter.
27.1 Algebraic and transcendental equationsThe problem of finding the real roots of an equation of the formf(x)=0,where
f(x) is an algebraic or transcendental function ofx, is one that can sometimes
be treated numerically, even if explicit solutions in closed form are not feasible.