0Introduction
IB Numerical Analysis
0 Introduction
Numerical analysis is the study of algorithms. There are many problems we
would like algorithms to solve. In this course, we will tackle the problems of
polynomial approximation, solving ODEs and solving linear equations. These
are all problems that frequently arise when we do (applied) maths.
In general, there are two things we are concerned with — accuracy and speed.
Accuracy is particularly important in the cases of polynomial approximation and
solving ODEs, since we are trying approximate things. We would like to make
good approximations of the solution with relatively little work. On the other
hand, in the case of solving linear equations, we are more concerned with speed
— our solutions will be exact (up to numerical errors due to finite precision of
calculations), but we would like to solve it quickly. We might have to deal with
huge systems, and we don’t want the computation time to grow too quickly.
In the past, this was an important subject, since they had no computers.
The algorithms had to be implemented by hand. It was thus very important to
find some practical and efficient methods of computing things, or else it would
take them forever to calculate what they wanted. So we wanted quick algorithms
that give reasonably accurate results.
Nowadays, this is still an important subject. While we have computers that
are much faster at computation, we still want our programs to be fast. We would
also want to get really accurate results, since we might be using them to, say,
send our rocket to the Moon. Moreover, with more computational power, we
might sacrifice efficiency for some other desirable properties. For example, if we
are solving for the trajectory of a particle, we might want the solution to satisfy
the conservation of energy. This would require some much more complicated and
slower algorithms that no one would have considered in the past. Nowadays, with
computers, these algorithms become more feasible, and are becoming increasingly
more popular.