.We will start by showing a type of problem which can only be solved by means of numerical methods, although you may have learned earlier that analytic methods can do it. Consider the initial value problem for the ordinary, linear, homogeneous, differential equation with constant coefficients
You have been taught to write down the characteristic equation
find the roots
obtain the general solution
and proceed to the particular solution of the problem by determination of the arbitrary constants A and B from the system of linear equations
which yield
This example shows that in the general case of a linear differential equation with constant coefficients of order n you must find the roots of a polynomial of order n and solve a system of n linear equations. Obviously, such tasks can rarely be completed by guessing or simple arithmetic operations!
Moreover, if a problem demands evaluation of a solution for a given value of x, it becomes necessary to evaluate functions such as the exponential function. If the equation is non-homogeneous
evaluations of other functions may be required.
Even in the case
of a quadratic
characteristic equation for which
the roots cannot be guessed, we must use formulae involving
square roots, so that the evaluation of the solution demands
special numerical work. In the case of a cubic characteristic equation, you may turn to the rather complicated formulae of the
Italian mathematician Facio Cardan (1444 - 1524), when
again you will have to evaluate roots. You may do the same thing
in the case of a fourth
order equation, although the
formulae are even more complicated than those for the third order
characteristic equation.
For higher order
equations, you will not know where to turn.The German
mathematician Karl Friedrich Gauss (1777 - 1855) has proved that
for such equations explicit formulae are impossible. This means
that you must find the roots numerically.
Finally, your equation may involve variable coefficients, i.e., given functions of x,
when it may not even be soluble in terms of elementary or even special functions. In that case, you will have to employ numerical methods of integration.
This is not an exhaustive description of the subject of numerical analysis. However, it mentions those of its areas with which we will deal here. Before we start, however, it will be worthwhile to think over what has been said above. We have seen that numerical methods will have to be employed in many practical situations. We have not gone into details regarding what we mean by a numerical method. Nevertheless, it will be clear that a numerical solution of a problem cannot be unique. It is one of an infinite number of approximations. Hence we will always be concerned with its quality, i.e., its inherent error. In real problems of physics, engineering, economics, etc., we will then be concerned with an approximation's accuracy. Thus, we will use the error to seek better approximations until it drops below a certain value. Error analysis becomes an important part of our work.
A numerical method which leads to a required result is often referred to as an algorithm, named after al-Khowârizmî, a Persian mathematician of the 9th Century. More often than not, algorithms are iterative, i.e., they involve cycles of identical computations, starting with the results of the preceding cycle. At the end of a cycle, the result will be examined to find out whether is has the required accuracy. The algorithm will stop, when the error becomes as small as desired.
