Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

0. 20. 4 0. 6 0. 81. 0 1. 2 1. 4 1. 6 1. 8

− 4

− 2

0

2

4

6

8

10

12

14

x

f(x)

f(x)=x^5 − 2 x^2 − 3

Figure 27.1 A graph of the functionf(x)=x^5 − 2 x^2 −3forxin the range

0 ≤x≤ 1 .9.

Examples of the types of equation mentioned are the quartic equation,

ax^4 +bx+c=0,

and the transcendental equation,

x−3 tanhx=0.

The latter type is characterised by the fact that it contains, in effect, a polynomial

of infinite order on the LHS.

We will discuss four methods that, in various circumstances, can be used to

obtain the real roots of equations of the above types. In all cases we will take as

the specific equation to be solved the fifth-order polynomial equation

f(x)≡x^5 − 2 x^2 −3=0. (27.1)

The reasons for using the same equation each time were discussed in the intro-

duction to this chapter.

For future reference, and so that the reader may follow some of the calculations

leading to the evaluation of the real root of (27.1), a graph off(x) in the range

0 ≤x≤ 1 .9 is shown in figure 27.1.

Equation (27.1) is one for which no solution can be found in closed form, that

is in the formx=a,whereadoes not explicitly containx. The general scheme to

be employed will be an iterative one in which successive approximations to a real

root of (27.1) will be obtained, each approximation, it is to be hoped, being better

than the preceding one; certainly, we require that the approximations converge

and that they have as their limit the sought-for root. Let us denote the required