Computer Aided Engineering Design

340 COMPUTER AIDED ENGINEERING DESIGN

ofg′(x)≡ f(x) = 0. The roots may be none, one or many and multiple as well. The methods devised
to find these roots may depend on their capabilities to find single or all roots (multiple or distinct) at
a time. They may also be limited to a class of functions, that is, whether they are applicable to only
polynomials, or any generic function. The conventional approach is to plot f(x) and obtain the
value(s) of x where it intersects the x axis. A drawback of the plot-and-find method is that the values
obtained are not usually very accurate. However, as a quick check, the graphical technique can be
employed to determine the initial guesses for many computational procedures. Consider, for instance,
a plot of f(x) = x^3 – 4x + 3 in Figure 12.2. We may note that at xl, the value of x to the left of any root
xr, and at xu, that to the right of xr, the function value changes sign, that is, f(xl) f(xu) < 0. Many
bracketing methods discussed next are based on this observation.

f(x) f′(x) = 0

f′(x) = 0

x

Figure 12.1 Optimality condition for a function of a single variable

x 1

x 2

f′(x) > 0

f′(x) < 0

(^3) x

• 4

x + 3

8

6

4

2

0

–2

–4

–6

–8

–10

–12

–3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 15 2

x

Figure 12.2 A candidate plot of f(x)

xl xr xu

12.2.1 Bracketing Methods

(a) Method of Bisection
This is implemented by first assigning the lower and upper bound xl and xu such that f (xl)f (xu) < 0.
A candidate value of the root is determined as xr =^12 (xl + xu), that is, the root bisects the chosen