Computer Aided Engineering Design

346 COMPUTER AIDED ENGINEERING DESIGN

h(x)

x

x h(x)

α α

(a) | h′(xi) | < 1 (b) | h′(xi) | > 1

Figure 12.6 Convergence issues in the method of successive substitution

As an alternative relation for successive substitutions, this method can be used for a suitable value
ofk in case the original relation fails to converge. Here the condition for convergence is
|(1 – k) + kh′(x) | < 1 in the neighborhood of the root.

Example 12.4. We determine the root of e–x – x = 0 using the method of successive substitutions with
results shown below. Note that |h′(x)| = e–x < 1 for x> 0.

xe–x

2.00 0.14

0.14 0.87

0.87 0.42

0.42 0.66

0.66 0.52

0.52 0.60

0.60 0.55

0.55 0.58

0.58 0.56

0.56 0.57

0.57 0.57

(b) Newton-Raphson method
Ifg(x) is the function for which the zero is to be determined, from the Taylor series expansion about
the guess xi, we have

g(xi+1) = g(xi) + g′(xi)(xi+1 – xi) +^1 / 2 g′′(ξ)(xi+1 – xi)^2

whereξ∈ [xi,xi+1]. An approximate value of xi+1 is obtained by considering the Taylor series up to
the first derivative and treating it at the root in which case g(xi+1) = 0. Thus

0 = g(xi) + g′(xi)(xi+1 – xi)

or xx

gx

i i gx

i

i

+1 = –

()

′()

(12.6)

Graphically, the above Newton-Raphson relation may be interpreted as shown in Figure 12.7.