27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
1. 0
1. 0 1. 0
1. 0 1. 2
1. 2 1. 2
1. 2 1. 4
1. 4 1. 4
1. 4 1. 6
1. 6 1. 6
1. 6
− 4
− 4 − 4
− 4
− 2
− 2 − 2
− 2
2
2 2
2
4
4 4
4
6
6 6
6
x 1x 1x 2x 2
x 3x 3x 1 x 4x (^2) x 2 x 1
x 3
x 3
ξ
ξ
ξ
ξ
(a) (b)
(c) (d)
Figure 27.2 Graphical illustrations of the iteration methods discussed in
the text: (a) rearrangement; (b) linear interpolation; (c) binary chopping;
(d) Newton–Raphson.
withn=1.Next,f(x 1 ) is evaluated and the process repeated after replacing
eitherA 1 orB 1 byx 1 , according to whetherf(x 1 ) has the same sign asf(A 1 )or
f(B 1 ), respectively. In figure 27.2(b),A 1 is the one replaced.
As can be seen in the particular example that we are considering, with this
method there is a tendency, if the curvature off(x) is of constant sign near
the root, for one of the two ends of the successive chords to remain un-
changed.
Starting with the initial valuesA 1 = 1 andB 1 =1.7, the results of the first
five iterations using (27.8) are given in table 27.2 and indicated in graph (b) of
figure 27.2. As with the rearrangement method, the improvement in accuracy,
as measured byf(xn)andxn−ξ, is a fairly constant factor at each iteration
(approximately 3 in this case), and for our particular example there is little to
choose between the two. Both tend to their limiting value ofξmonotonically,
from either higher or lower values, and this makes it difficult to estimate limits
within whichξcan safely be presumed to lie. The next method to be described
gives at any stage a range of values within whichξisknownto lie.