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or
k ln.ı/
ln."/;
(4.106)
so we need the number of iterations to satisfy
4
ln."/
ln.ı/
:
k
(4.107)
Theestimatein(
4.107
) states that the number of iterations grows as " is reduced.
This is very reasonable: We need to work harder in order to get a more accurate
approximation. But it is equally important to note that the number of iterations also
increases as ı
!
1.Infact,asmallerı gives faster convergence. This is also clear
from the basic estimate (
4.103
).
4.5.3
Existence and Uniqueness of a Solution
Given an equation of the form
v
D
h.
v
/;
(4.108)
it is reasonable to ask the following questions:
(a) Does there exist a value
v
such that
v
D
h.
v
/‹
(b) If so, is
v
unique?
(c) How can we compute
v
?
We now assume that h is a contractive mapping on a closed interval I such that
j
h.
v
/
h.
w
/
j
ı
j
v
w
j
;
where 0<ı<1;
(4.109)
v
2
I
)
h.
v
/
2
I
(4.110)
for all
v
,
w
. As noted above, we have a straightforward way of computing approxi-
mations of
v
. We know that fixed-point iteration
v
kC1
D
h.
v
k
/
(4.111)
4
We assume that 0<ı<<1and thus ln.ı/ < 0.
