Information Technology Reference
In-Depth Information
provides a method of solving the equation numerically, provided that h is a contrac-
tive mapping.
4.5.2
Speed of Convergence
If h is a contractive mapping on I and
v
solves
v
D
h.
v
/;
(4.97)
then we have seen that
e
e
0
ı
k
;
(4.98)
with e
k
D j
v
k
v
j
. Let us now consider the equation
x
D
sin.x=10/:
(4.99)
We have seen that
1
10
;
ı
D
(4.100)
cf. (
4.89
). Hence the error is
1
10
k
;
e
e
0
(4.101)
which is in agreement with our heuristic arguments on page
117
.
If we want
e
e
0
10
6
;
(4.102)
we need k
6 iterations. More generally, when
e
e
0
ı
k
;
(4.103)
we have
e
e
0
"
(4.104)
if
ı
k
"
(4.105)
