Information Technology Reference
In-Depth Information
Let us consider generally the fixed-point iteration
v
kC1
D
h.
v
k
/
(4.91)
as a method for finding
v
such that
v
D
h.
v
/:
(4.92)
Here we assume that h is a contractive mapping, i.e., we assume that for any
v
;
w
in
aclosedintervalI we have
j
h.
v
/
h.
w
/
j
ı
j
v
w
j
;
where 0<ı<1;
(4.93)
v
2
I
)
h.
v
/
2
I:
(4.94)
We define the error
e
k
D j
v
k
v
j
(4.95)
and observe that
j
v
kC1
v
j
D
e
kC1
j
h.
v
k
/
h.
v
/
j
(
4.91
),(
4.92
)
D
ı
j
v
k
v
j
(
4.93
)
ıe
k
:
(
4.95
)
D
It now follows by induction on k,that
e
k
ı
k
e
0
;
(4.96)
and since 0<ı<1, we know that e
k
!
0 as k
!1
and thus we have
convergence:
v
k
D
v
:
lim
k!1
We have seen that if
v
is the solution of an equation of the form
v
D
h.
v
/;
then the iteration
v
nC1
D
h.
v
n
/
