Information Technology Reference
In-Depth Information
Let us choose x
0
D
1:0 and compute approximations to the solution of (
4.77
)
using the iteration
x
kC1
D
sin.x
k
=10/:
(4.78)
We get the following results:
x
0
D
1:0;
x
1
D
0:09983;
x
2
D
0:00998;
x
3
D
0:00099;
which seems to converge nicely toward x
D
0 being the solution of (
4.77
). For this
particular problem, we can understand this behavior by observing that, for small x,
3
sin.x=10/
x=10;
(4.79)
and thus, from (
4.78
),
x
kC1
x
k
=10;
(4.80)
so
x
k
.1=10/
k
;
(4.81)
which certainly converges toward zero.
4.5.1
Convergence of Fixed-Point Iterations
We have seen that an equation of the form
v
D
h.
v
/
(4.82)
can sometimes be solved by an iteration of the form
v
kC1
D
h.
v
k
/:
(4.83)
3
The Taylor series of the sine function gives
y
C
O
.y
3
/:
sin.y/
D
