Aitken's Method
The linear convergence of the iterative methods can be improved
with the help of the Aitken's method. The error in two successive approximations
for linear convergence methods (for details please see Order
of Convergence) can be written as
ei+1 = Aei
ei+2 = Aei+1
elimination of A gives
|
(-s + xi+2) =
|
(-s + xi+1)2
|
|
| (-s + xi) |
(s2 - s(xi+xi+2)
+ xixi+2) = s2 - 2sxi+1+ xi+12
| ®
-s = |
xi+12 - xixi+2 |
|
| xi+ xi+2 - 2xi+1 |
| ®
s = xi - |
( xi+1 - xi )2 |
|
| xi+ xi+2 - 2xi+1 |
s gives an improved value of the approximation xi+2
|
Algorithm - Aitken's Method
Choose initial approximation x0
Do
Calculate x3i+1 and x3i+2 using any linear
iterative method
Modify x3i+2 using
| x3i+2 = x3i - |
( x3i+1 - x3i )2 |
i = 0, 1, 2, . . . |
|
| x3i+ x2i+2 - 2x3i+1 |
while (none of the convergence criterion C1 or C2 is met) |
C1. Fixing apriori the total number of iterations N.
C2. By testing the condition | xi+1 - xi |
(where i is the iteration number) less than some tolerance limit, say epsilon,
fixed apriori.
Example :
Find a root of cos[x] - x * exp[x] = 0 with x0 = 0.0
Let the linear iterative process be
xi+1 = xi + 1/2(cos[xi]-
xi * exp[xi] ) i = 0, 1,
2 . . .
|
i
|
xi
|
Aitken's Modification
|
|
0
|
0.00000000
|
0.52810686 |
|
1
|
0.50000000
|
|
2
|
0.52661096
|
|
0.52810686
|
0.51770956 |
|
3
|
0.51222771
|
|
4
|
0.52059979
|
|
0.51770956
|
|
|
