Extrapolation Techniques
- The errors of a central-difference approximation to were of . In effect, suggests that the errors are proportional to although that is true only in the limit as
. Unless is quite large, we can assume the proportionality.
- So, from two computations with being half as large in the second, we can estimate the proportionality factor, . For example, in Table 2;
h |
Approximation |
0.05 |
4.15831 |
0.025 |
4.16361 |
If errors were truly , we can write two equations:
from which we can solve for the true value, eliminating the unknown constant , getting;
which is very close to the exact value for , 4.165382.
- You can easily derive the general formula for improving the estimate, when errors decrease by
|
(4) |
where more and less in the last term are the two estimates at and . More accurate is the estimate at the smaller value of and is the power of in the order of the errors.
- As example, apply this to values from Table 1 which were from forward-difference approximations. Here the errors are .
h |
Approximation |
0.05 |
4.05010 |
0.025 |
4.10955 |
Using Eq. 4, we have
which shows considerable improvement but not as good as from the central differences.
2004-12-21