Apply Richardson extrapolation to compute a third-order approximation for the derivative of arctan(x) at x = 0.3. Use forward differences in your first-order approximation.
1. Write all formulas symbolically in terms of f, x, h, and r:
The first column with first-order (i.e.: O(h)) approximations is:
f(x+h) - f(x)
--------------- = Df(x,h)
h
f(x+r*h) - f(x)
----------------- = Df(x,r*h)
r*h
f(x+r^2*h) - f(x)
------------------- = Df(x,r^2*h)
r^2*h
We extrapolate to obtain second-order (i.e.: O(h^2)) approximations:
Df(x,r*h) - Df(x,h)*r
----------------------- = Df(x,h,r*h)
1 - r
Df(x,r^2*h) - Df(x,r*h)*r
--------------------------- = Df(x,r*h,r^2*h)
1 - r
Extrapolating once more, we obtain the third column with the formula
for the third-order (i.e.: O(h^3)) approximation:
Df(x,r*h,r^2*h) - Df(x,h,r*h)*r^2
----------------------------------- = Df(x,h,r*h,r^2*h)
1 - r^2
2. Evaluate all the formulas above numerically, f = arctan,
x = 0.3, h = 0.1, r = 0.5.
Show the intermediate results with six decimal places.
Continue the calculations with the intermediate rounded results.
0.890496
0.904360 0.918224
0.911004 0.917648 0.917456