Expert: Ahmed Salami - 10/28/2004

Question
using forward and central differences to estimate the first derivative of f(x) at x=1. start with step h=0.5 then reduce h until the error trend alters(record h at the minimum error for each formula). what is happening and why?
Explain why the optimal values of h in part 1 are significantly different.

if you can help me with any of this i wud be very grateful
thanx
skot

Answer
Hi Skot,
I really think you would do well by reading up the topic of numerical differentiation on a good text because i can't do all the explaining.
Forward and central differences are used to estimate the value of a derivative at a point using the values of the function in that vicinity. The distance between succesive x values chosen is referred to as the step size h.
I really can't explain the processes here, but the error in the approximation decreases as h is reduced. So for each value of h used, there is a different error recorded in the value of the derivative. Try reading about these processes.
One thing to note is that for the forward difference method, the error decreases in the same manner as h i.e when h is halved, the error is halved. But in the centre difference method, the error is quartered when h is halved i.e the error is proportional to the square of the change in h.
I hope you make something out of this. You can always get back to me.
Regards.