The central symmetric finite difference estimate of order 2 for the first deriva

Question

The central symmetric finite difference estimate of order 2 for the first derivative is given by:

f'(a) = [f(a+h)-f(a-h)] / 2h  + O(h^2)

The error denoted by O(h^2) is bounded in absolute value by Kh^2 for some constant K>0 when h is small. First prove that if  | f(x) | <= M in a neighborhood of a, then the roundoff error in the fraction is bounded in absolute value by (u2M) / (2h) where u measures the round off error. Then show that the error is estimating f'(a) with the above method is bounded by:

g(h) = (uM) / h + Kh^2

Finally, compute the optimal value of the step size h in order to minimize g(h)

Explanation / Answer

The central symmetric finite difference estimate of order 2 for the first deriva

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