The central-difference approximation to is

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Computational Physics, Ch Letting the computer do the algebra

Problem

The central-difference approximation to

f'(x)

D_h f(x) = \dfrac{f(x+h)-f(x-h)}{2h}

. Take

f(x) = \ln x

. (a) Expand

f(x+h)

and

f(x-h)

as Taylor series in

h

about

x

, keeping every term up to and including the one in

h^3

, and combine the two series to show that

D_hf(x) - f'(x) = \tfrac{1}{6}h^2 f'''(x) + \mathcal{O}(h^4).

(b) Given

f'''(x) = 2/x^3

, evaluate the coefficient of

h^2

in the error at

x=2

. (c) A ratio table built by halving

h

for

f(x)=\ln x

x=2

gives a measured coefficient (the number multiplying

h^2

in the error) of

0.0417

. State whether the symbolic result agrees, and to how many figures.