Consider the linearly convergent sequence where the first 5

Consider the linearly convergent sequence where the first 5 terms are listed below: Compute the first three terms p_0, p_1, p_2 using Aitken\'s method. This sequence was actually generated using the following formula: where h = 2^-n. Use Richardson extrapolation to obtain a O(h^4) estimate using the above data.

Solution

Aitken\'s

Method uses ^
pn=pn-(pn+1-pn)^2/pn+2-2pn+1+pn

so p0=p0-(p1+1-p0)^2/p2-2p1+1+p0
=4.67-(3.52+1-4.67)^2/3.08-2*3.52+1+4.67
=4.67-0.0225/3.08-7.04+1+4.67
=4.64/1.71
=2.713

Similarly P1 and P2 can be calculated