Richardson Extrapolation Suppose that $$ L=\lim_{h\rightarrow 0}f(h) $$ and that $$L-f(h)=c_{6}h^{6}+c_{9}h^{9}+\cdots $$ What combination of f(h) and f(h/2) should be the best estimate of L ?

$L-f(h)=ah^6+bh^9$

$L-f(h/2)=a(h/2)^6+b(h/2)^9$

Now cancel the $a$ term between the two

$[L-f(h)]-64[L-f(h/2)]=bh^9-64b(h/2)^9$

$64f(h/2)-f(h)-63L=7b/8h^9$

$L=(64f(h/2)-f(h))/63-b/72h^9$

Hence

the best approximation for $L$ from this step of Richardson is $(64f(h/2)-f(h))/63$.

You can see that if the first exponent is $n$ (in place of $6$) then $\displaystyle L\approx {2^nf(h/2)-f(h) \over 2^n-1}$