Higher order corrections

If the formula can be expressed as

> restart:expr1:=Q=Appr(h)+C1*h^2+C2*h^4+O(h^6);

We can apply Richardson Extrapolation to obtain

> expr2:=subs(h=h/2,expr1);

> expr3:=Q=(4*rhs(expr2)-rhs(expr1))/3;

which can be rewritten as

> expr4:=Q=N[2](h)-C2*h^4/4+O(h^6);

with

> def:=N[2](h)=(4*Appr(h/2)-Appr(h))/3;

is the value obtained by Richardson Extrapolation. Replacing by yields

> expr5:=subs(h=h/2,expr4);

so that the combination

> expr6:=Q=(16*rhs(expr5)-rhs(expr4))/15;

has only error terms. We can define

> def2:=N[3](h)=(16*N[2](h/2)-N[2](h))/15;

One may note that this can be written using powers of 4:

> def2a:=N[3](h)=(`4^2`*N[2](h/2)-N[2](h))/(`4^2`-1);

Indeed, when the formula can be written as

,

Richardson Extrapolation can be carried on with

, .