Derivation

Consider the approximation with step size :

> expr1:=Q=Appr(h)+C*h^2+O(h^4);

and then recalculate with half the step size:

> expr2:=Q=Appr(h/2)+C*(h/2)^2+O((h/2)^4);

Both approximations have errors of the order . We can eliminate the terms in by combining the two equations:

> expr3:=4*lhs(expr2)-lhs(expr1)=4*rhs(expr2)-rhs(expr1);

which yields the formula:

.

Therefore, two approximations generated by a formula with an error of order can be combined in a rather simple way to yield an error of the order .