Romberg's Method

Romberg's method uses extrapolation techniques to improve the answer [Press, et al., 1986]. If we let I₁ be the value of the integral obtained using interval size h = x, and I₂ be the value of I obtained when using interval size h/2, and I₀ the true value of I, then the error in a method is approximately h^m, or

replacing the by an equality (an approximation) and solving for c and I₀ gives

This process can also be used to obtain I₁, I₂,..., by halving h each time, and then calculating new estimates from each pair, calling them J₁, J₂, ..., i.e. in the formula above replace I₀ with J₁. The formulas are reapplied for each pair of J's to obtain K₁, K₂,... The process continues until the required tolerance is obtained.

Romberg's method is most useful for a low-order method (m small) because significant improvement is then possible.