Romberg's method

Romberg's method generates a triangular array consisting of numerical estimates of the definite integral

$\int _{a}^{b}f(x)dx$

The method can be defined by:

$R(0,0)={\frac {1}{2}}(b-a)(f(a)+f(b))$

$R(n,0)={\frac {1}{2}}r(n-1,0)+h\sum _{k=1}^{2^{n-1}}f(a+(2k-1)h)$

$R(n,m)=R(n,m-1)+{\frac {1}{4^{m}-1}}(R(n,m-1)-R(n-1,m-1))$

where

$n\geq 1$

$m\geq 1$

$h={\frac {b-a}{2^{n}}}$

In big O notation, the error for R(n,m) is:

$O\left(h^{2^{m+1}}\right)$