Romberg integration

Method

The zeroth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2 + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2 + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2 + 1 points. The further extrapolations differ from Newton-Cotes formulas. In particular further Romberg extrapolations expand on Boole's rule in very slight ways, modifying weights into ratios similar as in Boole's rule. In contrast, further Newton-Cotes methods produce increasingly differing weights, eventually leading to large positive and negative weights. This is indicative of how large degree interpolating polynomial Newton-Cotes methods fail to converge for many integrals, while Romberg integration is more stable.