Method
The method can be defined inductively
or
where
In big O notation, the error for R(n, m) is (Mysovskikh 2002):
The zeroeth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2n + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2n + 1 points. Further extrapolations differ from Newton Cote's 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.
When function evaluations are expensive, it may be preferable to replace the polynomial interpolation of Richardson with the rational interpolation proposed by Bulirsch & Stoer (1967).
Read more about this topic: Romberg's Method
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