A Short Proof for Romberg Integration

Abstract

The classical proof of the Romberg method on an interval uses the EulerMaclaurin formula to derive an asymptotic expansion of the error of the composite trapezoid rule (e.g., [2]). The convergence rate of the Romberg extrapolations then follows from this expansion and the fact that it contains only even powers of N. The proof of the Euler-Maclaurin formula is elementary. But the proof is based on certain recursion properties of the Bernoulli polynomials and it is not intuitively obvious what it is that makes the Romberg method work. The convergence properties of the Romberg method can be understood by using homogeneity and symmetry principles, see e.g. [1] and the references given there. Here we want to give a simple proof which only uses these two basic principles (and Taylor's theorem). We will only derive the convergence rates of the extrapolated values based on the sequence of 1, 2, 4, 8,. .. subintervals. We do not obtain a general asymptotic expansion or formulae for the constants in the estimates. For results of this type see [2], [1] and the references given there. THE ROMBERG INTEGRATION METHOD. Let f be a continuous function on the interval [a, b], and let (f) = fabf(x) dx. The trapezoid rule on [a, b] is defined