Romberg's method

In numerical analysis, Romberg's method (Romberg 1955) is used to estimate the definite integral

by applying Richardson extrapolation (Richardson 1911) repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate.

The method is named after Werner Romberg () (1909–2003), who published the method in 1955.

Using

the method can be inductively defined by

or

where ${\displaystyle n\geq m\,}$ and ${\displaystyle m\geq 1\,}$. In big O notation, the error for R(n, m) is (Mysovskikh 2002):

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