Romberg's method

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

<math> \int_a^b f(x) \, dx. </math>

by using Richardson extrapolation repeatedly on the trapezium rule.

The method can be defined inductively in this way:

<math>R(0,0) = \frac{1}{2} (b-a) (f(a) + f(b))</math>

<math>R(n,0) = \frac{1}{2} R(n-1,0) + h\sum_{k=1}^{2^{n-1}} f(a + (2k-1)h)</math>

<math>R(n,m) = R(n,m-1) + \frac{1}{4^m-1} ( 4^m R(n,m-1) - R(n-1,m-1))</math>

where

<math> n \ge 1 </math>

<math> m \ge 1 </math>

<math> h = \frac{b-a}{2^n}. </math>

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

<math> O\left(h^{2^{m+1}}\right). </math>

External links

• ROMBINT -- code for MATLAB (author: Martin Kacenak)de: Romberg-Integration

pl: Metoda Romberga