Romberg's method

Summary

Romberg's method computes a definite integral by applying Richardson extrapolation again and again to the trapezoidal rule with step sizes $h, h/2, h/4, \dots$. It works because the error of the trapezoidal rule for a smooth function has an expansion in even powers of $h$. This expansion comes from the Euler–Maclaurin formula. Each extrapolation step removes one more term of this expansion. The results are arranged in a triangular table whose diagonal converges fast for smooth integrands.