# Numerical Analysis/Romberg's method

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Romberg's method approximates a definite integral by applying Richardson extrapolation to the results of either the trapezoid rule or the midpoint rule.

The initial approximations are obtained by applying either the trapezoid or midpoint rule with points. In the case of the trapezoid rule on ,

where

For , we can reduce the number of places the function is evaluated by using our previously obtained approximations instead of re-sampling. For the trapezoid rule, this improvement gives

and

Richardson's extrapolation is then applied recursively, giving

Each successive level of improvement increases the order of error term from to at the expense of doubling the number of places the function is evaluated.