Romberg's Extrapolation Method

Use case

It is used to improve the accuracy of a Numerical Method to a higher order by combining results from different step sizes.

Mathematically

$$\boxed{A_e=\frac{2^k\left[A(h/2)-A(h)\right]}{2^k-1}}$$

Here,

• $k\to$is the order of convergence.

Note: The $k$ is increased by two for every iteration.

Quadrature Method	Order of Convergence
Left Hand Rule	$O(h)$
Midpoint Rule	$O(h^2)$
Trapezoidal Rule	$O(h^2)$
Simpson’s Rule	$O(h^4)$