If an iterative formula $x_{r+1}=f(x_r)$ is to be used to solve an equation, Aitken’s method of accelerating convergence uses the initial value and the first two values obtained by the formula to calculate a better approximation than the iterative formula would produce. This can then be used as a new starting point from which to repeat the process until the required accuracy has been reached. While this is computationally intensive, it is the sort of process which spreadsheets handle very easily.

If $x_0,x_1,x_2$ are the initial value and the first two iterations and $\Delta x_r=x_{r+1}-x_r,{\Delta }^2{x}_r=\Delta x_{r+1}-\Delta x_r$ are the forward differences then $x_4=x_3-\frac{{\left(\Delta x_2\right)}^2}{{\Delta }^2{x}_1}$. More generally this will be expressed as $x_{r+1}=x_r-\frac{{\left(\Delta x_{r-1}\right)}^2}{{\Delta }^2{x}_{r-2}}$.