Editing Richardson extrapolation (section)

=== Properties ===
The Richardson extrapolation can be considered as a linear [[sequence transformation]].

Additionally, the general formula can be used to estimate <math>k_0</math> (leading order step size behavior of [[Truncation error]]) when neither its value nor <math>A^*</math> is known ''a priori''.  Such a technique can be useful for quantifying an unknown [[rate of convergence]].  Given approximations of <math>A^*</math> from three distinct step sizes <math>h</math>, <math>h / t</math>, and <math>h / s</math>, the exact relationship<math display="block">A^*=\frac{t^{k_0}A_i\left(\frac{h}{t}\right) - A_i(h)}{t^{k_0}-1} + O(h^{k_1}) = \frac{s^{k_0}A_i\left(\frac{h}{s}\right) - A_i(h)}{s^{k_0}-1} + O(h^{k_1})</math>yields an approximate relationship (please note that the notation here may cause a bit of confusion, the two O appearing in the equation above only indicates the leading order step size behavior but their explicit forms are different and hence cancelling out of the two {{math|''O''}} terms is only approximately valid)
<math display="block">A_i\left(\frac{h}{t}\right) + \frac{A_i\left(\frac{h}{t}\right) - A_i(h)}{t^{k_0}-1} \approx A_i\left(\frac{h}{s}\right) +\frac{A_i\left(\frac{h}{s}\right) - A_i(h)}{s^{k_0}-1}</math>
which can be solved numerically to estimate <math>k_0</math> for some arbitrary valid choices of <math>h</math>, <math>s</math>, and <math>t</math>.

As <math>t \neq 1</math>, if <math>t>0</math> and <math>s</math> is chosen so that <math>s = t^2</math>, this approximate relation reduces to a quadratic equation in <math>t^{k_0}</math>, which is readily solved for <math>k_0</math> in terms of <math>h</math> and <math>t</math>.