Editing Richardson extrapolation (section)

=== Process ===
Using the step sizes <math>h</math> and <math>h / t</math> for some constant <math>t</math>, the two formulas for <math>A^*</math> are:
{{NumBlk||<math display="block">A^* = A_0(h)+ a_0h^{k_0} + a_1h^{k_1} + a_2h^{k_2} + O(h^{k_3}) </math>|{{EquationRef|1}}}}
{{NumBlk||<math display="block">A^* = A_0\!\left(\frac{h}{t}\right) + a_0\left(\frac{h}{t}\right)^{k_0} + a_1\left(\frac{h}{t}\right)^{k_1} + a_2\left(\frac{h}{t}\right)^{k_2} + O(h^{k_3}) </math>|{{EquationRef|2}}}}

To improve our approximation from <math>O(h^{k_0})</math> to <math>O(h^{k_1})</math> by removing the first error term, we multiply {{EquationNote|2|equation 2}} by <math>t^{k_0}</math> and subtract {{EquationNote|1|equation 1}} to give us
<math display="block"> (t^{k_0}-1)A^* = \bigg[t^{k_0}A_0\left(\frac{h}{t}\right) - A_0(h)\bigg] + \bigg(t^{k_0}a_1\bigg(\frac{h}{t}\bigg)^{k_1}-a_1h^{k_1}\bigg)+ \bigg(t^{k_0}a_2\bigg(\frac{h}{t}\bigg)^{k_2}-a_2h^{k_2}\bigg) + O(h^{k_3}). </math>
This multiplication and subtraction was performed because <math display="inline">\big[t^{k_0}A_0\left(\frac{h}{t}\right) - A_0(h)\big]</math> is an <math>O(h^{k_1})</math> approximation of <math>(t^{k_0}-1)A^*</math>. We can solve our current formula for <math>A^*</math> to give
<math display="block">A^* = \frac{\bigg[t^{k_0}A_0\left(\frac{h}{t}\right) - A_0(h)\bigg]}{t^{k_0}-1}
+ \frac{\bigg(t^{k_0}a_1\bigg(\frac{h}{t}\bigg)^{k_1}-a_1h^{k_1}\bigg)}{t^{k_0}-1}
+ \frac{\bigg(t^{k_0}a_2\bigg(\frac{h}{t}\bigg)^{k_2}-a_2h^{k_2}\bigg)}{t^{k_0}-1}
+O(h^{k_3}) </math>
which can be written as <math>A^* = A_1(h)+O(h^{k_1})</math> by setting
<math display="block">A_1(h) = \frac{t^{k_0}A_0\left(\frac{h}{t}\right) - A_0(h)}{t^{k_0}-1} .</math>