Richardson extrapolation

Richardson extrapolation enables accurate limits of sequences to be achieved more accurately and quickly. We are trying to estimate some constant $A^{\ast }$ from a sequence $A_N$ or $A_h$ where $N\to \mathrm{\infty }$ or $h\to 0$. We hope that as the limit is approached, the error goes to zero:

$$A^{\ast }=A_N+\mathrm{\Delta }{A}_N=A_h+\mathrm{\Delta }{A}_h$$

Further, we might hope that the behaviour is a simple power of the parameter, negative in the case of $N$, positive for $h$:

$$A^{\ast }=A_N+\frac{a_N}{N^{\alpha }}=A_h+b_h{h}^{\beta }$$

If the powers have been chosen correctly the multiplicative factors should tend towards constants in the appropriate limit. We can estimate these constants by evaluating the $A$’s appropriately, for example:

$$A^{\ast }=A_N+\frac{a^{\ast }}{N^{\alpha }}=A_{2N}+\frac{a^{\ast }}{{\left(2N\right)}^{\alpha }}\Rightarrow a^{\ast }\approx N^{\alpha }\frac{A_{2N}-A_N}{1-(1/2{)}^{\alpha }}\Rightarrow A^{\ast }\approx \frac{A_{2N}-A_N/2^{\alpha }}{1-(1/2{)}^{\alpha }}=\frac{2^{\alpha }{A}_{2N}-A_N}{2^{\alpha }-1}$$

A similar calculation gives:

$$A^{\ast }\approx \frac{2^{\beta }{A}_{h/2}-A_h}{2^{\beta }-1}$$

We are looking to improve the estimate to $e$ given by:

$$e_N={(1+\frac{1}{N})}^N$$

On the page Basic analysis, the limit as $\mathrm{\Delta }x=1/N\to 0$ appears to be linear, so here $\alpha ,\beta =1$. So:

$$e_N^{(1)}=\frac{2e_{2N}-e_N}{2-1}=2e_{2N}-e_N$$

should give us an improvement.

Moving to negative $\mathrm{\Delta }x,N$:

Both Richardson extrapolations seem to converge from below, and may suggest a quadratic (power 2) error. They both seem to be approaching a value around 2.718281... As a reminder Javascript in your browser gives:

Let us assume a quadratic error in $e_N^{(1)}$:

$$e_N^{(2)}=\frac{2^2{e}_{2N}^{(1)}-e_N^{(1)}}{2^2-1}$$

Now the estimate seems to be between 2.7182818283 and 2.7182818286. Some further investigation can be made using a simple average of the even-order extrapolations for $e_N$ and $e_{-N}$ with error of the two values from the average:

With the smallest $N$ of $2^1$, one finds the error reduces to 6.661338147750939e-14 when the order is 8, but then starts increasing, and even becoming erratic with going negative. This is indicative of an unreliable result. The best result at this level is therefore 2.718281828459084. Only the last two figures are off from the Javascript version. I was not able to improve on this value by increasing $N$ or by varying the order. Even orders are preferred since the error should then have an odd power, which crosses at $e$, rather than just kissing it. This means that the estimates for positive and negative $N$ bracket the value we are interested in.