Aitken Delta-Squared Acceleration

The Aitken delta-squared process is a sequence acceleration method. It transforms a slowly converging sequence into one that converges faster. It uses three consecutive terms of the sequence to extrapolate the limit, based on the assumption that the error decays roughly geometrically. This requires computing the first and second forward differences, but notably does not require prior knowledge of the geometric decay ratio.

Experiment

Observe how Aitken acceleration removes a pure geometric error term in the Aitken Delta-Squared Acceleration module:

1. Select the Single mode sequence and set the Ratio to 0.5. Notice that the green accelerated sequence instantly jumps to the exact limit (2.0) because the error is purely geometric.
2. Change the sequence to Two modes with the same ratio. The accelerated sequence is no longer exact but still converges to the limit faster than the original sequence.
3. Change the Ratio to -0.5. The original sequence oscillates around the limit, but the accelerated method still effectively damps this oscillation and converges faster.
4. Change the Ratio to 1. The denominator of the Aitken correction becomes zero, making the estimate unavailable, which demonstrates that the sequence must actually be converging (or diverging with q ≠ 1) for the method to apply safely.