Small Solver Program

[Albert Chan]

(02-15-2019 06:26 PM)Thomas Klemm Wrote:  

(02-15-2019 12:07 AM)Albert Chan Wrote:  It would be nice if we can temper the oscillation, or slow convergence.

We can also use Aitken's delta-squared process to accelerate the speed of convergence:

\(a_{n}=x_{n+2}-\frac {(x_{n+2}-x_{n+1})^{2}}{(x_{n+2}-x_{n+1})-(x_{n+1}-x_{n})}\)

Amazingly, my rate formula is same as Aitken extrapolation formula !
Assuming we have 3 values, x0,x1,x2 and tried to guess where it should end up.

My rate analysis: r = (x2-x1)/(x1-x0) = Δx1 / Δx0

We need this for the proof:
(Δx0)²
= ((Δx0 - Δx1) + Δx1)²
= (Δx0 - Δx1)² + 2 * Δx1 (Δx0 - Δx1) + (Δx1)²

If same trend continue, where it will ends up
= x0 + Δx0 * 1/(1-r)
= x0 + (Δx0)² / (Δx0 - Δx1)
= x0 + (Δx0 - Δx1) + 2 * Δx1 + (Δx1)² / (Δx0 - Δx1)
= x0 + (x1-x0) + (x1-x2) + 2*(x2-x1) − (Δx1)² / (Δx1 - Δx0)
= x2 − (Δx1)² / (Δx1 - Δx0)