Writing
Fun maths Friday - Newton's method and chaos
Constructing a pathological function for Newton's method using the logistic map in chaos mode.
Fun maths Friday!
I was preparing a lecture for next week where I will be introducing Newton's method. I managed to completely distract myself by thinking, "what is the most pathological function I could think of for poor convergence?"
It occured to me that if I construct the iterations from a known chaotic map I might get something interesting.
What I landed on was the logistic map with r =4, so in full Feigenbaum chaos mode, i.e.
xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) = 4xₙ(1 − xₙ)
which you can solve for f to give the function
f(x) = ∛((4x − 3)/x)
which is what is shown in the animation. It has a real root in the interval but Newton iterations will keep missing with a chaotic pattern for each step. Too fun not to share!
