# Conditioning of roots¶

Consider first the function

using Plots

f  = x -> (x-1)*(x-2);


At the root $$r=1$$, we have $$f'(r)=-1$$. If the values of $$f$$ were perturbed at any point by noise of size, say, $$0.05$$, we can imagine finding the root of the function as though drawn with a thick line, whose edges we show here.

interval = [0.8,1.2]

plot(f,interval...,ribbon=0.03,legend=:none,aspect_ratio=1,
xlabel="x", yaxis=("f(x)",[-0.2,0.2],[0]), title="Well-conditioned root")
scatter!([1],[0])


The possible values for a perturbed root all lie within the interval where the ribbon intersects the $$x$$ axis. The width of that zone is about the same as the vertical distance between the lines.

By contrast, consider the function

f  = x -> (x-1)*(x-1.01);


Now $$f'(1)=-0.01$$, and the graph of $$f$$ will be much shallower near $$x=1$$. Look at the effect this has on our thick rendering:

plot(f,interval...,ribbon=0.03,legend=:none,aspect_ratio=1,
xlabel="x", yaxis=("f(x)",[-0.2,0.2],[0]), title="Poorly conditioned root")
scatter!([1],[0])


The vertical displacements in this picture are exactly the same as before. But the potential horizontal displacement of the root is much wider. In fact, if we perturb the function upward by the amount drawn here, the root disappears entirely!