clear
format compact

%%
% Consider first the function
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];
fplot(f, interval)
grid on 
hold on
fplot(@(x) f(x)+0.02, interval, 'k')
fplot(@(x) f(x)-0.02, interval, 'k')
axis equal  
xlabel('x') 
ylabel('f(x)')
title('Well-conditioned root')

%%
% The possible values for a perturbed root all lie within the interval
% where the black lines intersect 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:
axis(axis)
cla
fplot(f, interval)
grid on
hold on
fplot(@(x) f(x)+0.02, interval, 'k')
fplot(@(x) f(x)-0.02, interval, 'k')
title('Poorly conditioned root')

%%
% 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!