Let's convert the roots of a quadratic polynomial 
to a fixed point problem.
to a fixed point problem.f = @(x) x.^2 - 4*x + 3.5;
r = roots([1 -4 3.5])
We'll define 
. Intersections of its graph with the line 
are fixed points of g and thus roots of f. (Only one is shown in the chosen plot range.)
. Intersections of its graph with the line are fixed points of g and thus roots of f. (Only one is shown in the chosen plot range.) g = @(x) x - f(x);
fplot(g,[2 3])
hold on, plot([2 3],[2 3],'k')
title('Finding a fixed point'), axis equal % ignore this line
xlabel('x'), ylabel('y') % ignore this line
If we evalaute 
, we get a value of almost 2.6.
, we get a value of almost 2.6. x = 2.1; y = g(x)
So 
is considerably closer to a fixed point than x was. The value 
ought to become our new x value! Changing the x coordinate in this way is the same as following a horizontal line over to the graph of .
is considerably closer to a fixed point than x was. The value ought to become our new x value! Changing the x coordinate in this way is the same as following a horizontal line over to the graph of .plot([x y],[y y],'-')
x = y;
Now we can compute a new value for y. We leave x alone here, so we travel along a vertical line to the graph of g.
y = g(x)
plot([x x],[x y],'-')
You see that we are in a position to repeat these steps as often as we like. Let's apply them a few times and see the result.
for k = 1:5
plot([x y],[y y],'-'), x = y; % y --> new x
y = g(x); plot([x x],[x y],'-') % g(x) --> new y
end
The process spirals in beautifully toward the fixed point we seek. Our last estimate has almost 4 accurate digits.
abs(y-r(1))/r(1)
Now let's try to find the other fixed point in the same way. We'll use 1.3 as a starting approximation.
cla
fplot(g,[1 2])
hold on, plot([1 2],[1 2],'k'), ylim([1 2])
x = 1.3; y = g(x);
for k = 1:5
plot([x y],[y y],'-'), x = y; % y --> new x
y = g(x); plot([x x],[x y],'-') % g(x) --> new y
end
title('No convergence') % ignore this line
We started near the true fixed point, which is at about 1.293. But this time, the iteration is pushing us away from the correct answer.