Comments on: Rooting Around in MATLAB – Part 3

By: Loren

Loren — Thu, 25 Jun 2009 10:43:03 +0000

Ben-

Yes, you have listed the 3rd way to factor the equation. Yi Cao mentioned it as well in a comment on the first part of this series.

–Loren

By: Ben

Ben — Thu, 25 Jun 2009 00:11:34 +0000

Loren, as you said there are many ways to write the fixed point equation. Another interesting way is to run the iterations backwards if possible. Here we can solve g1inv=@(x) (1-x).^(1/3) which converges for x0 in (0,1).

This technique is also useful for finding unstable fixed points and limit cycles of dynamical systems.

By: Loren

Loren — Wed, 24 Jun 2009 10:57:31 +0000

Matt-

Animations can definitely help. I was trying to illustrate the steps for teaching purposes.

–loren

By: matt fig

matt fig — Tue, 23 Jun 2009 22:03:58 +0000

I often find with these types of things that a simple animation is worth a thousand words. For instance:

 
g2 = @(x) 1./(x.^2+1);
fplot(g2,[.4 .9]);
hold on
straightLine = @(x) x;
fplot(straightLine, [.4 .9], 'g')
legend('g2','x','Location','SouthEast')
grid on
axis equal, axis([.4 .9 .4 .9])


xo = .5; % Initial Guess.
yo = .5;
plot(xo,yo,'*r','markersize',6)
T = title(['Root: ',num2str(yo,'%.6f')],'fonts',14,'fontw','b');
pause(.4)

for n = 1:22
    x = yo;
    y = g2(xo);
    plot(x,y,'*r','markersize',5)
    set(T,'string',['Root: ',num2str(y,'%.6f')])
    xo = x;
    yo = y;
    pause(.4)
end

hold off