Using Newton's Method to plot basin of attraction?

Hi all,

So I have this question and these two codes: "The basin of attraction for a particular root p s the set of all numbers that, if used as initial guesses in Newton’s method, will cause the algorithm to converge to p. The polynomial f(x) = x 3 − 2x 2 − 11x + 12 has roots 4, −3, and 1. Write a function that creates an array x of 1, 000 initial guesses on the interval [−3, 4] and runs Newton’s method with tolerance 10−8 for each initial guess (remember to comment out any plot commands or print to screen commands in your newton.m or you may have 1000 plots come up). Store the root for each initial guess in the array p, and plot p against x (use a ylim([-5 5]) command to set the y axis). From your plot, what, approximately, is the basin of attraction for the root 1?"

This is my code for this assignment:

x = linspace(-3,4,1000);
tol = 10^-8;
f = @(x) (x^3-2*x^2-11*x+12);
df = @(x) (3*x^2-4*x-11);
p = zeros(length(x));
for j=-3:length(x)
  p(j) = newton(p(j),tol,f,df);
end
ylim([-5,5]);
plot(x,p)

and here is my Newton Method code which works:

function [root,count] = newton(p0,tol,f,df)
dist = tol+1;
count = 0;
while dist>tol
  if df(p0)==0
      disp('root not found');
      return
  end 
  p1 = p0-(f(p0)/df(p0));
  dist = abs(p1-p0);
  p0 = p1;
  count = count + 1;
end
root = p0;
count

My basinofattraction code does not work. It does not plot nor does it run through the loop. What am I missing?

Thanks!

Answer 1

[Matt J]

You do not appear to be using 'x' anywhere in the root calculations.

Also, starting your loop at j=-3 is probably a typo.

[Erin W]

I'm confused as to what you mean by I'm not using x anywhere in the root calculations. My function newton.m accepts a f(x) and a df(x) (the derivative of f(x)) in the forms of x...

Also, should I start my loop at 1 and extend it to length(x)?

[Matt J]

The first line of your code defines a series of initial guesses at which to start Newton's method,

x = linspace(-3,4,1000);

But you never use this data anywhere later in the code, as far as I can see. Aren't you meant to be looping over these x(j) and doing something with them?

[Erin W]

Oh yea you're right. I guess I don't know what to do with it but I know I'm supposed to have it.

[Matt J]

Well, maybe you should consider again this line from your assignment,

Write a function that creates an array x of 1, 000 initial guesses on the interval [−3, 4] and runs Newton’s method with tolerance 10−8 for each initial guess

[Erin W]

So I need to loop my function through each of these guesses and store them in a vector p and then plot p against x. I'm not sure why but for some reason I cannot think of how to do this. I understand it conceptually but the code is bogging me down.

[Erin W]

Oh nevermind. I had it. I had one small problem. No worries!

[Christina Reed]

Erin W, what did you do to get your code working? Doing similar problem but with for loops.