Iteration method

Question

I am working on finding the initial points of convergence using newton's iteration method in mathematica. newton function works now I would like to show which initial points from a grid produce Newton iterations that converge to -1, same for points that converge to (1 + (3)^1/2)/2i, given that:

f(x) = x^3+1

newton[x0_] := (
  x = x0;
  a1 = {};
  b1 = {};
  c1 = {};
  counter = 0;
  error  = Abs[f[x]];
  While[counter < 20 && error > 0.0001,
   If[f'[x] != 0, x = x - N[f[x]/f'[x]]];
   counter = counter + 1;
   error = Abs[f[x]]];
  x)

I created a grid to show which initial points of a+bi converge to the roots.

grid = Table[a + b I, {a, -2, 2, 0.01}, {b, -2, 2, 0.01}];

Then I created a fractal, but whenever I plot it gives me a blank graph on the axis. There's got to be a way for me to be able to identify the converge points from the grid but so far I have not been successful. I tried using the Which[] method but when comparing the value its returns false. Any help will appreciate it

show the plot command you tried that doesnt work – agentp Feb 24 '17 at 11:53 — agentp, Feb 24 '17 at 11:53
have you tried plotting the real part? – agentp Feb 25 '17 at 13:22 — agentp, Feb 25 '17 at 13:22

score 2 · Answer 1 · answered Feb 24 '17 at 06:05

Your code is not optimal, to put it mildly, but to give you a head start, why don't you start with something like this:

f[x_] := x^3 + 1;

newton[x0_] := (x = x0;
  a1 = {};
  b1 = {};
  c1 = {};
  counter = 0;
  error = Abs[f[x]];
  While[counter < 20 && error > 0.0001, 
   If[f'[x] != 0, x = x - N[f[x]/f'[x]]];
   counter = counter + 1;
   error = Abs[f[x]]];
  {x, counter})

Table[Re@newton[a + b I], {a, -2, 2, 0.01}, {b, -2, 2, 0.01}] // Image

Mathematica graphics

Iteration method

1 Answers1