function newtonx3
% Roots of the equation x^3 - 1 = 0
r1 = 1.0;
r2 = 0.5*(-1+i*sqrt(3));
r3 = 0.5*(-1-i*sqrt(3));
% Tolerance
tol = 1.0E-08;

re=linspace(-2,2,200);
im=linspace(-2,2,200);
[RE,IM]=meshgrid(re,im);
C = zeros(size(RE));
for i1=1:size(RE,1),
    for i2=1:size(IM,2),
        r=newtonsolve(RE(i1,i2)+i*IM(i1,i2));
        if abs(r-r1) < tol,
            C(i1,i2) = 1;
        end
        if abs(r-r2) < tol,
            C(i1,i2) = 2;
        end
        if abs(r-r3) < tol,
            C(i1,i2) = 3;
        end        
    end
end
imagesc(re,im,C,[0,3]);
colormap('jet(4)'); colorbar;
hold on;
plot(real(r1),imag(r1),'xk','LineWidth',2,'MarkerSize',10);
plot(real(r2),imag(r2),'xk','LineWidth',2,'MarkerSize',10);
plot(real(r3),imag(r3),'xk','LineWidth',2,'MarkerSize',10);
hold off;
xlabel('real');  ylabel('imag');

% solve equation x^3 - 1 = 0 using Newton's method
function [x]=newtonsolve(x)
N = 100;
tol = 1.0E-10;
for i=1:N,
    rk = x^3-1;
    if abs(rk)<tol,
        break;
    end
    gk = 3*x^2;
    x  = x - rk/gk;
    if ~isfinite(x)
        break;
    end
end