function roots1(n,xmin,xmax,ymin,ymax)
%ROOTS1 Roots of unity, basin of attraction.
%   ROOTS1(N) depicts the basin of attraction for the Nth root of unity
%   by solving Z^N - 1 = 0 using Newton's method with initial conditions
%   in the complex square [-1,1] x [-1 1].
%
%   ROOTS1(N,XMIN,XMAX,YMIN,YMAX) depicts basin of attraction in the
%   complex rectangle [XMIN XMAX] x [YMIN YMAX].

%   Francisco J. Beron-Vera
%   $Revision: 1.0 $  $Date: 2006/09/13 10:07:44 $

if nargin<2
   xmin=-1; xmax=1;
   ymin=-1; ymax=1;
end

% Defining z^n - 1 and derivative...
f=@(z,n) z.^n-1;
dfdz=@(z,n) n*z.^(n-1);

% n roots of 1.
r=exp(2*i*[0:n-1]*pi/n);

% Initial conditions (guesses) on a dense rectangular grid.
N=250;
x=linspace(xmin,xmax,N);
y=linspace(ymin,ymax,N);
[x y]=meshgrid(x,y);
z=x+i*y;

% Computing roots from above initial guesses using Newton method...
for I=1:50
   z=z-f(z,n)./dfdz(z,n);
end

% Coloring according to attractor reached...
w=z;
z=repmat(NaN,N);
for I=1:n
   J=find(abs(w-r(I))<1e-3); 
   z(J)=repmat(I,size(J));
end

% Plotting...
pcolor(x(1,:),y(:,1),z)
shading flat
axis([xmin xmax ymin ymax])
axis squar