% The fractal visualization of the convergence for Newton's method

% range from which we choose complex numbers
min_re=-10;
max_re=10;
min_im=-10;
max_im=10;

% degree of the interval comminution
 n_re=300;
 n_im=300;

% Accuracy of calculations
tol=0.01;

format compact;

% matrix, in which every point will be assigned
% a number of Newton's methid iteration
R=zeros (n_re, n_im);

% maximum number of steps
max_steps=25;


% stepsize
delta_re=(max_re-min_re)/n_re;
delta_im=(max_im-min_im)/n_im;

% w-polynomial coefficents
% wp-polynomial derivative coefficents

% np w=[1,0,0,0,-1] is x^4-1

%w=[1,-2 + 3*i,4 +15*i, 164,- 27*i,-155,-135*i];
%w=[1,0,0,0,-1];
%w=[1,-4,6,-4,1];
%w=[1,0,1];
%w=[7 2 0 49];
w=ones(1,10);
%w=[-5 5 1 2 3 -3 0 4];
%w=randn(1,10);
%w=ones(1,5)+i*[-1, 2, 5, -3, 4];

deg=length (w);
wp=zeros (deg-1,1);
for m=1:(deg-1)
    wp (m)=(deg-m)*w (m);
end

for j=1:n_re
    for k=1:n_im
        x=min_re+k*delta_re + i*(min_im + j*delta_im);
        if x==0
            x=tol;
        end
        % l-number of iterations from range [0,max_steps]
        l=0;
        flag=0;
        while flag==0
            % Newton's algorithm
            poprawka=horner11 (w,x)/horner11(wp,x);
            x=x-poprawka;
            if norm (poprawka)<=tol
                flag=1;
            end
            % number of steps
            if l>max_steps
                flag=1;
            end
            l=l+1;
        end
        % number attribution from range [0,max_steps]
        %assigning colour according to number of steps
        R (j, k)=l;
    end
end

% coloristic map
%colormap (hot);
colormap ('default');
brighten (0.5);

% function displays matrix R as an image
% value of point (i,j) is the color of this point on the graph
imagesc (R);
colorbar;

% turning off the backround, signatures etc.
axis off;