Question

Write a MATLAB function cobweb that plots a cobweb diagram to visualise the iterations per-
formed by an iterative method of the form

xn+1 = g(xn).

the cobweb function should take as its inputs a function handle g of a function that implements
g, an initial estimate x0, the number of iterations to perform numits, values xmin, xmax, ymin
and ymax, and a one dimensional array ud. The function g may be undefined or discontinuous
at a finite number of points between xmin and xmax. The first entry of ud will contain the
number of points between xmin and xmax at which g is undefined or discontinuous. The
remaining entries of ud will contain these points in increasing order. The function should
plot y = x in red and y = g(x) in blue. The horizontal axis should be labelled x and the
vertical axis should be labelled y. A legend which identifies y=x and y=g(x) should be on
your plot. The axes should then be adjusted so that the horizontal axis goes from xmin to
xmax and the vertical axis goes from ymin to ymax. Henceforth, the range of the axes and
the legend should not change. Your cobweb function should then plot vertical and horizontal
lines in black to visualise the iterations performed by the iterative method
xn+1 = g(xn).
Before plotting each horizontal and vertical line your function should wait for 1 second. The
function should not print anything in the command window.

Answer 1

function cobweb(g,x0,numits,xmin,xmax,ymin,ymax,ud)
figure

x = x0;
for i=1:numits

safe = false;

if isempty(ud)||numel(ud)<2||ud(1)==0
safe = true;   
else
% checking x in ud
num_ud = ud(1);
values_ud = ud(2:end);
for j = 1:num_ud
if isempty(find(values_ud==x))
safe = true;
end
end

end

if safe % x is not in ud
  
plot(x,x, 'o r') % plotting y = x
hold on
plot(x,g(x),'o b') % plotting y = g(x)

% getting new x
x = g(x);

hold on

else % x is in ud
% try getting out of discontinuity by incrementing x slightly
x = x+0.001;
end

hold on
end
legend('y = x','y = g(x)','location','best')
xlim([xmin xmax])
ylim([ymin ymax])
xlabel('x')
ylabel('y')

% showing iterations atarting from initial value of x
x = x0;
hold on

for i=1:numits
  
safe = false;

if isempty(ud)||numel(ud)<2||ud(1)==0
safe = true;   
else
% checking x in ud
num_ud = ud(1);
values_ud = ud(2:end);
for j = 1:num_ud
if isempty(find(values_ud==x))
safe = true;
end
end

end

if safe   

plot( [x x], [x g(x)], 'k') % plotting vertical line
pause(1)
hold on
plot( [x g(x)], [g(x) g(x)], 'k') %plotting horizontal line
x = g(x);
hold on

else
% try getting out of discontinuity by incrementing x slightly
x = x+0.001;
end

end

hold off
end

----------------------------------------------------------------

%driver file

close all
clc
clear

g = @(x) (x+x.^2)./((x-2).*(x+2));
%g = @(x) cos(x)
x0 = -2.5;
numits = 6;
xmin = -4; xmax = 5;
ymin = -4 ; ymax = 3;

ud = [2,-2,2]; % 2 discontinuties at x = -2 and at x = 2
cobweb(g,x0,numits,xmin,xmax,ymin,ymax,ud)
title('cobweb for (x+x^2)/((x-2)*(x+2))')
%}

%{
% getting cobweb for another function
g = @(x) cos(x);
x0 = 0;
numits = 10;
xmin = 0; xmax = 1;
ymin = 0.4 ; ymax = 1;
ud = [];
%figure
cobweb(g,x0,numits,xmin,xmax,ymin,ymax,ud)
title('cobweb for cos(x)')

%}

cobweb for (x+x2)/((x-2)(x+2)) O y=g(x) -1 -2 -4 3 210 1 3 4 5

cobweb for cos(x) 10 O y g(x) 0.9 0.8 0.7 0.6 0.5 0.4 0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 0.9 1

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