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How can I draw the following blue color tone intensity figure using the Newtons Raphson basin of attraction code?

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Atique Khan
Atique Khan on 21 Aug 2020
Commented: J. Alex Lee on 24 Aug 2020
The figure below shows Newtons Raphson Iteration to achieve the desired accuracy along x axis using the intensity of blue color tone . How can I make such figure I have the following code for achieving the basin of attraction. Please also guide that how can I find that how many iterations the program is doing to reach a certain attractor for each initial condition.
clc ; clear all
warning('off') % To off the warning which shows "Matrix is close to singular
% badly scaled" when algorithm passes through a point where the Jacobian
% matrix is singular
% The roots of the given governing equations
r1 = [-1 ;-1] ;
r2 = [0 ;0] ;
r3 = [1 ;1] ;
% Initial conditions
x = linspace(-2,2,200) ;
y = linspace(-2,2,200) ;
% Initialize the required matrices
Xr1 = [] ; Xr2 = [] ; Xr3 = [] ; Xr4 = [] ;
tic
for i = 1:length(x)
for j = 1:length(y)
X0 = [x(i);y(j)] ;
% Solve the system of Equations using Newton's Method
X = NewtonRaphson(X0) ;
% Locating the initial conditions according to error
if norm(X-r1)<1e-8
Xr1 = [X0 Xr1] ;
elseif norm(X-r2)<1e-8
Xr2 = [X0 Xr2] ;
elseif norm(X-r3)<1e-8
Xr3 = [X0 Xr3] ;
else % if not close to any of the roots
Xr4 = [X0 Xr4] ;
end
end
end
toc
warning('on') % Remove the warning off constraint
% Initialize figure
figure
set(gcf,'color','w')
hold on
plot(Xr1(1,:),Xr1(2,:),'.','color','r') ;
plot(Xr2(1,:),Xr2(2,:),'.','color','b') ;
plot(Xr3(1,:),Xr3(2,:),'.','color','g') ;
plot(Xr4(1,:),Xr4(2,:),'.','color','k') ;
title('Basin of attraction for f(x,y) = x^3-y = 0 and y^3-x=0')
and the Newton Raphson Method file is
function X = NewtonRaphson(X)
NoIter = 10 ;
% Run a loop for given number of iterations
for j=1:NoIter
% Governing equations
f=[X(1)^3 - X(2); X(2)^3 - X(1)] ;
% Jacobian Matrix
Jf=[3*X(1)^2 -1; -1 3*X(2)^2];
% Updating the root
X=X-Jf\f;
end

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Answers (1)

J. Alex Lee
J. Alex Lee on 21 Aug 2020
Organize your "list" of initial guesses (x,y) as a meshgrid
[X,Y] = meshgrid(x,y)
Then easier to loop through the "unwrapped" arrays
for k = 1:numel(X)
X0 = [X(k);Y(k)];
end
Then you can plot any output of your solution as an "image".
Also, not sure your current code actually does what you are asking...you don't seem to be recording the number of iterations, and it doesn't seem to me that you need to know which roots you are converging to in order to create the first plot...

  2 Comments

Atique Khan
Atique Khan on 24 Aug 2020
I a bit new to MATLAB iteration, kindly help it to put it together that how can I generate the blue tone picture given ablove. Thanks for the help
J. Alex Lee
J. Alex Lee on 24 Aug 2020
No better time to learn than through a problem that you have to solve! Give it a try yourself and come back if you run into specific questions.

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