Matlab Newton Raphson - matlab

Question

I asked for help about Matlab task few weeks ago but removed it since it took me some time to solve. Unfortunately, I still have a problem.
Task is: Find a real root of the function f(x)=tanh(x^2 - 9) using at least 3 iterations, using Newton-Raphson method. x= 3.2 show each iteration graphically.
In code "pog" means min. mistake, "br" is counter.
If I don't put a counter, it immediately gives me "NaN", with counter it does write first few calculations.
My code:
clc
clear all
x=3.2;
fx=tanh(x^2-9);
iter=5;
pog=0.01;
br=1;
while br<10;
xk= x-((tanh(x^2-9))/(-2*x*(tanh(x^2 - 9)^2 - 1)));
fprintf ('x=%g\txk=%g\t%g\n', x,xk, abs(xk-x))
if pog>abs(xk-x);
break
end
x=xk;
br=br+1;
end
Thank you in advance!

Answer 1

As far as showing the iterations graphically, this is the best I can do:
clc
clear all
close
G=zeros(20,10);
X=linspace(2.5,3.5,20)
G(:,1)=X;
for i=1:length(X)
x=X(i)
fx=tanh(x^2-9);
pog=0.0001;
br=1;
while br<10;
xk= x-((tanh(x^2-9))/(2*x*sech(9-x^2)^2));
G(i,br+1)=xk;
x=xk;
br=br+1;
end
end
I=tanh(G(:,end).^2-9)<1e-5;
X(I)
plot(G,1:10)
hold off
axis([2.5 3.5 0 5])
X(I) is a list of starting values the converge to the root, and the plot shows iteration number on the y-axis, and the guess at that iteration on the x-axis. You can follow each starting value through and see what happens.
Here's another way of visualising Newton's method. It shows the tangent line that is constructed, and where it passes through 0 which gives you the new x values, from which a vertical line gives you the new function value, which defines a new tangent line for the next iteration. It might help.
clc
clear all
close
G=zeros(20,10);
X=linspace(2.75,3.25,20)
G(:,1)=X;
x2=2:.01:4;f2=#(x) tanh(x.^2-9); %// to help with plotting
for i=1:length(X)
x=X(i)
fx=tanh(x^2-9);
pog=0.0001;
br=1;
xk=x;
while br<10;
%// Newton method step
dx=((tanh(x^2-9))/(2*x*sech(9-x^2)^2));
xk=x-dx;
%// plotting everything
G(i,br+1)=xk;
plot(x2,f2(x2))
hold all
plot(G(i,br:br+1),f2(G(i,br:br+1)),'.','MarkerSize',16)
plot(x2,f2(x)+(x2-x)./(xk-x).*(-f2(x)))
plot(xk,0,'.','MarkerSize',16)
plot(x2,(x2-xk)*(2*xk*sech(9-xk^2)^2)+f2(xk))
plot([xk xk],[-1 1])
plot([2 4],[0 0],'k')
axis([2 4 -1 1])
drawnow
pause
hold off
%// finishing Newton step
x=xk;
br=br+1;
end
hold off
end