Non-Linear Dynamical Systems - Iteration Help? (Beginer)

Hi,

I am a total novice MATLAB user and I feel as if I am banging my head against a desk with this problem. I am aware that it is VERY simple, but as I mentioned I am very new to MATLAB, any help at all would be much appreciated.

I am studying a non-linear dynamical systems course and I'm required to write MATLAB files that will solve a non-linear system of equations of the form: f(x)=0

I am also told that the function should implement the newton iteration:

x(k+1) =  x(k) - ( [df/dx(x(k))]^-1  ( f( x(k) ) )

The expression [df/dx(x(k))] is the Jacobian of f in the previous point x(k) and the initial guess x0 is provided by the user of mysolve (MySolve.m is the name we are told to call our file) I have written a program that finds the Jacobian of any matrix:

    function df=MyJacobian(f,x,h)

n = length(x);

xplus = x;
xminus = x;

for i=1:n

      xplus(i)=xplus(i)+h;
      xminus(i)=xminus(i)-h;    

      df(:,i)=(f(xplus)-f(xminus))/(2.*h);    

      xplus(i)=x(i);
      xminus(i)=x(i);

end

I know this works.

Finally we are told that the iteration should stop if both the correction (e(k+1)=x(k+1)-x(k)) and the residual (r=f(x(k))) are less than some specified tolerance 'tol' (for example 1e-10). After maxit iterations without convergence one should exit anyway, and set converged=0 (for example, maxit could be 10).

I am sorry for such a long question but once again would appreciate any help what so ever,

Answer 1

[Andrew Newell]

I don't want to just do your homework for you, so here are a few hints that will help with the Matlab syntax. First, make a loop for k=1:maxit, so if you exceed the maximum number of iterations you just exit the loop. Inside the loop, update x and use an IF condition to test for convergence:

converged = correction < tol && residual < tol;
if converged
   xsolution = current_value_of_x;
   return
end

If converged is true, it is set to 1 and you finish.

[Joseph Hughes]

got it working thanks for the help it is much appreciated :)