Make sure that all output is clearly labeled so that it is clear what each number means and by what line in the .m file it was generated.

• Problems 3 and 4: Stop the Newton iteration when ||x^(k)-x^(k-1)||₂ becomes smaller than some tolerance.
• Problem 3: In Matlab you can plot the contours where, e.g., cos(x²) - sin(xy) - .5 = 0 for x in [-5,5] and y in [-4,4] as follows:

  [X,Y]=meshgrid(-5:.1:5,-4:.1:4);
  contour(X,Y,cos(X.^2)-sin(X.*Y)-.5,[0 0])  % use .*,./,.^ !

  You can superimpose several contours by using hold on after the first contour command.

• Problem 4: You need to write m-files f.m and J.m for the function and the Jacobian. The file f.m should look like this:

  function w=f(v)
  w = zeros(19,1);
  w(1) = ...
  for i=2:18
    w(i) = ...
  end
  w(19) = ...

  The file J.m should look like this:

  function a=J(v)
  a = sparse(19,19); % 19x19 zero matrix as sparse structure
  ...          % row 1 
  for j=2:18   % rows 2,...,18
    a(j,j-1) = ...
    a(j,j)   = ...
    a(j,j+1) = ...
  end
  ...          % row 19

  Use plot(0:.05:1,[0;v;0]) to plot the angle of the rod vs. the arclength.