Implementing forward Euler method

So I'm working on part(b) and I'm unsure how to plot the error versus step size on a log-log scale. Here is the code I have thus far...

function [tgrid, Y] = euler_method(fun, y_0, n, T)

if nargin(fun) ~=2
    error('fun must take two inputs, t and y.');
end 

if ~all(size(y_0) == size(fun(0, y_0)))
    error('You have not passed appropriate fun or y_0.');
end 

%Set up the time grid. ***NOTE THE n+1***
tgrid = linspace(0, T, n+1);

%Compute h from the time grid.
h = tgrid(2) - tgrid(1);

%Orient tgrid as a column vector.
tgrid = reshape(tgrid, n+1, 1);

%How many equations?
m = length(y_0);

%Orient y0 as a row vector.
y_0 = reshape(y_0, 1, m);

% Preallocate an array to hold the approximate solution. Each row
% corresponds to a point in the time grid.
Y = zeros(n+1, m);

% Set the initial conditions.
Y(1,:) = y_0;

% Euler loop
for i = 1: n

      % Store the point in time as a temporary variable
      t_i = tgrid(i);

      % Take the Euler step into the temporary variable
      y_1 = y_0 + h * fun(t_i, y_0);

      % Store the Euler step
      Y(i+1,:) = y_1;

      % Update the temporary variable
      y_0 = y_1;

end

Answer 1

[Torsten]

Call "euler_method" in a loop for n = 8*2^k (k=1,...,15) and store Y(n+1,1) for each run.

Then make the plot.

Best wishes

Torsten.

[Matthew Kaplan]

So you mean like this? and how do you store Y(n+1,1)?

 for n = 8*2.^-(1:15) 

   for i = 1: n

      % Store the point in time as a temporary variable
      t_i = tgrid(i);

      % Take the Euler step into the temporary variable
      y_1 = y_0 + h * fun(t_i, y_0);

      % Store the Euler step
      Y(i+1,:) = y_1;

      % Update the temporary variable
      y_0 = y_1;

end

[Matthew Kaplan]

for k = 1:15
    h = 2.^-k;

% Euler loop
for i = 1: n

      % Store the point in time as a temporary variable
      t_i = tgrid(i);

      % Take the Euler step into the temporary variable
      y_1 = y_0 + h * fun(t_i, y_0);

      % Store the Euler step
      Y(i+1,:) = y_1;

      % Update the temporary variable
      y_0 = y_1;

end

loglog();

end

[Matthew Kaplan]

Am I on the right path at all?

[Torsten]

y_0 = 0;
tend = 8.0;
fun=@(t,y) sin(t)-y;

for k = 1:15
  n = 8*2^k;
  [T Y] = euler_method(fun, y_0, n, tend);
  Yend(k) = Y(n+1,1);
end

Now add the plot.

Best wishes

Torsten.