What does this code say? (Linear Algebra related) (Poisson)

This is my first time using MATLab and I have no idea what this code is saying. Can anyone translate this for me? I have proficient skills in Java, but it's not helping me out here.

p = 10; n = 2.^(1:p)'; % grid sizes

err = zeros(p,1); % error for each n

for i = 1:p

    h = 1/n(i); x = (1:n(i)-1)'*h;    
    A = (1/h^2)*(diag(2*ones(n(i)-1,1))-diag(ones(n(i)-2,1), 1) ...
                                    -diag(ones(n(i)-2,1),-1));
    f = exp(x);  % right-hand side
    v = A\f;     % solve

    u = -exp(x)+(exp(1)-1)*x+1; % exact soln
    err(i) = max(abs(u-v));     % infinity-norm error
end
format short e
disp(' ') 
disp('       h           err        err/h^2 ')
disp('     -----       -------    -----------')
disp(' ') 
disp([1./n err err./(1./n.^2)])

I need a step by step explanation, if possible. For example, I have no idea what's going on with n on the first line. What does 2.^(1:p)' do? Like the period, the carrot, the parentheses, the colon, specifically? When it runs, it looks like this:

       h           err        err/h^2 
     -----       -------    -----------

   5.0000e-01   4.3295e-03   1.7318e-02
   2.5000e-01   1.0925e-03   1.7480e-02
   1.2500e-01   2.7377e-04   1.7521e-02
   6.2500e-02   6.8827e-05   1.7620e-02
   3.1250e-02   1.7234e-05   1.7647e-02
   1.5625e-02   4.3098e-06   1.7653e-02
   7.8125e-03   1.0776e-06   1.7655e-02
   3.9063e-03   2.6940e-07   1.7655e-02
   1.9531e-03   6.7351e-08   1.7656e-02
   9.7656e-04   1.6838e-08   1.7656e-02

Answer 1

[sixwwwwww]

Dear Guile, I placed references for all the things which maybe confusing for you. You can try to access those references to find out how the code works:

p = 10; 
% References for below code line:
% for (:) http://www.mathworks.de/de/help/matlab/ref/colon.html
% for (') http://www.mathworks.de/de/help/matlab/ref/transpose.html
% for (.^) http://www.mathworks.de/de/help/matlab/ref/power.html
n = 2.^(1:p)'; 
% References for below code line:
% for (zeros) http://www.mathworks.de/de/help/matlab/ref/zeros.html
err = zeros(p,1); % error for each n
% References for below code line:
% for (for) http://www.mathworks.de/de/help/matlab/ref/for.html
for i = 1:p
    h = 1 / n(i); 
    x = (1:n(i) - 1)' * h;
    % References for below code line:
    % for (diag) http://www.mathworks.de/de/help/matlab/ref/diag.html
    % for (ones) http://www.mathworks.de/de/help/matlab/ref/ones.html
    A = (1 / h^2) * (diag(2*ones(n(i)-1,1))-diag(ones(n(i)-2,1), 1)-diag(ones(n(i)-2,1),-1));
    % References for below code line:
    % for (exp) http://www.mathworks.de/de/help/matlab/ref/exp.html
    f = exp(x);  
    v = A\f;     
    u = -exp(x)+(exp(1)-1)*x+1;
    % References for below code line:
    % for (max) http://www.mathworks.de/de/help/fixedpoint/ref/max.html
    err(i) = max(abs(u-v));    
end
% References for below code line:
% for (format) http://www.mathworks.de/de/help/matlab/ref/format.html
format short e
% References for below code line:
% for (disp) http://www.mathworks.de/de/help/matlab/ref/disp.html
disp(' ') 
disp('       h           err        err/h^2 ')
disp('     -----       -------    -----------')
disp(' ')
% References for below code line:
% for (./) http://www.mathworks.de/de/help/matlab/ref/rdivide.html
disp([1./n err err./(1./n.^2)])

I hope it can help you somehow. Good luck!