function [Q,R] = mgs(A)
% function [Q,R] = mgs(A)
% 
% Math 5334, Fall 2009, HW5
% Trefethen & Bau problem 8.2
% 10/22/2009 VEH
%
% Computes a reduced QR factorization of A using modified Gram-Schmidt.
%   A is mxn (m >=n)
%   Q is mxn with ON columns
%   R is nxn upper triangular
%
% Note that any comments I put like this at the beginning of a function
% appear when I use help in MATLAB. Type "help mgs" in MATLAB to see.

% Get the dimensions of A and initialize Q and R to be zero matrices of the 
% correct sizes.
[m,n] = size(A);
R = zeros(n,n);
Q = zeros(m,n);
if n > m
  disp('Error: Input matrix must be m x n with m >= n.') 
  disp(sprintf('Given matrix has m = %g and n = %g', m, n));
  return;
end

% No need to loop over the columns when I can just set V to be A.
% Then the columns of V are my v_i vectors
V = A;

for i = 1:n
  
  R(i,i) = norm(V(:,i));
  
  Q(:,i) = V(:,i) / R(i,i);
  % Note that Q(:,i) is MATLAB shorthand for Q(1:m,i); 
  % it means "all rows and column i"
  
  for j = i+1:n
    
    R(i,j) = Q(:,i)'*V(:,j);
    V(:,j) = V(:,j) - R(i,j)*Q(:,i);
    
  end
  
end