function [Q, H] = arnoldi(A, u, m)
% ARNOLDI   Arnoldi iteration for Krylov subspaces.
% Input:
%   A    square matrix (n by n)
%   u    initial vector
%   m    number of iterations
% Output: 
%   Q    orthonormal basis of Krylov space (n by m+1)
%   H    upper Hessenberg matrix, A*Q(:,1:m)=Q*H (m+1 by m)
%
    n = length(A);
    Q = zeros(n, m+1);  
    H = zeros(m+1, m);
    Q(:,1) = u/norm(u);
    for j = 1:m
        % Find the new direction that extends the Krylov subspace.
        v = A*Q(:, j);
        % Remove the projections onto the previous vectors.
        for i = 1:j
            H(i,j) = Q(:,i)'*v;
            v = v - H(i,j)*Q(:,i);
        end
        % Normalize and store the new basis vector.
        H(j+1, j) = norm(v);
        Q(:, j+1) = v/H(j+1, j);
    end
end