function [x, k] = steepest_descent(g, x0, step_size, tolerance, max_iterations)
  %
  % An implementation of the steepest-descent algorithm, with the
  % search direction d_{k} = -\nabla f(x_{k}).
  %
  % We should terminate when either,
  %
  %   a) The 2-norm of the gradient at x_{k} is greater than
  %   ``tolerance``.
  %
  %   b) We have performed ``max_iterations`` iterations.
  %
  % INPUT:
  %
  %   * ``g`` - the gradient of the function ``f`` we're minimizing.
  %
  %   * ``x0`` - the starting point for the search.
  %
  %   * ``step_size`` - a function of x which returns the optimal step
  %     size alpha_{k}. This will generally require more information
  %     than just x; for example it might need the function ``f`` or the
  %     gradient ``g``. However, our caller has this information so it
  %     should be incorporated into the step size algorithm that we
  %     receive.
  %
  %   * ``tolerance`` - the stopping tolerance. We stop when the norm
  %   of the gradient falls below this value.
  %
  %   * ``max_iterations`` - a safety net; we return x_{k}
  %   unconditionally if we exceed this number of iterations.
  %
  % OUTPUT:
  %
  %   * ``x`` - the solution at the final value of x_{k}.
  %
  %   * ``k`` - the value of k when we stop; i.e. the number of
  %   iterations.

  % The initial gradient at x_{0} is not supplied, so we compute it
  % here and begin the loop at k=1.
  x = x0;
  g_k = g(x);

  if (norm(g_k) < tolerance)
    % If x_0 is close enough to a solution, there's nothing for us to
    % do! We use g_k (the gradient of f at x_k) instead of d_k because
    % their 2-norms will be the same, and g_k is already stored.
    return;
  end

  for k = [1 : max_iterations]
    % Loop until either of our stopping conditions are met. If the
    % loop finishes, we have implicitly met the second stopping
    % condition (number of iterations).
    d_k = -g_k;
    alpha_k = step_size(x);
    x = x + (alpha_k * d_k);
    g_k = g(x);

    if (norm(g_k) < tolerance)
      return;
    end
  end

  % If we make it to the end of the loop, that means we've executed the
  % maximum allowed iterations. The caller should be able to examine the
  % return value ``k`` to determine what happened.
end
