function [x, k] = preconditioned_conjugate_gradient_method(A,
							   M,
							   b,
							   x0,
							   tolerance,
							   max_iterations)
  %
  % Solve,
  %
  %   Ax = b
  %
  % or equivalently,
  %
  %   min [phi(x) = (1/2)*<Ax,x> + <b,x>]
  %
  % using the preconditioned conjugate gradient method (14.56 in
  % Guler). If ``M`` is the identity matrix, we use the slightly
  % faster implementation in conjugate_gradient_method.m.
  %
  % INPUT:
  %
  %   - ``A`` -- The coefficient matrix of the system to solve. Must
  %     be positive definite.
  %
  %   - ``M`` -- The preconditioning matrix. If the actual matrix used
  %     to precondition ``A`` is called ``C``, i.e. ``C^(-1) * Q *
  %     C^(-T) == \bar{Q}``, then M=CC^T. However the matrix ``C`` is
  %     never itself needed. This is explained in Guler, section 14.9.
  %
  %   - ``b`` -- The right-hand-side of the system to solve.
  %
  %   - ``x0`` -- The starting point for the search.
  %
  %   - ``tolerance`` -- How close ``Ax`` has to be to ``b`` (in
  %     magnitude) before we stop.
  %
  %   - ``max_iterations`` -- The maximum number of iterations to
  %     perform.
  %
  % OUTPUT:
  %
  %   - ``x`` - The solution to Ax=b.
  %
  %   - ``k`` - The ending value of k; that is, the number of
  %   iterations that were performed.
  %
  % NOTES:
  %
  % All vectors are assumed to be *column* vectors.
  %
  % The cited algorithm contains a typo; in "The Preconditioned
  % Conjugate-Gradient Method", we are supposed to define
  % d_{0} = -z_{0}, not -r_{0} as written.
  %
  % REFERENCES:
  %
  %   1. Guler, Osman. Foundations of Optimization. New York, Springer,
  %   2010.
  %
  n = length(x0);

  if (isequal(M, eye(n)))
    [x, k] = conjugate_gradient_method(A, b, x0, tolerance, max_iterations);
    return;
  end

  zero_vector = zeros(n, 1);

  k = 0;
  x = x0; % Eschew the 'k' suffix on 'x' for simplicity.
  rk = A*x - b; % The first residual must be computed the hard way.
  zk = M \ rk;
  dk = -zk;

  for k = [ 0 : max_iterations ]
    if (norm(rk) < tolerance)
       % Success.
       return;
    end

    % Unfortunately, since we don't know the matrix ``C``, it isn't
    % easy to compute alpha_k with an existing step size function.
    alpha_k = (rk' * zk)/(dk' * A * dk);
    x_next = x + alpha_k*dk;
    r_next = rk + alpha_k*A*dk;
    z_next = M \ r_next;
    beta_next = (r_next' * z_next)/(rk' * zk);
    d_next = -z_next + beta_next*dk;

    k = k + 1;
    x = x_next;
    rk = r_next;
    zk = z_next;
    dk = d_next;
  end
end
