%
% min [phi(x) = (1/2)*<Qx,x> + <b,x>]
%
- % using the preconditioned conjugate gradient method (14.54 in
- % Guler).
+ % 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:
%
%
% - ``M`` -- The preconditioning matrix. If the actual matrix used
% to precondition ``Q`` is called ``C``, i.e. ``C^(-1) * Q *
- % C^(-T) == \bar{Q}``, then M=CC^T.
+ % 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.
%
%
% OUTPUT:
%
- % - ``x`` - The solution to Qx=b.
+ % - ``x`` - The computed solution to Qx=b.
%
% - ``k`` - The ending value of k; that is, the number of
% iterations that were performed.
%
% 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.
+ %
+ % The rather verbose name of this function was chosen to avoid
+ % conflicts with other implementations.
+ %
% REFERENCES:
%
% 1. Guler, Osman. Foundations of Optimization. New York, Springer,
% 2010.
%
- Ct = chol(M);
- C = Ct';
- C_inv = inv(C);
- Ct_inv = inv(Ct);
+ % Set k=0 first, that way the references to xk,rk,zk,dk which
+ % immediately follow correspond (semantically) to x0,r0,z0,d0.
+ k = 0;
+
+ xk = x0;
+ rk = Q*xk - b;
+ zk = M \ rk;
+ dk = -zk;
+
+ for k = [ 0 : max_iterations ]
+
+ if (norm(rk) < tolerance)
+ % Check our stopping condition. This should catch the k=0 case.
+ x = xk;
+ return;
+ end
+
+ % Used twice, avoid recomputation.
+ rkzk = rk' * zk;
+
+ % The term alpha_k*dk appears twice, but so does Q*dk. We can't
+ % do them both, so we precompute the more expensive operation.
+ Qdk = Q * dk;
+
+ % After substituting the two previously-created variables, the
+ % following algorithm occurs verbatim in the reference.
+ alpha_k = rkzk/(dk' * Qdk);
+ x_next = xk + (alpha_k * dk);
+ r_next = rk + (alpha_k * Qdk);
+ z_next = M \ r_next;
+ beta_next = (r_next' * z_next)/rkzk;
+ d_next = -z_next + beta_next*dk;
- Q_bar = C_inv * Q * Ct_inv;
- b_bar = C_inv * b;
+ k = k + 1;
+ xk = x_next;
+ rk = r_next;
+ zk = z_next;
+ dk = d_next;
+ end
- % The solution to Q_bar*x_bar == b_bar is x_bar = Ct*x.
- [x_bar, k] = conjugate_gradient_method(Q_bar, b_bar, x0, tolerance, max_iterations);
- x = Ct_inv * x_bar;
+ % The algorithm didn't converge, but we still want to return the
+ % terminal value of xk.
+ x = xk;
end