-function [x, k] = preconditioned_conjugate_gradient_method(A,
- M,
- b,
- x0,
- tolerance,
+function [x, k] = preconditioned_conjugate_gradient_method(Q, ...
+ M, ...
+ b, ...
+ x0, ...
+ tolerance, ...
max_iterations)
%
% Solve,
%
- % Ax = b
+ % Qx = b
%
% or equivalently,
%
- % min [phi(x) = (1/2)*<Ax,x> + <b,x>]
+ % min [phi(x) = (1/2)*<Qx,x> + <b,x>]
%
% using the preconditioned conjugate gradient method (14.56 in
% Guler). If ``M`` is the identity matrix, we use the slightly
%
% INPUT:
%
- % - ``A`` -- The coefficient matrix of the system to solve. Must
+ % - ``Q`` -- 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 *
+ % to precondition ``Q`` 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.
%
%
% - ``x0`` -- The starting point for the search.
%
- % - ``tolerance`` -- How close ``Ax`` has to be to ``b`` (in
+ % - ``tolerance`` -- How close ``Qx`` has to be to ``b`` (in
% magnitude) before we stop.
%
% - ``max_iterations`` -- The maximum number of iterations to
%
% OUTPUT:
%
- % - ``x`` - The solution to Ax=b.
+ % - ``x`` - The computed solution to Qx=b.
%
% - ``k`` - The ending value of k; that is, the number of
% iterations that were performed.
% 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.
+ % 2010.
+ %
+ % 2. Shewchuk, Jonathan Richard. An Introduction to the Conjugate
+ % Gradient Method Without the Agonizing Pain, Edition 1.25.
+ % August 4, 1994.
%
- 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);
+ % We use this in the inner loop.
+ sqrt_n = floor(sqrt(length(x0)));
+ % 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;
- x = x0; % Eschew the 'k' suffix on 'x' for simplicity.
- rk = A*x - b; % The first residual must be computed the hard way.
+
+ xk = x0;
+ rk = Q*xk - b;
zk = M \ rk;
dk = -zk;
- for k = [ 0 : max_iterations ]
- if (norm(rk) < tolerance)
- % Success.
- return;
+ while (k <= max_iterations && norm(rk, 'inf') > tolerance)
+ % 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;
+
+ % We're going to divide by this quantity...
+ dkQdk = dk' * Qdk;
+
+ % So if it's too close to zero, we replace it with something
+ % comparable but non-zero.
+ if (abs(dkQdk) < eps)
+ dkQdk = sign(dkQdk)*eps;
+ end
+
+ alpha_k = rkzk/dkQdk;
+ x_next = xk + (alpha_k * dk);
+
+ % The recursive definition of r_next is prone to accumulate
+ % roundoff error. When sqrt(n) divides k, we recompute the
+ % residual to minimize this error. This modification was suggested
+ % by the second reference.
+ if (mod(k, sqrt_n) == 0)
+ r_next = Q*x_next - b;
+ else
+ r_next = rk + (alpha_k * Qdk);
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);
+ beta_next = (r_next' * z_next)/rkzk;
d_next = -z_next + beta_next*dk;
+ % We potentially just performed one more iteration than necessary
+ % in order to simplify the loop. Note that due to the structure of
+ % our loop, we will have k > max_iterations when we fail to
+ % converge.
k = k + 1;
- x = x_next;
+ xk = x_next;
rk = r_next;
zk = z_next;
dk = d_next;
end
+
+ % If we make it here, one of the two stopping conditions was met.
+ x = xk;
end