-function [x, k] = preconditioned_conjugate_gradient_method(Q,
- M,
- b,
- x0,
- tolerance,
+function [x, k] = preconditioned_conjugate_gradient_method(Q, ...
+ M, ...
+ b, ...
+ x0, ...
+ tolerance, ...
max_iterations)
%
% 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.
% 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.
%
+ % 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 to x0,r0,z0,d0 respectively.
+ % immediately follow correspond (semantically) to x0,r0,z0,d0.
k = 0;
xk = x0;
zk = M \ rk;
dk = -zk;
- for k = [ 0 : max_iterations ]
+ while (k <= max_iterations)
+
if (norm(rk) < tolerance)
- x = xk;
- return;
+ % Check our stopping condition. This should catch the k=0 case.
+ x = xk;
+ return;
end
% Used twice, avoid recomputation.
alpha_k = rkzk/(dk' * Qdk);
x_next = xk + (alpha_k * dk);
- r_next = rk + (alpha_k * Qdk);
+
+ % 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 is due to the
+ % second reference.
+ if (mod(k, sqrt_n) == 0)
+ r_next = Q*x_next - b;
+ else
+ r_next = rk + (alpha_k * Qdk);
+ end
+
z_next = M \ r_next;
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;
xk = x_next;
rk = r_next;
dk = d_next;
end
+ % The algorithm didn't converge, but we still want to return the
+ % terminal value of xk.
x = xk;
end