X-Git-Url: http://gitweb.michael.orlitzky.com/?p=octave.git;a=blobdiff_plain;f=optimization%2Fpreconditioned_conjugate_gradient_method.m;h=35fecaa99cdb81b7f22836791797c0bb2fe1acac;hp=63943482c8c6dd1b47d916d9bdaf400521d6b992;hb=302a62bb7c0adf581916fcd1f93faa719e8d51e8;hpb=84b8fb9002d091f84d0205e923c3989d0138ec9e

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index 6394348..35fecaa 100644
--- a/optimization/preconditioned_conjugate_gradient_method.m
+++ b/optimization/preconditioned_conjugate_gradient_method.m
@@ -1,8 +1,8 @@
-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,
@@ -39,7 +39,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   % 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.
@@ -52,14 +52,24 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   % 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;
@@ -67,10 +77,12 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   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.
@@ -82,11 +94,25 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
 
     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;
@@ -94,5 +120,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
     dk = d_next;
   end
 
+  % The algorithm didn't converge, but we still want to return the
+  % terminal value of xk.
   x = xk;
 end