From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 22 Mar 2013 18:47:58 +0000 (-0400)
Subject: Add a second reference for the PCGM and make it more resistant to accumulated roundof... 
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=48a7e4e418ee26465a4d3a24e45e26cf7e90eb71;p=octave.git

Add a second reference for the PCGM and make it more resistant to accumulated roundoff error.
---

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index b7999d9..4e68ccb 100644
--- a/optimization/preconditioned_conjugate_gradient_method.m
+++ b/optimization/preconditioned_conjugate_gradient_method.m
@@ -58,8 +58,15 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q, ...
   % 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 (semantically) to x0,r0,z0,d0.
@@ -85,11 +92,19 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q, ...
     % 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);
+
+    % 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;