Update comments in the *conjugate_gradient_method() functions.

diff --git a/optimization/conjugate_gradient_method.m b/optimization/conjugate_gradient_method.m
index a6401e5ec21ff963883368576deb59ff0e1b36d0..0acf5b27998012a85e733a872cb88411913de2af 100644 (file)
--- a/optimization/conjugate_gradient_method.m
+++ b/optimization/conjugate_gradient_method.m
@@ -35,6 +35,9 @@ function [x, k] = conjugate_gradient_method(A, b, x0, tolerance, max_iterations)
   %
   % All vectors are assumed to be *column* vectors.
   %
   %
   % All vectors are assumed to be *column* vectors.
   %

+  % The rather verbose name of this function was chosen to avoid
+  % conflicts with other implementations.
+  %

   n = length(x0);
   M = eye(n);
 
   n = length(x0);
   M = eye(n);
 

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index 63943482c8c6dd1b47d916d9bdaf400521d6b992..eb2089f5b4e50d367c8237be9c166d5065f70982 100644 (file)
--- a/optimization/preconditioned_conjugate_gradient_method.m
+++ b/optimization/preconditioned_conjugate_gradient_method.m
@@ -39,7 +39,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   % OUTPUT:
   %
   %
   % 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.
   %
   %   - ``k`` - The ending value of k; that is, the number of
   %   iterations that were performed.


@@ -52,6 +52,9 @@ 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.
   %
   % 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,
   % REFERENCES:
   %
   %   1. Guler, Osman. Foundations of Optimization. New York, Springer,


@@ -59,7 +62,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
 
   % Set k=0 first, that way the references to xk,rk,zk,dk which
   %
 
   % 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;
   k = 0;
 
   xk = x0;


@@ -68,9 +71,11 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   dk = -zk;
 
   for k = [ 0 : max_iterations ]
   dk = -zk;
 
   for k = [ 0 : max_iterations ]

+

     if (norm(rk) < tolerance)
     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.
     end
 
     % Used twice, avoid recomputation.


@@ -80,6 +85,8 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
     % do them both, so we precompute the more expensive operation.
     Qdk = Q * dk;
 
     % 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);
     alpha_k = rkzk/(dk' * Qdk);
     x_next = xk + (alpha_k * dk);
     r_next = rk + (alpha_k * Qdk);


@@ -94,5 +101,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
     dk = d_next;
   end
 
     dk = d_next;
   end
 

+  % The algorithm didn't converge, but we still want to return the
+  % terminal value of xk.

   x = xk;
 end
   x = xk;
 end