From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 20 Mar 2013 05:16:40 +0000 (-0400)
Subject: Fix up preconditioned CGM code.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=84b8fb9002d091f84d0205e923c3989d0138ec9e;p=octave.git

Fix up preconditioned CGM code.
Rename 'A' to 'Q' in preconditioned CGM code.
Test the preconditioned CGM against the simple implementation.
---

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index 1442194..6394348 100644
--- a/optimization/preconditioned_conjugate_gradient_method.m
+++ b/optimization/preconditioned_conjugate_gradient_method.m
@@ -1,4 +1,4 @@
-function [x, k] = preconditioned_conjugate_gradient_method(A,
+function [x, k] = preconditioned_conjugate_gradient_method(Q,
 							   M,
 							   b,
 							   x0,
@@ -7,11 +7,11 @@ function [x, k] = preconditioned_conjugate_gradient_method(A,
   %
   % 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
@@ -19,11 +19,11 @@ function [x, k] = preconditioned_conjugate_gradient_method(A,
   %
   % 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.
   %
@@ -31,7 +31,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(A,
   %
   %   - ``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
@@ -39,7 +39,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(A,
   %
   % OUTPUT:
   %
-  %   - ``x`` - The solution to Ax=b.
+  %   - ``x`` - The solution to Qx=b.
   %
   %   - ``k`` - The ending value of k; that is, the number of
   %   iterations that were performed.
@@ -57,40 +57,42 @@ function [x, k] = preconditioned_conjugate_gradient_method(A,
   %   1. Guler, Osman. Foundations of Optimization. New York, Springer,
   %   2010.
   %
-  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);
 
+  % Set k=0 first, that way the references to xk,rk,zk,dk which
+  % immediately follow correspond to x0,r0,z0,d0 respectively.
   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.
+       x = xk;
        return;
     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;
+    % 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;
+
+    alpha_k = rkzk/(dk' * Qdk);
+    x_next = xk + (alpha_k * dk);
+    r_next = rk + (alpha_k * Qdk);
     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;
 
     k = k + 1;
-    x = x_next;
+    xk = x_next;
     rk = r_next;
     zk = z_next;
     dk = d_next;
   end
+
+  x = xk;
 end
diff --git a/tests/preconditioned_conjugate_gradient_method_tests.m b/tests/preconditioned_conjugate_gradient_method_tests.m
index c58eb55..d44ee42 100644
--- a/tests/preconditioned_conjugate_gradient_method_tests.m
+++ b/tests/preconditioned_conjugate_gradient_method_tests.m
@@ -3,12 +3,9 @@ A = [5,1,2; ...
      2,3,7];
 
 M = eye(3);
-
 b = [1;2;3];
-
 x0 = [1;1;1];
 
-## Solved over the rationals.
 cgm  = conjugate_gradient_method(A, b, x0, 1e-6, 1000);
 pcgm = preconditioned_conjugate_gradient_method(A, M, b, x0, 1e-6, 1000);
 diff = norm(cgm - pcgm);
@@ -17,6 +14,12 @@ unit_test_equals("PCGM agrees with CGM when M == I", ...
 		 true, ...
 		 norm(diff) < 1e-6);
 
+pcgm_simple = simple_preconditioned_cgm(A, M, b, x0, 1e-6, 1000);
+diff = norm(pcgm_simple - pcgm);
+
+unit_test_equals("PCGM agrees with SimplePCGM when M == I", ...
+		 true, ...
+		 norm(diff) < 1e-6);
 
 ## Needs to be symmetric!
 M = [0.97466, 0.24345, 0.54850; ...
@@ -29,3 +32,11 @@ diff = norm(cgm - pcgm);
 unit_test_equals("PCGM agrees with CGM when M != I", ...
 		 true, ...
 		 norm(diff) < 1e-6);
+
+
+pcgm_simple = simple_preconditioned_cgm(A, M, b, x0, 1e-6, 1000);
+diff = norm(pcgm_simple - pcgm);
+
+unit_test_equals("PCGM agrees with Simple PCGM when M != I", ...
+		 true, ...
+		 norm(diff) < 1e-6);