X-Git-Url: http://gitweb.michael.orlitzky.com/?p=octave.git;a=blobdiff_plain;f=optimization%2Fpreconditioned_conjugate_gradient_method.m;h=63943482c8c6dd1b47d916d9bdaf400521d6b992;hp=af70af5f14e90b819f7ef2c6b1dc980877a62d7e;hb=84b8fb9002d091f84d0205e923c3989d0138ec9e;hpb=af2083885af78b1290c21f2852c6fdba25820918

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index af70af5..6394348 100644
--- a/optimization/preconditioned_conjugate_gradient_method.m
+++ b/optimization/preconditioned_conjugate_gradient_method.m
@@ -13,8 +13,9 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   %   min [phi(x) = (1/2)*<Qx,x> + <b,x>]
   %
-  % using the preconditioned conjugate gradient method (14.54 in
-  % Guler).
+  % using the preconditioned conjugate gradient method (14.56 in
+  % Guler). If ``M`` is the identity matrix, we use the slightly
+  % faster implementation in conjugate_gradient_method.m.
   %
   % INPUT:
   %
@@ -23,7 +24,8 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   %   - ``M`` -- The preconditioning matrix. If the actual matrix used
   %     to precondition ``Q`` is called ``C``, i.e. ``C^(-1) * Q *
-  %     C^(-T) == \bar{Q}``, then M=CC^T.
+  %     C^(-T) == \bar{Q}``, then M=CC^T. However the matrix ``C`` is
+  %     never itself needed. This is explained in Guler, section 14.9.
   %
   %   - ``b`` -- The right-hand-side of the system to solve.
   %
@@ -46,21 +48,51 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   % All vectors are assumed to be *column* vectors.
   %
+  % The cited algorithm contains a typo; in "The Preconditioned
+  % Conjugate-Gradient Method", we are supposed to define
+  % d_{0} = -z_{0}, not -r_{0} as written.
+  %
   % REFERENCES:
   %
   %   1. Guler, Osman. Foundations of Optimization. New York, Springer,
   %   2010.
   %
 
-  Ct = chol(M);
-  C = Ct';
-  C_inv = inv(C);
-  Ct_inv = inv(Ct);
+  % 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;
+
+  xk = x0;
+  rk = Q*xk - b;
+  zk = M \ rk;
+  dk = -zk;
+
+  for k = [ 0 : max_iterations ]
+    if (norm(rk) < tolerance)
+       x = xk;
+       return;
+    end
+
+    % 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)/rkzk;
+    d_next = -z_next + beta_next*dk;
 
-  Q_bar = C_inv * Q * Ct_inv;
-  b_bar = C_inv * b;
+    k = k + 1;
+    xk = x_next;
+    rk = r_next;
+    zk = z_next;
+    dk = d_next;
+  end
 
-  % The solution to Q_bar*x_bar == b_bar is x_bar = Ct*x.
-  [x_bar, k] = conjugate_gradient_method(Q_bar, b_bar, x0, tolerance, max_iterations);
-  x = Ct_inv * x_bar;
+  x = xk;
 end