From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 20 Mar 2013 04:26:44 +0000 (-0400)
Subject: Revert "Commit a simpler version of the preconditioned CGM."
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=131e7ef4f30e03286e4a09fc77730a0ca91e7a22;p=octave.git

Revert "Commit a simpler version of the preconditioned CGM."

This reverts commit af2083885af78b1290c21f2852c6fdba25820918.
---

diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m
index af70af5..1442194 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(Q,
+function [x, k] = preconditioned_conjugate_gradient_method(A,
 							   M,
 							   b,
 							   x0,
@@ -7,29 +7,31 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   % Solve,
   %
-  %   Qx = b
+  %   Ax = b
   %
   % or equivalently,
   %
-  %   min [phi(x) = (1/2)*<Qx,x> + <b,x>]
+  %   min [phi(x) = (1/2)*<Ax,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:
   %
-  %   - ``Q`` -- The coefficient matrix of the system to solve. Must
+  %   - ``A`` -- The coefficient matrix of the system to solve. Must
   %     be positive definite.
   %
   %   - ``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.
+  %     to precondition ``A`` 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.
   %
   %   - ``b`` -- The right-hand-side of the system to solve.
   %
   %   - ``x0`` -- The starting point for the search.
   %
-  %   - ``tolerance`` -- How close ``Qx`` has to be to ``b`` (in
+  %   - ``tolerance`` -- How close ``Ax`` has to be to ``b`` (in
   %     magnitude) before we stop.
   %
   %   - ``max_iterations`` -- The maximum number of iterations to
@@ -37,7 +39,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(Q,
   %
   % OUTPUT:
   %
-  %   - ``x`` - The solution to Qx=b.
+  %   - ``x`` - The solution to Ax=b.
   %
   %   - ``k`` - The ending value of k; that is, the number of
   %   iterations that were performed.
@@ -46,21 +48,49 @@ 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.
   %
+  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);
+
+  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.
+  zk = M \ rk;
+  dk = -zk;
 
-  Ct = chol(M);
-  C = Ct';
-  C_inv = inv(C);
-  Ct_inv = inv(Ct);
+  for k = [ 0 : max_iterations ]
+    if (norm(rk) < tolerance)
+       % Success.
+       return;
+    end
 
-  Q_bar = C_inv * Q * Ct_inv;
-  b_bar = C_inv * b;
+    % 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;
+    z_next = M \ r_next;
+    beta_next = (r_next' * z_next)/(rk' * zk);
+    d_next = -z_next + beta_next*dk;
 
-  % 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;
+    k = k + 1;
+    x = x_next;
+    rk = r_next;
+    zk = z_next;
+    dk = d_next;
+  end
 end