X-Git-Url: http://gitweb.michael.orlitzky.com/?p=octave.git;a=blobdiff_plain;f=optimization%2Fsimple_preconditioned_cgm.m;fp=optimization%2Fsimple_preconditioned_cgm.m;h=31e0225ba7b05bf7f3c7c88ea372df2d2baff483;hp=0000000000000000000000000000000000000000;hb=5ef96eb244b2a52886a31fc9873db479f52a6483;hpb=131e7ef4f30e03286e4a09fc77730a0ca91e7a22

diff --git a/optimization/simple_preconditioned_cgm.m b/optimization/simple_preconditioned_cgm.m
new file mode 100644
index 0000000..31e0225
--- /dev/null
+++ b/optimization/simple_preconditioned_cgm.m
@@ -0,0 +1,71 @@
+function [x, k] = simple_preconditioned_cgm(Q,
+					    M,
+					    b,
+					    x0,
+					    tolerance,
+					    max_iterations)
+  %
+  % Solve,
+  %
+  %   Qx = b
+  %
+  % or equivalently,
+  %
+  %   min [phi(x) = (1/2)*<Qx,x> + <b,x>]
+  %
+  % using the preconditioned conjugate gradient method (14.54 in
+  % Guler).
+  %
+  % INPUT:
+  %
+  %   - ``Q`` -- 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. Must be symmetric positive-
+  %     definite. See for example Golub and Van Loan.
+  %
+  %   - ``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
+  %     magnitude) before we stop.
+  %
+  %   - ``max_iterations`` -- The maximum number of iterations to
+  %     perform.
+  %
+  % OUTPUT:
+  %
+  %   - ``x`` - The solution to Qx=b.
+  %
+  %   - ``k`` - The ending value of k; that is, the number of
+  %   iterations that were performed.
+  %
+  % NOTES:
+  %
+  % All vectors are assumed to be *column* vectors.
+  %
+  % REFERENCES:
+  %
+  %   1. Guler, Osman. Foundations of Optimization. New York, Springer,
+  %   2010.
+  %
+
+  % This isn't great in practice, since the CGM is usually used on
+  % huge sparse systems.
+  Ct = chol(M);
+  C = Ct';
+  C_inv = inv(C);
+  Ct_inv = inv(Ct);
+
+  Q_bar = C_inv * Q * Ct_inv;
+  b_bar = C_inv * b;
+
+  % But it sure is easy.
+  [x_bar, k] = vanilla_cgm(Q_bar, b_bar, x0, tolerance, max_iterations);
+
+  % The solution to Q_bar*x_bar == b_bar is x_bar = Ct*x.
+  x = Ct_inv * x_bar;
+end