optimization/preconditioned_conjugate_gradient_method.m

   1 function [x, k] = preconditioned_conjugate_gradient_method(Q,
   2                                                            M,
   3                                                            b,
   4                                                            x0,
   5                                                            tolerance,
   6                                                            max_iterations)
   7   %
   8   % Solve,
   9   %
  10   %   Qx = b
  11   %
  12   % or equivalently,
  13   %
  14   %   min [phi(x) = (1/2)*<Qx,x> + <b,x>]
  15   %
  16   % using the preconditioned conjugate gradient method (14.54 in
  17   % Guler).
  18   %
  19   % INPUT:
  20   %
  21   %   - ``Q`` -- The coefficient matrix of the system to solve. Must
  22   %     be positive definite.
  23   %
  24   %   - ``M`` -- The preconditioning matrix. If the actual matrix used
  25   %     to precondition ``Q`` is called ``C``, i.e. ``C^(-1) * Q *
  26   %     C^(-T) == \bar{Q}``, then M=CC^T.
  27   %
  28   %   - ``b`` -- The right-hand-side of the system to solve.
  29   %
  30   %   - ``x0`` -- The starting point for the search.
  31   %
  32   %   - ``tolerance`` -- How close ``Qx`` has to be to ``b`` (in
  33   %     magnitude) before we stop.
  34   %
  35   %   - ``max_iterations`` -- The maximum number of iterations to
  36   %     perform.
  37   %
  38   % OUTPUT:
  39   %
  40   %   - ``x`` - The solution to Qx=b.
  41   %
  42   %   - ``k`` - The ending value of k; that is, the number of
  43   %   iterations that were performed.
  44   %
  45   % NOTES:
  46   %
  47   % All vectors are assumed to be *column* vectors.
  48   %
  49   % REFERENCES:
  50   %
  51   %   1. Guler, Osman. Foundations of Optimization. New York, Springer,
  52   %   2010.
  53   %
  54
  55   Ct = chol(M);
  56   C = Ct';
  57   C_inv = inv(C);
  58   Ct_inv = inv(Ct);
  59
  60   Q_bar = C_inv * Q * Ct_inv;
  61   b_bar = C_inv * b;
  62
  63   % The solution to Q_bar*x_bar == b_bar is x_bar = Ct*x.
  64   [x_bar, k] = conjugate_gradient_method(Q_bar, b_bar, x0, tolerance, max_iterations);
  65   x = Ct_inv * x_bar;
  66 end