1 function [x, k] = preconditioned_conjugate_gradient_method(A,
2 M,
3 b,
4 x0,
5 tolerance,
6 max_iterations)
7 %
8 % Solve,
9 %
10 % Ax = b
11 %
12 % or equivalently,
13 %
14 % min [phi(x) = (1/2)*<Ax,x> + <b,x>]
15 %
16 % using the preconditioned conjugate gradient method (14.56 in
17 % Guler). If M is the identity matrix, we use the slightly
18 % faster implementation in conjugate_gradient_method.m.
19 %
20 % INPUT:
21 %
22 % - A -- The coefficient matrix of the system to solve. Must
23 % be positive definite.
24 %
25 % - M -- The preconditioning matrix. If the actual matrix used
26 % to precondition A is called C, i.e. C^(-1) * Q *
27 % C^(-T) == \bar{Q}, then M=CC^T. However the matrix C is
28 % never itself needed. This is explained in Guler, section 14.9.
29 %
30 % - b -- The right-hand-side of the system to solve.
31 %
32 % - x0 -- The starting point for the search.
33 %
34 % - tolerance -- How close Ax has to be to b (in
35 % magnitude) before we stop.
36 %
37 % - max_iterations -- The maximum number of iterations to
38 % perform.
39 %
40 % OUTPUT:
41 %
42 % - x - The solution to Ax=b.
43 %
44 % - k - The ending value of k; that is, the number of
45 % iterations that were performed.
46 %
47 % NOTES:
48 %
49 % All vectors are assumed to be *column* vectors.
50 %
51 % The cited algorithm contains a typo; in "The Preconditioned
52 % Conjugate-Gradient Method", we are supposed to define
53 % d_{0} = -z_{0}, not -r_{0} as written.
54 %
55 % REFERENCES:
56 %
57 % 1. Guler, Osman. Foundations of Optimization. New York, Springer,
58 % 2010.
59 %
60 n = length(x0);
62 if (isequal(M, eye(n)))
63 [x, k] = conjugate_gradient_method(A, b, x0, tolerance, max_iterations);
64 return;
65 end
67 zero_vector = zeros(n, 1);
69 k = 0;
70 x = x0; % Eschew the 'k' suffix on 'x' for simplicity.
71 rk = A*x - b; % The first residual must be computed the hard way.
72 zk = M \ rk;
73 dk = -zk;
75 for k = [ 0 : max_iterations ]
76 if (norm(rk) < tolerance)
77 % Success.
78 return;
79 end
81 % Unfortunately, since we don't know the matrix C, it isn't
82 % easy to compute alpha_k with an existing step size function.
83 alpha_k = (rk' * zk)/(dk' * A * dk);
84 x_next = x + alpha_k*dk;
85 r_next = rk + alpha_k*A*dk;
86 z_next = M \ r_next;
87 beta_next = (r_next' * z_next)/(rk' * zk);
88 d_next = -z_next + beta_next*dk;
90 k = k + 1;
91 x = x_next;
92 rk = r_next;
93 zk = z_next;
94 dk = d_next;
95 end
96 end