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 (14.56 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 % - Q -- 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 Q 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 Qx 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 computed solution to Qx=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 % The rather verbose name of this function was chosen to avoid
56 % conflicts with other implementations.
57 %
58 % REFERENCES:
59 %
60 % 1. Guler, Osman. Foundations of Optimization. New York, Springer,
61 % 2010.
62 %
63 % 2. Shewchuk, Jonathan Richard. An Introduction to the Conjugate
64 % Gradient Method Without the Agonizing Pain, Edition 1.25.
65 % August 4, 1994.
66 %
68 % We use this in the inner loop.
69 n = length(x0);
70 sqrt_n = floor(sqrt(n));
72 % Set k=0 first, that way the references to xk,rk,zk,dk which
73 % immediately follow correspond (semantically) to x0,r0,z0,d0.
74 k = 0;
76 xk = x0;
77 rk = Q*xk - b;
78 zk = M \ rk;
79 dk = -zk;
81 while (k <= max_iterations && norm(rk, 'inf') > tolerance)
82 % Used twice, avoid recomputation.
83 rkzk = rk' * zk;
85 % The term alpha_k*dk appears twice, but so does Q*dk. We can't
86 % do them both, so we precompute the more expensive operation.
87 Qdk = Q * dk;
89 % We're going to divide by this quantity...
90 dkQdk = dk' * Qdk;
92 % So if it's too close to zero, we replace it with something
93 % comparable but non-zero.
94 if (dkQdk < eps)
95 dkQdk = eps;
96 end
98 alpha_k = rkzk/dkQdk;
99 x_next = xk + (alpha_k * dk);
101 % The recursive definition of r_next is prone to accumulate
102 % roundoff error. When sqrt(n) divides k, we recompute the
103 % residual to minimize this error. This modification was suggested
104 % by the second reference.
105 if (mod(k, sqrt_n) == 0)
106 r_next = Q*x_next - b;
107 else
108 r_next = rk + (alpha_k * Qdk);
109 end
111 z_next = M \ r_next;
112 beta_next = (r_next' * z_next)/rkzk;
113 d_next = -z_next + beta_next*dk;
115 % We potentially just performed one more iteration than necessary
116 % in order to simplify the loop. Note that due to the structure of
117 % our loop, we will have k > max_iterations when we fail to
118 % converge.
119 k = k + 1;
120 xk = x_next;
121 rk = r_next;
122 zk = z_next;
123 dk = d_next;
124 end
126 % If we make it here, one of the two stopping conditions was met.
127 x = xk;
128 end