From af2083885af78b1290c21f2852c6fdba25820918 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 20 Mar 2013 00:11:23 -0400 Subject: [PATCH] Commit a simpler version of the preconditioned CGM. --- ...preconditioned_conjugate_gradient_method.m | 68 ++++++------------- 1 file changed, 19 insertions(+), 49 deletions(-) diff --git a/optimization/preconditioned_conjugate_gradient_method.m b/optimization/preconditioned_conjugate_gradient_method.m index 1442194..af70af5 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(A, +function [x, k] = preconditioned_conjugate_gradient_method(Q, M, b, x0, @@ -7,31 +7,29 @@ function [x, k] = preconditioned_conjugate_gradient_method(A, % % Solve, % - % Ax = b + % Qx = b % % or equivalently, % - % min [phi(x) = (1/2)* + ] + % min [phi(x) = (1/2)* + ] % - % 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. + % using the preconditioned conjugate gradient method (14.54 in + % Guler). % % INPUT: % - % - A -- The coefficient matrix of the system to solve. Must + % - Q -- The coefficient matrix of the system to solve. Must % be positive definite. % % - M -- The preconditioning matrix. If the actual matrix used - % 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. + % to precondition Q is called C, i.e. C^(-1) * Q * + % C^(-T) == \bar{Q}, then M=CC^T. % % - b -- The right-hand-side of the system to solve. % % - x0 -- The starting point for the search. % - % - tolerance -- How close Ax has to be to b (in + % - tolerance -- How close Qx has to be to b (in % magnitude) before we stop. % % - max_iterations -- The maximum number of iterations to @@ -39,7 +37,7 @@ function [x, k] = preconditioned_conjugate_gradient_method(A, % % OUTPUT: % - % - x - The solution to Ax=b. + % - x - The solution to Qx=b. % % - k - The ending value of k; that is, the number of % iterations that were performed. @@ -48,49 +46,21 @@ function [x, k] = preconditioned_conjugate_gradient_method(A, % % 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; - for k = [ 0 : max_iterations ] - if (norm(rk) < tolerance) - % Success. - return; - end + Ct = chol(M); + C = Ct'; + C_inv = inv(C); + Ct_inv = inv(Ct); - % 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; + Q_bar = C_inv * Q * Ct_inv; + b_bar = C_inv * b; - k = k + 1; - x = x_next; - rk = r_next; - zk = z_next; - dk = d_next; - end + % 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; end -- 2.33.1