function [X,rho,eta,F] = cgls(A,b,k,reorth,s) %CGLS Conjugate gradient algorithm applied implicitly to the normal equations. % % [X,rho,eta,F] = cgls(A,b,k,reorth,s) % % Performs k steps of the conjugate gradient algorithm applied % implicitly to the normal equations A'*A*x = A'*b. % % The routine returns all k solutions, stored as columns of % the matrix X. The corresponding solution and residual norms % are returned in the vectors eta and rho, respectively. % % If the singular values s are also provided, cgls computes the % filter factors associated with each step and stores them % columnwise in the matrix F. % % Reorthogonalization of the normal equation residual vectors % A'*(A*X(:,i)-b) is controlled by means of reorth: % reorth = 0 : no reorthogonalization (default), % reorth = 1 : reorthogonalization by means of MGS. % References: A. Bjorck, "Numerical Methods for Least Squares Problems", % SIAM, Philadelphia, 1996. % C. R. Vogel, "Solving ill-conditioned linear systems using the % conjugate gradient method", Report, Dept. of Mathematical % Sciences, Montana State University, 1987. % Per Christian Hansen, IMM, July 23, 2007. % The fudge threshold is used to prevent filter factors from exploding. fudge_thr = 1e-4; % Initialization. if (k < 1), error('Number of steps k must be positive'), end if (nargin==3), reorth = 0; end if (nargout==4 & nargin<5), error('Too few input arguments'), end if (reorth<0 | reorth>1), error('Illegal reorth'), end [m,n] = size(A); X = zeros(n,k); if (reorth==1), ATr = zeros(n,k+1); end if (nargout > 1) eta = zeros(k,1); rho = eta; end if (nargin==5) F = zeros(n,k); Fd = zeros(n,1); s2 = s.^2; end % Prepare for CG iteration. x = zeros(n,1); d = A'*b; r = b; normr2 = d'*d; if (reorth==1), ATr(:,1) = d/norm(d); end % Iterate. for j=1:k % Update x and r vectors. Ad = A*d; alpha = normr2/(Ad'*Ad); x = x + alpha*d; r = r - alpha*Ad; s = A'*r; % Reorthogonalize s to previous s-vectors, if required. if (reorth==1) for i=1:j, s = s - (ATr(:,i)'*s)*ATr(:,i); end ATr(:,j+1) = s/norm(s); end % Update d vector. normr2_new = s'*s; beta = normr2_new/normr2; normr2 = normr2_new; d = s + beta*d; X(:,j) = x; % Compute norms, if required. if (nargout>1), rho(j) = norm(r); end if (nargout>2), eta(j) = norm(x); end % Compute filter factors, if required. if (nargin==5) if (j==1) F(:,1) = alpha*s2; Fd = s2 - s2.*F(:,1) + beta*s2; else F(:,j) = F(:,j-1) + alpha*Fd; Fd = s2 - s2.*F(:,j) + beta*Fd; end if (j > 2) f = find(abs(F(:,j-1)-1) < fudge_thr & abs(F(:,j-2)-1) < fudge_thr); if ~isempty(f), F(f,j) = ones(length(f),1); end end end end