function [X,rho,eta,F] = pcgls(A,L,W,b,k,reorth,sm) %PCGLS "Preconditioned" conjugate gradients appl. implicitly to normal equations. % [X,rho,eta,F] = pcgls(A,L,W,b,k,reorth,sm) % % Performs k steps of the `preconditioned' conjugate gradient % algorithm applied implicitly to the normal equations %    (A*L_p)'*(A*L_p)*x = (A*L_p)'*b , % where L_p is the A-weighted generalized inverse of L.  Notice that the % matrix W holding a basis for the null space of L must also be specified. % % The routine returns all k solutions, stored as columns of the matrix X. % The solution seminorm and residual norm are returned in eta and rho, % respectively. % % If the generalized singular values sm of (A,L) are also provided, % pcgls 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. % P. C. Hansen, "Rank-Deficient and Discrete Ill-Posed Problems. % Numerical Aspects of Linear Inversion", SIAM, Philadelphia, 1997.   % Per Christian Hansen, IMM and Martin Hanke, Institut fuer % Praktische Mathematik, Universitaet Karlsruhe, April 8, 2001.  % 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==5), reorth = 0; end if (nargout==4 & nargin<7), error('Too few input arguments'), end if (reorth<0 | reorth>1), error('Illegal reorth'), end [m,n] = size(A); [p,n1] = size(L); X = zeros(n,k); if (nargout > 1)   eta = zeros(k,1); rho = eta; end if (nargin==7)   F = zeros(p,k); Fd = zeros(p,1); gamma = (sm(:,1)./sm(:,2)).^2; end  % Prepare for computations with L_p. [NAA,x_0] = pinit(W,A,b);  % Prepare for CG iteartion. x  = x_0; r  = b - A*x_0; s = (r'*A)';   % A'*r; q1 = ltsolve(L,s); q  = lsolve(L,q1,W,NAA); z  = q; dq = s'*q; if (nargout>2), z1 = q1; x1 = zeros(p,1); end if (reorth==1), Q1n = q1/norm(q1); end  % Iterate. for j=1:k    % Update x and r vectors; compute q1.   Az  = A*z; alpha = dq/(Az'*Az);   x   = x + alpha*z;   r   = r - alpha*Az; s = (r'*A)';   % A'*r;   q1  = ltsolve(L,s);    % Reorthogonalize q1 to previous q1-vectors, if required.   if (reorth==1)     for i=1:j, q1 = q1 - (Q1n(:,i)'*q1)*Q1n(:,i); end     Q1n = [Q1n,q1/norm(q1)];   end    % Update z vector.   q   = lsolve(L,q1,W,NAA);   dq2 = s'*q; beta = dq2/dq;   dq  = dq2;   z   = q + beta*z;   X(:,j) = x;   if (nargout>1), rho(j) = norm(r); end   if (nargout>2)     x1 = x1 + alpha*z1; z1 = q1 + beta*z1; eta(j) = norm(x1);   end    % Compute filter factors, if required.   if (nargin==7)     if (j==1)       F(:,1) = alpha*gamma;       Fd = gamma - gamma.*F(:,1) + beta*gamma;     else       F(:,j) = F(:,j-1) + alpha*Fd;       Fd = gamma - gamma.*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 (length(f) > 0), F(f,j) = ones(length(f),1); end     end   end  end