function [x_k,rho,eta] = ttls(V1,k,s1) %TTLS Truncated TLS regularization. % % [x_k,rho,eta] = ttls(V1,k,s1) % % Computes the truncated TLS solution %    x_k = - V1(1:n,k+1:n+1)*pinv(V1(n+1,k+1:n+1)) % where V1 is the right singular matrix in the SVD of the matrix %    [A,b] = U1*diag(s1)*V1' . % % If k is a vector, then x_k is a matrix such that %    x_k = [ x_k(1), x_k(2), ... ] . % If k is not specified, k = n is used. % % The solution norms and TLS residual norms corresponding to x_k are % returned in eta and rho, respectively.  Notice that the singular % values s1 are required to compute rho.  % Reference: R. D. Fierro, G. H. Golub, P. C. Hansen and D. P. O'Leary, % "Regularization by truncated total least squares", SIAM J. Sci. Comput. 18 % (1997), 1223-1241,  % Per Christian Hansen, IMM, 03/18/93.  % Initialization. [n1,m1] = size(V1); n = n1-1; if (m1 ~= n1), error('The matrix V1 must be square'), end if (nargin == 1), k = n; end lk = length(k); if (min(k) < 1 | max(k) > n)   error('Illegal truncation parameter k') end x_k = zeros(n,lk); if (nargout > 1)   if (nargin < 3)     error('The singular values must also be specified')   end   ns = length(s1); rho = zeros(lk,1); end if (nargout==3), eta = zeros(lk,1); end  % Treat each k separately. for j=1:lk   i = k(j);   v = V1(n1,i+1:n1); gamma = 1/(v*v');   x_k(:,j) = - V1(1:n,i+1:n1)*v'*gamma;   if (nargout > 1), rho(j) = norm(s1(i+1:ns)); end   if (nargout == 3), eta(j) = sqrt(gamma - 1); end end