function [x_k,rho,eta] = tgsvd(U,sm,X,b,k)%TGSVD Truncated GSVD regularization.%% [x_k,rho,eta] = tgsvd(U,sm,X,b,k) ,  sm = [sigma,mu]%% Computes the truncated GSVD solution%            [ 0              0                 0    ]%    x_k = X*[ 0  inv(diag(sigma(p-k+1:p)))     0    ]*U'*b .%            [ 0              0             eye(n-p) ]% If k is a vector, then x_k is a matrix such that%    x_k = [ x_k(1), x_k(2), ... ] .%% The solution seminorm and the residual norm are returned in eta and rho.% Reference: P. C. Hansen, "Regularization, GSVD and truncated GSVD",% BIT 29 (1989), 491-504.% Per Christian Hansen, IMM, Feb. 24, 2008.% Initializationm = size(U,1);n = size(X,1);p = size(sm,1);lk = length(k);if (min(k)<0 | max(k)>p)  error('Illegal truncation parameter k')endx_k = zeros(n,lk);eta = zeros(lk,1); rho = zeros(lk,1);beta = U'*b;xi = beta(1:p)./sm(:,1);if (nargout==3), mxi = sm(:,2).*xi; endif (m>=n)  % The overdetermined or square case. Treat each k separately.  if (p==n)    x_0 = zeros(n,1);  else    x_0 = X(:,p+1:n)*(U(:,p+1:n)'*b);  end  for j=1:lk    i = k(j); pi1 = p-i+1;    if(i==0)      x_k(:,j) = x_0;    else      x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0;    end    if (nargout>1), rho(j) = norm(beta(1:p-i)); end    if (nargout==3), eta(j) = norm(mxi(pi1:p)); end  end  if (nargout > 1 & size(U,1) > n)    rho = sqrt(rho.^2 + norm(b - U(:,1:n)*beta(1:n))^2);  endelse  % The underdetermined case. Treat each k separately.  if (p==m)    x_0 = zeros(n,1);  else    x_0 = X(:,p+1:m)*(U(:,p+1:m)'*b);  end  for j=1:lk    i = k(j); pi1 = p-i+1;    if(i==0)      x_k(:,j) = x_0;    else      x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0;    end    if (nargout>1), rho(j) = norm(beta(1:p-i)); end    if (nargout==3), eta(j) = norm(mxi(pi1:p)); end  endend