function [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0) 
%TIKHONOV Tikhonov regularization. 
% 
% [x_lambda,rho,eta] = tikhonov(U,s,V,b,lambda,x_0) 
% [x_lambda,rho,eta] = tikhonov(U,sm,X,b,lambda,x_0) ,  sm = [sigma,mu] 
% 
% Computes the Tikhonov regularized solution x_lambda.  If the SVD 
% is used, i.e. if U, s, and V are specified, then standard-form 
% regularization is applied: 
%    min { || A x - b ||^2 + lambda^2 || x - x_0 ||^2 } . 
% If, on the other hand, the GSVD is used, i.e. if U, sm, and X are 
% specified, then general-form regularization is applied: 
%    min { || A x - b ||^2 + lambda^2 || L (x - x_0) ||^2 } . 
% 
% If x_0 is not specified, then x_0 = 0 is used.
%
% Note that x_0 cannot be used if A is underdetermined and L ~= I.
% 
% If lambda is a vector, then x_lambda is a matrix such that 
%    x_lambda = [ x_lambda(1), x_lambda(2), ... ] . 
% 
% The solution norm(standard-form case) or seminorm (general-form
% case) and the residual norm are returned in eta and rho. 
 
% Per Christian Hansen, IMM, April 14, 2003. 
 
% Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of 
% Ill-Posed Problems", Wiley, 1977. 
 
% Initialization. 
if (min(lambda)<0) 
  error('Illegal regularization parameter lambda') 
end 
m = size(U,1);
n = size(V,1);
[p,ps] = size(s); 
beta = U(:,1:p)'*b; 
zeta = s(:,1).*beta; 
ll = length(lambda); x_lambda = zeros(n,ll); 
rho = zeros(ll,1); eta = zeros(ll,1); 
 
% Treat each lambda separately. 
if (ps==1) 
  
  % The standard-form case.
  if (nargin==6), omega = V'*x_0; end 
  for i=1:ll 
    if (nargin==5) 
      x_lambda(:,i) = V(:,1:p)*(zeta./(s.^2 + lambda(i)^2)); 
      rho(i) = lambda(i)^2*norm(beta./(s.^2 + lambda(i)^2)); 
   else 
      x_lambda(:,i) = V(:,1:p)*... 
        ((zeta + lambda(i)^2*omega)./(s.^2 + lambda(i)^2)); 
      rho(i) = lambda(i)^2*norm((beta - s.*omega)./(s.^2 + lambda(i)^2)); 
    end 
    eta(i) = norm(x_lambda(:,i)); 
  end 
  if (nargout > 1 & size(U,1) > p) 
    rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2); 
  end
  
elseif (m>=n)
      
  % The overdetermined or square general-form case.
  gamma2 = (s(:,1)./s(:,2)).^2; 
  if (nargin==6), omega = V\x_0; omega = omega(1:p); end 
  if (p==n) 
    x0 = zeros(n,1); 
  else 
    x0 = V(:,p+1:n)*U(:,p+1:n)'*b;  
  end 
  for i=1:ll 
    if (nargin==5)
      xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
      x_lambda(:,i) = V(:,1:p)*xi + x0; 
      rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2)); 
    else
      xi = (zeta + lambda(i)^2*(s(:,2).^2).*omega)./(s(:,1).^2 + lambda(i)^2*s(:,2).^2)
      x_lambda(:,i) = V(:,1:p)*xi + x0; 
      rho(i) = lambda(i)^2*norm((beta - s(:,1).*omega)./(gamma2 + lambda(i)^2)); 
    end 
    eta(i) = norm(s(:,2).*xi); 
  end
  if (nargout > 1 & size(U,1) > p) 
    rho = sqrt(rho.^2 + norm(b - U(:,1:n)*[beta;U(:,p+1:n)'*b])^2); 
  end
  
else
      
  % The underdetermined general-form case.
  gamma2 = (s(:,1)./s(:,2)).^2;
  if (nargin==6), error('x_0 not allowed'), end 
  if (p==m) 
    x0 = zeros(n,1); 
  else 
    x0 = V(:,p+1:m)*U(:,p+1:m)'*b;  
  end 
  for i=1:ll
    xi = zeta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2);
    x_lambda(:,i) = V(:,1:p)*xi + x0; 
    rho(i) = lambda(i)^2*norm(beta./(gamma2 + lambda(i)^2)); 
    eta(i) = norm(s(:,2).*xi); 
  end
  
end