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