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')endm = 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);  endelseif (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);  endelse  % 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);  endend