function [x_lambda,rho,eta] = dsvd(U,s,V,b,lambda)%DSVD Damped SVD and GSVD regularization.%% [x_lambda,rho,eta] = dsvd(U,s,V,b,lambda)% [x_lambda,rho,eta] = dsvd(U,sm,X,b,lambda) ,  sm = [sigma,mu]%% Computes the damped SVD solution defined as%    x_lambda = V*inv(diag(s + lambda))*U'*b .% If lambda is a vector, then x_lambda is a matrix such that%    x_lambda = [ x_lambda(1), x_lambda(2), ... ] .%% If sm and X are specified, then the damped GSVD solution:%    x_lambda = X*[ inv(diag(sigma + lambda*mu)) 0 ]*U'*b%                 [            0                 I ]% is computed.%% The solution norm (standard-form case) or seminorm (general-form% case) and the residual norm are returned in eta and rho.% Reference: M. P. Ekstrom & R. L. Rhoads, "On the application of% eigenvector expansions to numerical deconvolution", J. Comp.% Phys. 14 (1974), 319-340.% The extension to GSVD is by P. C. Hansen.% Per Christian Hansen, IMM, April 14, 2003.% 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;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.  for i=1:ll    x_lambda(:,i) = V(:,1:p)*(beta./(s + lambda(i)));    rho(i) = lambda(i)*norm(beta./(s + lambda(i)));    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.  x0 = V(:,p+1:n)*U(:,p+1:n)'*b;  for i=1:ll    xi = beta./(s(:,1) + lambda(i)*s(:,2));    x_lambda(:,i) = V(:,1:p)*xi + x0;    rho(i) = lambda(i)*norm(beta./(s(:,1)./s(:,2) + lambda(i)));    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.  x0 = V(:,p+1:m)*U(:,p+1:m)'*b;  for i=1:ll    xi = beta./(s(:,1) + lambda(i)*s(:,2));    x_lambda(:,i) = V(:,1:p)*xi + x0;    rho(i) = lambda(i)*norm(beta./(s(:,1)./s(:,2) + lambda(i)));    eta(i) = norm(s(:,2).*xi);  endend