function [x_delta,lambda] = discrep(U,s,V,b,delta,x_0) %DISCREP Discrepancy principle criterion for choosing the reg. parameter. % % [x_delta,lambda] = discrep(U,s,V,b,delta,x_0) % [x_delta,lambda] = discrep(U,sm,X,b,delta,x_0)  ,  sm = [sigma,mu] % % Least squares minimization with a quadratic inequality constraint: %    min || x - x_0 ||       subject to   || A x - b || <= delta %    min || L (x - x_0) ||   subject to   || A x - b || <= delta % where x_0 is an initial guess of the solution, and delta is a % positive constant.  Requires either the compact SVD of A saved as % U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X. % The regularization parameter lambda is also returned. % % If delta is a vector, then x_delta is a matrix such that %    x_delta = [ x_delta(1), x_delta(2), ... ] . % % If x_0 is not specified, x_0 = 0 is used.  % Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed % Problems", Springer, 1984; Chapter 26.  % Per Christian Hansen, IMM, 12/29/97.  % Initialization. [n,p] = size(V);    [p,ps] = size(s);      ld  = length(delta); x_k = zeros(n,ld);  lambda = zeros(ld,1);  rho = zeros(p,1); if (min(delta)<0)   error('Illegal inequality constraint delta') end if (nargin==5), x_0 = zeros(n,1); end if (ps == 1), omega = V'*x_0; else, omega = V\x_0; end  % Compute residual norms corresponding to TSVD/TGSVD. beta = U'*b; nb   = norm(b); snz  = length(find(s(:,1)>0)); if (ps == 1)   delta_0 = norm(b - U*beta);   rho(n) = delta_0^2;   for i=n:-1:2     rho(i-1) = rho(i) + (beta(i) - s(i)*omega(i))^2;   end else   delta_0 = norm(b - U*beta);   rho(1) = delta_0^2;   for i=1:p-1     rho(i+1) = rho(i) + (beta(i) - s(i,1)*omega(i))^2;   end end  % Check input. if (min(delta) < delta_0)   error('Irrelevant delta < || (I - U*U'')*b ||') end  % Determine the initial guess via rho-vector, then solve the nonlinear % equation || b - A x ||^2 - delta_0^2 = 0 via Newton's method. if (ps == 1)   s2 = s.^2;   for k=1:ld     if (delta(k)^2 >= norm(beta - s.*omega)^2 + delta_0^2)       x_delta(:,k) = x_0;     else       [dummy,kmin] = min(abs(rho - delta(k)^2));       lambda_0 = s(kmin);       lambda(k) = newton(lambda_0,delta(k),s,beta,omega,delta_0);       e = s./(s2 + lambda(k)^2); f = s.*e;       x_delta(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega);     end   end else   omega = omega(1:p); gamma = s(:,1)./s(:,2);   x_u   = V(:,p+1:n)*beta(p+1:n);   for k=1:ld     if (delta(k)^2 >= norm(beta(1:p) - s(:,1).*omega)^2 + delta_0^2)       x_delta(:,k) = V*[omega;U(:,p+1:n)'*b];     else       [dummy,kmin] = min(abs(rho - delta(k)^2));       lambda_0 = gamma(kmin);       lambda(k) = newton(lambda_0,delta(k),s,beta(1:p),omega,delta_0);       e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e;       x_delta(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ...                                (1-f).*s(:,2).*omega) + x_u;     end   end end