function [reg_corner,rho,eta,reg_param] = l_curve(U,sm,b,method,L,V) %L_CURVE Plot the L-curve and find its "corner". % % [reg_corner,rho,eta,reg_param] = %                  l_curve(U,s,b,method) %                  l_curve(U,sm,b,method)  ,  sm = [sigma,mu] %                  l_curve(U,s,b,method,L,V) % % Plots the L-shaped curve of eta, the solution norm || x || or % semi-norm || L x ||, as a function of rho, the residual norm % || A x - b ||, for the following methods: %    method = 'Tikh'  : Tikhonov regularization   (solid line ) %    method = 'tsvd'  : truncated SVD or GSVD     (o markers  ) %    method = 'dsvd'  : damped SVD or GSVD        (dotted line) %    method = 'mtsvd' : modified TSVD             (x markers  ) % The corresponding reg. parameters are returned in reg_param.  If no% method is specified then 'Tikh' is default.  For other methods use plot_lc.%% Note that 'Tikh', 'tsvd' and 'dsvd' require either U and s (standard-% form regularization) or U and sm (general-form regularization), while% 'mtvsd' requires U and s as well as L and V.% % If any output arguments are specified, then the corner of the L-curve % is identified and the corresponding reg. parameter reg_corner is % returned.  Use routine l_corner if an upper bound on eta is required. % % If the Spline Toolbox is not available and reg_corner is requested, % then the routine returns reg_corner = NaN for 'tsvd' and 'mtsvd'.  % Reference: P. C. Hansen & D. P. O'Leary, "The use of the L-curve in % the regularization of discrete ill-posed problems",  SIAM J. Sci. % Comput. 14 (1993), pp. 1487-1503.  % Per Christian Hansen, IMM, Sep. 13, 2001. % Set defaults. if (nargin==3), method='Tikh'; end  % Tikhonov reg. is default. npoints = 200;  % Number of points on the L-curve for Tikh and dsvd. smin_ratio = 16*eps;  % Smallest regularization parameter.  % Initialization. [m,n] = size(U); [p,ps] = size(sm); if (nargout > 0), locate = 1; else locate = 0; end beta = U'*b; beta2 = norm(b)^2 - norm(beta)^2; if (ps==1)   s = sm; beta = beta(1:p); else   s = sm(p:-1:1,1)./sm(p:-1:1,2); beta = beta(p:-1:1); end xi = beta(1:p)./s;  if (strncmp(method,'Tikh',4) | strncmp(method,'tikh',4))    eta = zeros(npoints,1); rho = eta; reg_param = eta; s2 = s.^2;   reg_param(npoints) = max([s(p),s(1)*smin_ratio]);   ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));   ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));   for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end   for i=1:npoints     f = s2./(s2 + reg_param(i)^2);     eta(i) = norm(f.*xi);     rho(i) = norm((1-f).*beta(1:p));   end   if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end   marker = '-'; pos = .8; txt = 'Tikh.';  elseif (strncmp(method,'tsvd',4) | strncmp(method,'tgsv',4))    eta = zeros(p,1); rho = eta;   eta(1) = abs(xi(1))^2;   for k=2:p, eta(k) = eta(k-1) + abs(xi(k))^2; end   eta = sqrt(eta);   if (m > n)     if (beta2 > 0), rho(p) = beta2; else rho(p) = eps^2; end   else     rho(p) = eps^2;   end   for k=p-1:-1:1, rho(k) = rho(k+1) + abs(beta(k+1))^2; end   rho = sqrt(rho);   reg_param = [1:p]'; marker = 'o'; pos = .75;   if (ps==1)     U = U(:,1:p); txt = 'TSVD';   else     U = U(:,1:p); txt = 'TGSVD';   end  elseif (strncmp(method,'dsvd',4) | strncmp(method,'dgsv',4))    eta = zeros(npoints,1); rho = eta; reg_param = eta;   reg_param(npoints) = max([s(p),s(1)*smin_ratio]);   ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));   for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end   for i=1:npoints     f = s./(s + reg_param(i));     eta(i) = norm(f.*xi);     rho(i) = norm((1-f).*beta(1:p));   end   if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end   marker = ':'; pos = .85;   if (ps==1), txt = 'DSVD'; else txt = 'DGSVD'; end  elseif (strncmp(method,'mtsv',4))    if (nargin~=6)     error('The matrices L and V must also be specified')   end   [p,n] = size(L); rho = zeros(p,1); eta = rho;   [Q,R] = qr(L*V(:,n:-1:n-p),0);   for i=1:p     k = n-p+i;     Lxk = L*V(:,1:k)*xi(1:k);     zk = R(1:n-k,1:n-k)\(Q(:,1:n-k)'*Lxk); zk = zk(n-k:-1:1);     eta(i) = norm(Q(:,n-k+1:p)'*Lxk);     if (i < p)       rho(i) = norm(beta(k+1:n) + s(k+1:n).*zk);     else       rho(i) = eps;     end   end   if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end   reg_param = [n-p+1:n]'; txt = 'MTSVD';   U = U(:,reg_param); sm = sm(reg_param);   marker = 'x'; pos = .7; ps = 2;  % General form regularization.   else, error('Illegal method'), end  % Locate the "corner" of the L-curve, if required.  If the Spline % Toolbox is not available, return NaN for reg_corner. if (locate)   SkipCorner = ( (strncmp(method,'tsvd',4) | strncmp(method,'tgsv',4) | ...                   strncmp(method,'mtsv',4)) & exist('splines')~=7 );   if (SkipCorner)     reg_corner = NaN;   else     [reg_corner,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,sm,b,method);   end end  % Make plot. plot_lc(rho,eta,marker,ps,reg_param); if (locate & ~SkipCorner)   ax = axis;   HoldState = ishold; hold on;   loglog([min(rho)/100,rho_c],[eta_c,eta_c],':r',...          [rho_c,rho_c],[min(eta)/100,eta_c],':r')   title(['L-curve, ',txt,' corner at ',num2str(reg_corner)]);   axis(ax)   if (~HoldState), hold off; end end