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16 Apr 1998 (Updated )

Analysis and Solution of Discrete Ill-Posed Problems.

l_curve(U,sm,b,method,L,V)
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.

% 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, July 26, 2007.

% 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));
  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 = '-'; 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';
  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 = ':';
  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'; ps = 2;  % General form regularization.

else
  error('Illegal method')
end

% Locate the "corner" of the L-curve, if required.
if (locate)
  [reg_corner,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,sm,b,method);
end

% Make plot.
plot_lc(rho,eta,marker,ps,reg_param);
if locate
  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

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