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16 Apr 1998 (Updated )

Analysis and Solution of Discrete Ill-Posed Problems.

corner(rho,eta,fig)
function [k_corner,info] = corner(rho,eta,fig)
%CORNER Find corner of discrete L-curve via adaptive pruning algorithm.
%
% [k_corner,info] = corner(rho,eta,fig)
%
% Returns the integer k_corner such that the corner of the log-log
% L-curve is located at ( log(rho(k_corner)) , log(eta(k_corner)) ).
%
% The vectors rho and eta must contain corresponding values of the
% residual norm || A x - b || and the solution's (semi)norm || x ||
% or || L x || for a sequence of regularized solutions, ordered such
% that rho and eta are monotonic and such that the amount of
% regularization decreases as k increases.
%
% The second output argument describes possible warnings.
% Any combination of zeros and ones is possible.
%   info = 000 : No warnings - rho and eta describe a discrete 
%                   L-curve with a corner.
%   info = 001 : Bad data - some elements of rho and/or eta are
%                   Inf, NaN, or zero.
%   info = 010 : Lack of monotonicity - rho and/or eta are not
%                   strictly monotonic.
%   info = 100 : Lack of convexity - the L-curve described by rho
%                   and eta is concave and has no corner.
%
% The warnings described above will also result in text warnings on the
% command line. Type 'warning off Corner:warnings' to disable all
% command line warnings from this function.
%
% If a third input argument is present, then a figure will show the discrete
% L-curve in log-log scale and also indicate the found corner.

% Reference: P. C. Hansen, T. K. Jensen and G. Rodriguez, "An adaptive
% pruning algorithm for the discrete L-curve criterion," J. Comp. Appl.
% Math., 198 (2007), 483-492.
  
% Per Christian Hansen and Toke Koldborg Jensen, IMM, DTU;
% Giuseppe Rodriguez, University of Cagliari, Italy; March 22, 2006.

% Initialization of data
rho = rho(:);       % Make rho and eta column vectors.
eta = eta(:);

if (nargin < 3) | isempty(fig)
    fig = 0;        % Default is no figure.
elseif fig < 0,
    fig = 0;
end

info = 0;

fin = isfinite(rho+eta);    % NaN or Inf will cause trouble.
nzr = rho.*eta~=0;          % A zero will cause trouble.
kept = find(fin & nzr);
if isempty(kept)
	error('Too many Inf/NaN/zeros found in data')
end
if length(kept) < length(rho)
	info = info + 1;
	warning('Corner:warnings', ...
           ['Bad data - Inf, NaN or zeros found in data\n' ...
            '         Continuing with the remaining data'])
end
rho = rho(kept);            % rho and eta with bad data removed.
eta = eta(kept);

if any(rho(1:end-1)<rho(2:end)) | any(eta(1:end-1)>eta(2:end))
	info = info + 10;
	warning('Corner:warnings', 'Lack of monotonicity')
end

% Prepare for adaptive algorithm.
nP = length(rho);           % Number of points.
P = log10([rho eta]);       % Coordinates of the loglog L-curve.
V = P(2:nP,:)-P(1:nP-1,:);  % The vectors defined by these coordinates.
v = sqrt(sum(V.^2,2));      % The length of the vectors.
W = V./repmat(v,1,2);       % Normalized vectors.
clist = [];                 % List of candidates.
p = min(5, nP-1);           % Number of vectors in pruned L-curve.
convex = 0;                 % Are the pruned L-curves convex?

% Sort the vectors according to the length, the longest first.
[Y,I] = sort(v);
I = flipud(I);

% Main loop -- use a series of pruned L-curves. The two functions
% 'Angles' and 'Global_Behavior' are used to locate corners of the
% pruned L-curves. Put all the corner candidates in the clist vector.
while p < (nP-1)*2
	elmts = sort(I(1:min(p, nP-1)));
    
    % First corner location algorithm
	candidate = Angles( W(elmts,:), elmts);
	if candidate>0,
		convex = 1;
	end
	if candidate & ~any(clist==candidate)
		clist = [clist;candidate];
	end
    
    % Second corner location algorithm
	candidate = Global_Behavior(P, W(elmts,:), elmts);
	if ~any(clist==candidate)
		clist = [clist; candidate];
	end
    
	p = p*2;
end

% Issue a warning and return if none of the pruned L-curves are convex.
if convex==0
	k_corner = [];
	info = info + 100;
	warning('Corner:warnings', 'Lack of convexity')
	return
end

% Put rightmost L-curve point in clist if not already there; this is
% used below to select the corner among the corner candidates.
if sum(clist==1) == 0
	clist = [1;clist];
end

% Sort the corner candidates in increasing order.
clist = sort(clist);

% Select the best corner among the corner candidates in clist. 
% The philosophy is: select the corner as the rightmost corner candidate
% in the sorted list for which going to the next corner candidate yields
% a larger increase in solution (semi)norm than decrease in residual norm,
% provided that the L-curve is convex in the given point. If this is never
% the case, then select the leftmost corner candidate in clist.

vz = find(diff(P(clist,2)) ...  % Points where the increase in solution
     >= abs(diff(P(clist,1)))); % (semi)norm is larger than or equal
                                % to the decrease in residual norm.
if length(vz)>1
	if(vz(1) == 1),  vz = vz(2:end);  end
elseif length(vz)==1
	if(vz(1) == 1),  vz = [];  end
end

if isempty(vz)		
        % No large increase in solution (semi)norm is found and the
        % leftmost corner candidate in clist is selected.
	index = clist(end);
else
        % The corner is selected as described above.
	vects = [P(clist(2:end),1)-P(clist(1:end-1),1) ...
			P(clist(2:end),2)-P(clist(1:end-1),2)];
	vects = sparse(diag(1./sqrt(sum(vects.^2,2)))) * vects;
	delta = vects(1:end-1,1).*vects(2:end,2) ...
			- vects(2:end,1).*vects(1:end-1,2);
	vv = find(delta(vz-1)<=0);
	if isempty(vv)
		index = clist(vz(end));
	else
		index = clist(vz(vv(1)));
	end
end

% Corner according to original vectors without Inf, NaN, and zeros removed.
k_corner = kept(index);

if fig  % Show log-log L-curve and indicate the found corner.
    figure(fig); clf
    diffrho2 = (max(P(:,1))-min(P(:,1)))/2;
    diffeta2 = (max(P(:,2))-min(P(:,2)))/2;
    loglog(rho, eta, 'k--o'); hold on; axis square;
    % Mark the corner.
    loglog([min(rho)/100,rho(index)],[eta(index),eta(index)],':r',... 
           [rho(index),rho(index)],[min(eta)/100,eta(index)],':r') 
    % Scale axes to same number of decades.
    if abs(diffrho2)>abs(diffeta2),
        ax(1) = min(P(:,1)); ax(2) = max(P(:,1));
        mid = min(P(:,2)) + (max(P(:,2))-min(P(:,2)))/2;
        ax(3) = mid-diffrho2; ax(4) = mid+diffrho2;
    else
        ax(3) = min(P(:,2)); ax(4) = max(P(:,2));
        mid = min(P(:,1)) + (max(P(:,1))-min(P(:,1)))/2;
        ax(1) = mid-diffeta2; ax(2) = mid+diffeta2;
    end
    ax = 10.^ax; ax(1) = ax(1)/2; axis(ax);
	xlabel('residual norm || A x - b ||_2')
	ylabel('solution (semi)norm || L x ||_2');
    title(sprintf('Discrete L-curve, corner at %d', k_corner));
end

% =========================================================================
% First corner finding routine -- based on angles

function index = Angles( W, kv)

% Wedge products
delta = W(1:end-1,1).*W(2:end,2) - W(2:end,1).*W(1:end-1,2);

[mm kk] = min(delta);
if mm < 0		% Is it really a corner?
	index = kv(kk) + 1;
else			% If there is no corner, return 0.
	index = 0;
end

% =========================================================================
% Second corner finding routine -- based on global behavior of the L-curve

function index = Global_Behavior(P, vects, elmts)

hwedge = abs(vects(:,2));  % Abs of wedge products between
                           % normalized vectors and horizontal,
                           % i.e., angle of vectors with horizontal.
[An, In] = sort(hwedge);   % Sort angles in increasing order.

% Locate vectors for describing horizontal and vertical part of L-curve.
count = 1;
ln = length(In);
mn = In(1);
mx = In(ln);
while(mn>=mx)
	mx = max([mx In(ln-count)]);
	count = count + 1;
	mn = min([mn In(count)]);
end
if count > 1
	I = 0; J = 0;
	for i=1:count
		for j=ln:-1:ln-count+1
			if(In(i) < In(j))
				I = In(i); J = In(j); break
			end
		end
		if I>0, break; end
	end
else
	I = In(1); J = In(ln);
end

% Find intersection that describes the "origin".
x3 = P(elmts(J)+1,1)+(P(elmts(I),2)-P(elmts(J)+1,2))/(P(elmts(J)+1,2) ...
			-P(elmts(J),2))*(P(elmts(J)+1,1)-P(elmts(J),1));
origin = [x3 P(elmts(I),2)];

% Find distances from the original L-curve to the "origin".  The corner
% is the point with the smallest Euclidian distance to the "origin".
dists = (origin(1)-P(:,1)).^2+(origin(2)-P(:,2)).^2;
[Y,index] = min(dists);

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