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Algorithm: External Approximation of Minkowski-Sum

Add as much inputarguments as you need.

Additionally you can add 2 more input parameters:
In the n-1th parameter you can specify the criterion to be optimized. This can either be 'trace' or 'determinant'.
In the last parameter you can use an optional weighting that will be used for optimalization ... (show more)
Tags: ellipsoid
Usage: Algorithm is public.
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Florian Pfaff
08/21/2012 12:31 a.m. (version #1)

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Source Code


 1 function Xresult=minkExternalApprox(varargin)
 2 % Returns an external approximatization with (among those that can be
 3 % created using the formula) minizes a certain criterion (optimality is
 4 % not always guaranteed)
 5 % Syntax minkExternalApprox(Q1,Q2,...Qn,'trace',weight)
 6 %
 7 % @author Florian Pfaff
 8 % @date 2011
10     %test if weighting and criterion was passed
11     if nargin>2 && ischar(varargin{end-1})
12             criterion=varargin{end-1};
13             weighting=varargin{end};
14             varargin=varargin(1:end-2);
15     %test if only criterion was passed
16     elseif ischar(varargin{end})
17         criterion=varargin{end};
18         varargin=varargin(1:end-1);
19         weighting=eye(size(varargin{1}));
20     else
21         criterion='trace';
22         weighting=eye(size(varargin{1}));
23 %         criterion='volume';
24     end
25     %remove empty to prevent div / 0!
26     notEmpty=cellfun(@(X)max(max(X))~=0,varargin);
27     varargin=varargin(notEmpty);
28     %if only single matrix do not approx
29     if length(varargin)<=1
30         warning('ELLIPSOIDAPPROX:TOOFEW','At most one nonzero Matrix passed.')
31         Xresult=varargin{1};
32         return;
33     end
35     if strncmpi(criterion,'trace',3)||strncmpi(criterion,'spur',3)
36         q_i=(ones(length(varargin))-eye(length(varargin)))*cellfun(@(X)sqrt(trace(weighting*X*weighting')),varargin)'./(cellfun(@(X)sqrt(trace(weighting*X*weighting')),varargin)');
37         Xresult=sum(reshape(cell2mat(arrayfun(@(i){(1+q_i(i))*varargin{i}},1:length(varargin))),[size(varargin{1}),length(varargin)]),3);
38     elseif strncmpi(criterion,'volume',3)
39         if length(varargin)>2
40             error('Volume formula currently supports only 2 at a time');
41         end
42         %solution using syms
43         syms lambda real
44         lambdas=solve(det(varargin{1}-lambda*varargin{2}));
45         disp(real(double(lambdas)));
46         syms p positive        
47         pmin=real(double(solve(sum(1./(lambdas+p))-2/(p*(p+1)),'Real',true)));
48         Xresult=(pmin^-1+1)*varargin{1}+(pmin+1)*varargin{2};
50     else
51         error('Unknown mode');
52     end
54     %guarantee symmetric matrix
55     Xresult=triu(Xresult)+triu(Xresult)'-diag(diag(Xresult));
57  end
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