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Create local repository for matrix toolboxes. Step #0 done.
author Antonio Pino Robles <data.script93@gmail.com>
date Wed, 06 May 2015 14:56:53 +0200
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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
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1 function y = dual(x, p)
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2 %DUAL Dual vector with respect to Holder p-norm.
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3 % Y = DUAL(X, p), where 1 <= p <= inf, is a vector of unit q-norm
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4 % that is dual to X with respect to the p-norm, that is,
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5 % norm(Y, q) = 1 where 1/p + 1/q = 1 and there is
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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6 % equality in the Holder inequality: X'*Y = norm(X, p)*norm(Y, q).
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7 % Special case: DUAL(X), where X >= 1 is a scalar, returns Y such
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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8 % that 1/X + 1/Y = 1.
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9
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10 % Called by PNORM.
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11
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12 warns = warning;
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13 warning('off')
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14
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15 if nargin == 1
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16 if length(x) == 1
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17 y = 1/(1-1/x);
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18 return
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19 else
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20 error('Second argument missing.')
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21 end
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22 end
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23
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24 q = 1/(1-1/p);
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25
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26 if norm(x,inf) == 0, y = x; return, end
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27
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28 if p == 1
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29
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30 y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real).
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32 elseif p == inf
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33
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34 [xmax, k] = max(abs(x));
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35 f = find(abs(x)==xmax); k = f(1);
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36 y = zeros(size(x));
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37 y(k) = sign(x(k)); % y is a multiple of unit vector e_k.
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39 else % 1 < p < inf. Dual is unique in this case.
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41 x = x/norm(x,inf); % This scaling helps to avoid under/over-flow.
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42 y = abs(x).^(p-1) .* ( sign(x) + (x==0) );
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43 y = y / norm(y,q); % Normalize to unit q-norm.
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45 end
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46 warning(warns)