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annotate toolbox/dual.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
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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
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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
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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 if max(size(x)) == 1 & nargin == 1
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13 p = x;
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14 end
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16 % The following test avoids a `division by zero message' when p = 1.
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17 if p == 1
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18 q = inf;
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19 else
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20 q = 1/(1-1/p);
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21 end
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23 if max(size(x)) == 1 & nargin == 1
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24 y = q;
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25 return
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26 end
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28 if norm(x,inf) == 0, y = x; return, end
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30 if p == 1
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32 y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real).
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34 elseif p == inf
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35
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36 [xmax, k] = max(abs(x));
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37 f = find(abs(x)==xmax); k = f(1);
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38 y = zeros(size(x));
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39 y(k) = sign(x(k)); % y is a multiple of unit vector e_k.
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41 else % 1 < p < inf. Dual is unique in this case.
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43 x = x/norm(x,inf); % This scaling helps to avoid under/over-flow.
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44 y = abs(x).^(p-1) .* ( sign(x) + (x==0) );
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45 y = y / norm(y,q); % Normalize to unit q-norm.
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47 end