Mercurial > matrix-functions
comparison matrixcomp/dual.m @ 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
| author | Antonio Pino Robles <data.script93@gmail.com> |
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| date | Wed, 06 May 2015 14:56:53 +0200 |
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| -1:000000000000 | 0:8f23314345f4 |
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| 1 function y = dual(x, p) | |
| 2 %DUAL Dual vector with respect to Holder p-norm. | |
| 3 % Y = DUAL(X, p), where 1 <= p <= inf, is a vector of unit q-norm | |
| 4 % that is dual to X with respect to the p-norm, that is, | |
| 5 % norm(Y, q) = 1 where 1/p + 1/q = 1 and there is | |
| 6 % equality in the Holder inequality: X'*Y = norm(X, p)*norm(Y, q). | |
| 7 % Special case: DUAL(X), where X >= 1 is a scalar, returns Y such | |
| 8 % that 1/X + 1/Y = 1. | |
| 9 | |
| 10 % Called by PNORM. | |
| 11 | |
| 12 warns = warning; | |
| 13 warning('off') | |
| 14 | |
| 15 if nargin == 1 | |
| 16 if length(x) == 1 | |
| 17 y = 1/(1-1/x); | |
| 18 return | |
| 19 else | |
| 20 error('Second argument missing.') | |
| 21 end | |
| 22 end | |
| 23 | |
| 24 q = 1/(1-1/p); | |
| 25 | |
| 26 if norm(x,inf) == 0, y = x; return, end | |
| 27 | |
| 28 if p == 1 | |
| 29 | |
| 30 y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real). | |
| 31 | |
| 32 elseif p == inf | |
| 33 | |
| 34 [xmax, k] = max(abs(x)); | |
| 35 f = find(abs(x)==xmax); k = f(1); | |
| 36 y = zeros(size(x)); | |
| 37 y(k) = sign(x(k)); % y is a multiple of unit vector e_k. | |
| 38 | |
| 39 else % 1 < p < inf. Dual is unique in this case. | |
| 40 | |
| 41 x = x/norm(x,inf); % This scaling helps to avoid under/over-flow. | |
| 42 y = abs(x).^(p-1) .* ( sign(x) + (x==0) ); | |
| 43 y = y / norm(y,q); % Normalize to unit q-norm. | |
| 44 | |
| 45 end | |
| 46 warning(warns) |