Mercurial > matrix-functions
view toolbox/dual.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
children |
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function y = dual(x, p) %DUAL Dual vector with respect to Holder p-norm. % Y = DUAL(X, p), where 1 <= p <= inf, is a vector of unit q-norm % that is dual to X with respect to the p-norm, that is, % norm(Y, q) = 1 where 1/p + 1/q = 1 and there is % equality in the Holder inequality: X'*Y = norm(X, p)*norm(Y, q). % Special case: DUAL(X), where X >= 1 is a scalar, returns Y such % that 1/X + 1/Y = 1. % Called by PNORM. if max(size(x)) == 1 & nargin == 1 p = x; end % The following test avoids a `division by zero message' when p = 1. if p == 1 q = inf; else q = 1/(1-1/p); end if max(size(x)) == 1 & nargin == 1 y = q; return end if norm(x,inf) == 0, y = x; return, end if p == 1 y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real). elseif p == inf [xmax, k] = max(abs(x)); f = find(abs(x)==xmax); k = f(1); y = zeros(size(x)); y(k) = sign(x(k)); % y is a multiple of unit vector e_k. else % 1 < p < inf. Dual is unique in this case. x = x/norm(x,inf); % This scaling helps to avoid under/over-flow. y = abs(x).^(p-1) .* ( sign(x) + (x==0) ); y = y / norm(y,q); % Normalize to unit q-norm. end