function [deltAlp,Stepmax] = LagrangeUpdate(DirSigma,Sigma,Xsup,Alpsup,C,pow);%LAGRANGEUPDATE Compute gradient and maximal change in Lagrange multipliers%       [DELTALP,STEPMAX] = LAGRANGEUPDATE(DIRSIGMA,SIGMA,XSUP,ALPSUP,C,POW)%       provides the descent direction DELTALP and the maximal update STEPMAX%       for Lagrange multipliers based on %       - the secent direction DIRSIGMA for Sigma;%       - the current Sigma value SIGMA;%       - the SVM parameters XSUP,ALPSUP;%       - the SVM hyper-parameters C and POW%       27/01/03 Y. Grandvalet% initialization[nsup d] = size(Xsup);deltAlp  = zeros(nsup,1);Stepmax  = realmax;% support vectors of the first categoryindsup1 = find(abs(Alpsup)<C);nsup1   = length(indsup1);if nsup1>0,   % 1) compute distances      Xsup = Xsup.*repmat(Sigma,nsup,1);   Dist = Xsup(indsup1,:)*Xsup';   dist = 0.5*sum(Xsup.^2,2) ;   Dist = Dist - repmat(dist(indsup1),1,nsup) - repmat(dist',nsup1,1) ; % -1/2 (xi-xj)T Sigma^2 (xi-xj)   Dist = exp(Dist) ;   % 2) solve linear systems (d nsup1+1 Ax_k = b_k systems (A symmetric (not def. pos.): factorize ? ))   A = [ [Dist(:,indsup1) ones(nsup1,1)] ; [ones(1,nsup1) 0] ];   b = zeros(nsup1+1,d);   for k=1:d;      Distk  = Xsup(indsup1,k)*Xsup(:,k)';      distk  = Xsup(:,k).^2;      Distk  = -2*Distk + repmat(distk(indsup1),1,nsup) + repmat(dist',nsup1,1) ; % (xik-xjk)^2      b(1:nsup1,k) = (Distk.*Dist) * Alpsup * Sigma(k) ;   end;   ind = find(Sigma~=0);   b(1:nsup1,ind) = b(1:nsup1,ind).*repmat((1/pow*abs(real(Sigma(ind).^(1-pow)))),nsup1,1);   x = (A+1e-10*eye(nsup1+1))\b;   gradAlp          = x(1:nsup1,:);   deltAlp(indsup1) = gradAlp*DirSigma';   % 3) compute maximal Alpsup change   % 3.1) no sign change   ind = find((deltAlp(indsup1).*Alpsup(indsup1))<0);   if ~isempty(ind)      Stepmax = min(-Alpsup(indsup1(ind))./deltAlp(indsup1(ind))) ;   end;       % 3.2) max(Alpsup) = +-C;   ind = find((deltAlp(indsup1).*Alpsup(indsup1))>0);   if ~isempty(ind)      Stepmax = min(Stepmax,min((C-abs(Alpsup(indsup1(ind))))./abs(deltAlp(indsup1(ind))))) ;   end;end;