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Demos for MATLAB/Example_3_c_hom_norm/approx_hnorm_weight.m

+function Q=approx_hnorm_weight(Gd,P,rb,Nmax)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% The function Q=approx_hnorm_weight(Gd,P,rb) computes the (n x n) matrix Q
+%% for an explicit homogenenous norm 
+%%
+%%          ||x||_rb=(y'*Q*y)^(1/rb),     y=(J*x).^(rb./L)  
+%%
+%%  which approximates the canonical homogeneous norm induced by the norm 
+%%          
+%%                   ||x||=sqrt(x'*P*x) 
+%% 
+%% The (n x n) matrix Gd is a diagonalizable anti-Hurwitz generator of 
+%% a linear dilation in R^n being strictly monotone with respect to ||x||
+%%
+%%                     Gd=inv(J)*diag(L)*J
+%%
+%% The symmetric positive definite (n x n) matrix P defines a shape of
+%% the weighted Euclidean norm ||x|| 
+%%
+%% rb - a positive smoothing parameter
+%%
+%% Important: The function needs YALMIP toolbox with some LMI solver 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+if nargin<4 Nmax=10; end;
+if nargin<3 rb=1; end;
+
+n=size(P,1); 
+
+[J L]=eig(Gd);
+L=diag(L); J=inv(J);
+
+%genaration of the array
+z=[];
+lin=[-1:n/Nmax:1];k=size(lin,2);
+ind=ones(1,n); ok=1;
+while ok==1
+ x=[];if ind*ind'==n*k^2 ok=0; end;
+ for i=1:n x=[x;lin(ind(i))]; end;
+ if (min(ind)==1) || (max(ind)==k) z=[z J*x/sqrt(x'*P*x)]; end;
+ if ind(1,1)<k ind(1,1)=ind(1,1)+1;
+ else j=1;while (ind(1,j)==k) && (j<n) ind(1,j)=1; j=j+1; end;ind(1,j)=ind(1,j)+1;end;
+end;
+
+
+Lz=L*ones(1,size(z,2));
+Phiz=abs(z).^(rb./L).*sign(z);
+ 
+Q=sdpvar(n);
+cost=@(Q)sum(sum((Q*Phiz).*Phiz).^2)-2*sum(sum((Q*Phiz).*Phiz));
+ops = sdpsettings('verbose',0);
+optimize([Q>=0],cost(Q),ops);
+
+Q=double(Q);
+
+
+