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Demos for MATLAB/Example_3_c_hom_norm/hnorm_weight.m

+function nx=hnorm_weight(x,L,rb,Q,J)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% The function nx=hnorm_weight(x, L) computes (for a given vector x)                
+%% a weighted homogeneous norm with the weights L=(r_1; r_2; ... r_n)>0
+%% 
+%% The function nx=hnorm_weight(x, L, rb) uses the smooting parameter rb: 
+%% the larger rb, the smoother norm  (by default, rb=1)
+%%
+%%  
+%% The function nx=hnorm_weight(x, L, rb, Q) uses the matrix  Q - (n x n)
+%% to define the shape of the norm's level set (by default, Q is identity).
+%%
+%%  
+%% The function nx=hnorm_weight(x, L, rb, Q, J) uses the matrix J - (n x n)
+%% for the coordinate transformation x->Jx (by default, J is identity).
+%%
+%% Remark: To compute a homogeneous norm in the case of a diagonilizable 
+%% linear dilation, the matrix J must be a real Jordan transformation of 
+%% its generator Gd = J^{-1} * diag(L) * J  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+n=size(x,1);
+if nargin<3 rb=1; end;
+if nargin<4 Q=eye(n); end;
+if nargin<5 J=eye(n); end;
+
+
+y=J*x;
+y=abs(y).^(rb./L).*sign(y);
+nx=(y'*Q*y)^(1/(2*rb));
+