Homogeneous Norm $\|\cdot\|_{\mathbf{ d }} $

Computation of homogeneous norm

The function ${\color{red} { \texttt{hnorm }} } $ computes homogeneous norm of the vector $x$.

$\textbf{Input parameters}: {\color{blue}x}, {\color{blue}{G_{\mathbf{ d } }}}, {\color{blue}P} $

$ \textbf{Output parameters}: {\color{magenta}{\|x\|_{\mathbf{ d } }}} $

${\color{red} { \texttt{demo_hnorm.m}} } $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Example of compuation of homogeneous norm and homogeneous projection %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tol=1e-6; % compuational tolerance P=[1 -1; -1 2]; % the shape matrix of the wighted Euclidean norm if norm(P-P')>tol disp('Error: shape matrix P must be symmetric'); return; end; if min(real(eig(P)))<tol disp('Error: shape matrix P must be positive definite'); return; end; Gd=[2 0; 1 1]; % a dilation in R^2 %monotonicy check if min(real(eig(P*Gd+Gd'*P)))<tol disp('Error: dilation must be monotone'); return; end; x=[1;-1]; % vector x in R^2 nx=hnorm(x,Gd,P); % homogeneous projection of the vector x in R^2 disp(['Hom. norm = ', num2str(nx)]); z=hproj(x,Gd,P); % homogeneous projection of the vector x on the unit sphere x'*P*x=1 disp(['Hom. projection = [', num2str(z(1)),';',num2str(z(2)),']']);