Homogeneous Norm $\|\cdot\|_{\mathbf{ d }} $
Let a linear dilation $\mathbf{d}$ in $\mathbb{R}^n$ be monotone with respect to a norm $\|\cdot\|$. A function $\|\cdot\|_{\mathbf{d}}: \mathbb{R}^n\to \mathbb{R}_+$ defined as $\|\mathbf{0}\|_{\mathbf{d}}=0$ and$\quad \|x\|_{\mathbf{d}}=e^s \quad : \quad \|\mathbf{d}(-s)x\|=1 \quad \text{ for } x\neq \mathbf{0}$
is the canonical $\mathbf{d}$-homogeneous norm in $\mathbb{R}^n$ by the norm $\|\cdot\|$.
Computation of the canonical homogeneous norm
The function ${\color{red} { \texttt{hnorm }} } $ computes the canonical homogeneous norm $\|x\|_{\mathbf{d}}$ of the vector $x$ induced by the weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$.
- $\textbf{Input parameters}: {\color{blue}x}\text{ (vector)}, {\color{blue} {G_{\mathbf{ d }}} } \text{ (generator of linear dilation)}, {\color{blue}P} \text{ (a shape matrix of the weighted Euclidean norm)}$
- $ \textbf{Output parameters}: {\color{magenta}{\|x\|_{\mathbf{ d } }}} $
${\color{red} { \texttt{demo_hnorm.m}} } $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Example of compuation of homogeneous norm induced by a linear dilation %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tol=1e-6; % compuational tolerance %% %% Shape matrix of the wighted Euclidean norm %% P=[1 -1; -1 2]; 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; %% %% Generator of the linear dilation in R^2 %% Gd=[3 0; 1 1]; %monotonicy check if min(real(eig(P*Gd+Gd'*P)))< tol disp('Error: dilation must be monotone'); return; end; %% %% Computation of homogeneous norm of a vector x %% x=[1;-1]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); disp(['Norm ||x||=sqrt(x^{T}*P*x) =', num2str(sqrt(x'*P*x),'%.8f')]); % weighted Euclidean norm of x nx=hnorm(x,Gd,P); % homogeneous norm of x disp(['Hom. norm (bisection method) of x = ', num2str(nx,'%.8f')]); % computed by bisection method nx=hnorm_newton(x,Gd,P); % homogeneous norm of x disp(['Hom. norm (Newton method) of x = ', num2str(nx,'%.8f')]); % computed by Newton method %% %% Computation of homogeneous norm of the scaled vector y=d(ln(2))*x %% y=expm(Gd*log(2))*x; % scaled vector y=d(ln2)x disp(['Scaled vector d(ln(2))x = [', num2str(y(1)),';',num2str(y(2)),']']); disp(['Norm ||d(ln2)x||=',num2str(sqrt(y'*P*y),'%.8f')]); % weighted Euclidean norm of y ny=hnorm(y,Gd,P); % homogeneous norm of y disp(['Hom. norm (bisection method) of d(ln2)x = ', num2str(ny,'%.8f')]); % computed by bisection method ny=hnorm_newton(y,Gd,P); % homogeneous norm of y disp(['Hom. norm (Newton method) of d(ln2)x = ', num2str(ny,'%.8f')]); % computed by Newton method %% %% Computation of homogeneous norm of the scaled vector z=2*x %% z=2*x; % scaled vector z=2x disp(['Scaled vector 2x = [', num2str(z(1)),';',num2str(z(2)),']']); disp(['Norm ||2x||=', num2str(sqrt(z'*P*z),'%.8f')]); % weighted Euclidean norm of y nz=hnorm(z,Gd,P); % homogeneous norm of y disp(['Hom. norm (bisection method) of 2x = ', num2str(nz,'%.8f')]); % computed by bisection method nz=hnorm_newton(z,Gd,P); % homogeneous norm of y disp(['Hom. norm (Newton method) of 2x = ', num2str(nz,'%.8f')]); % computed by Newton method