Homogeneous Homeomorphism $\Phi$

Let a linear continuous dilation $\mathbf{d}$ in $\mathbb{R^n}$ be strictly monotone with respect to a weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$. Let $\|\cdot\|_{\mathbf{d}}$ be the canonical homogeneous norm induced by $\|\cdot\|$. The homogeneous homeomorphism on $\mathbf{R}^n$ is defined as $$ \Phi(x)=\|x\|_{\mathbf{d}} \mathbf{d}(-\ln \|x\|_{\mathbf{d}})x \quad \text{ for } \quad x\neq \mathbf{0} $$ with $\Phi(\mathbf{0})=\mathbf{0}$.

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Computation of the homogeneous homeomorphism

The function ${\color{red} { \texttt{hphi}} } $ computes the value of the homogeneous homeomorphism $\Phi(x)$ for a given vector $x$.

$\textbf{Input parameters}: {\color{blue}x} \text{ (vector)}, {\color{blue}{G_{\mathbf{ d } }}} \text{ (generator of linear dilation)}, {\color{blue}P}\text{ (shape matrix of weighted Euclidean norm)} $

$ \textbf{Output parameters}: {\color{magenta}{\Phi(x)}} $