Inverse Homogeneous Homeomorphism $\Phi^{-1}$

Let a linear continuous dilation $\mathbf{d}$ in $\mathbb{R^n}$ be strictly monotone with respect to a weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$. The inverse homogeneous homeomorphism on $\mathbf{R}^n$ is defined as $$ \Phi^{-1}(z)= \mathbf{d}(-\ln \|z\|)\frac{z}{\|z\|} \quad \text{ for } \quad z\neq \mathbf{0} $$ with $\Phi^{-1}(\mathbf{0})=\mathbf{0}$.

$$ \quad $$

Computation of the inverse homogeneous homeomorphism

The function ${\color{red} { \texttt{hphi_inv }} } $ computes the value of the inverse homogeneous homeomorphism $\Phi^{-1}(z)$ for a given vector $z$.

$\textbf{Input parameters}: {\color{blue}z} \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^{-1}(z)}} $