iI have the following equation which I want solve, what is the most efficient way? I could not find closed form solution.
$\Sigma$ is a diagonal positive definite matrix. $s, \lambda$ are positive scalars, and $v$ is arbitrary vector. I need to solve for $\delta$.
$$ \left(\Sigma \frac{ \sqrt{\delta^T \delta}}{s} + \frac{s}{\sqrt{\delta^T \delta}} I\right) \delta = - \frac{1}{\lambda} v$$
Let $w = -v/\lambda$, $r = \sqrt{\delta^T \delta}$ and $u = \delta/r$. Your equation becomes
$$ \left(\dfrac{r^2}{s} \Sigma + s I\right) u = w$$
where you want $u$ to be a unit vector. Since $r^2/s$ and $s$ are positive and $\Sigma$ is positive definite, $\frac{r^2}{s} \Sigma + s I$ is invertible and
$$ u = \left(\dfrac{r^2}{s}\Sigma + s I\right)^{-1} w $$
Thus you need
$$ w^T \left(\dfrac{r^2}{s} \Sigma + s I\right)^{-2} w = 1$$
The left side is a rational function in $r^2$. Moreover, it's decreasing in $r$, so the solution (if it exists) is unique.