I was wondering if anyone has any ideas for a closed-form solution to the equation

$$Ax + \exp(x) +b =0$$ where $x,b \in \mathbb{R}^n$, $A$ is a symmetric positive definite matrix and $\exp$ denotes elementwise exponentiation (i.e., $\exp(x) =(\exp(x_1),\exp(x_2), \dots, \exp(x_n) $). The case where $n=1$,

$$ ax+\exp(x)+b =0$$

has the solution $$ x= -W_{n}\left(e^{\frac{-b}{a}}\right)+\dfrac{b}{a}, $$ where $a\neq 0$, $n \in \mathbb{Z}$, and $W_{n}$ is the $n$th branch of the Lambert W function. I'm looking for unique solutions of the vector version.

  • $\begingroup$ This is a system of nonlinear equations. I doubt a closed form solution exists for a general matrix $A$. It could be solved numerically with vector form of the Newton's method, or any other $\endgroup$ – Yuriy S Jan 14 at 11:34

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