Let $A_{ij}$ be a square, symmetric, and positive definite real matrix. I am interested in finding vectors $x_i$, with real components, that are solutions of the following equation:
$$b_{i}e^{ x_{i} } + \sum_{j}A_{ij}x_{j} = 0$$
where $b_i$ is another real vector of appropriate dimensions, with non-negative components.
I know this is not an eigenvector equation, but looks (perhaps?) related. I have two questions:
- Does this kind of equation that involve a nonlinearity with a matrix product, have a name? A keyword would help me to find relevant papers / textbooks.
- How could one solve these equations, numerically? And perhaps understand properties of the solutions (e.g., how does $x_i$ scale with increasing $b_i$)?