We have this iteration
$$X_{k+1}=(G\cdot Jf+H)X_k+C$$
with $G$ is symmetric and nonnegative, $H$ is nonnegative. $Jf$ is the jacobian matrix of some function $f$ and we can assume it satisfy certain conditions if needed.
To show the iteration converges, it is equivalent to show the spectral radius of $G\cdot Jf+H$ is less than $1$. We already have a bound on spectral radius of $G$ and $H$.
I tried to use the spectrum radius of $G\cdot Jf+H$ less than summation of spectrum radius of $G\cdot Jf$ and $H$, however, $G\cdot Jf$ is not always nonnegative and this theorem seems only true for nonnegative matrix.
My question is is there any bound for the spectrum radius of $G\cdot Jf+H$ suppose I can bound $G$ and $H$?
Any ideas and references are appreciated. Thanks!
