I have an iterative sequence for optimizing an EM (Expectation Maximization) algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and $C$ are positive semi-definite matrices. Also the diagonal entries in $A$ are the inverse of the diagonal elements of $C$. i.e, $A=Diag^{-1}(C)$.
I would like to compute the convergence rate (root-convergence rate) of this algorithm as $t\to \infty$. Am assuming it has got to do with taylor expansions, spectral radii and fixed point theorems. How is this approached or done for iterative schemes?