System identification, part E: iterative search principle and identification methods Recursive identification and iterative identification are two important parameter estimation methods. The recursive index in the recursive identification is a time variable and the recursive identification can be used for online estimating system parameters; the iterative index in the iterative identification is a natural number and independent of time and the iterative identification is generally used for off-line estimating system parameters. The auxiliary model identification idea, multi-innovation identification theory, hierarchical identification principle and coupling identification concept based methods can be realized through recursive algorithms and iterative algorithms. Iterative methods can be traced to hundreds of years ago Jacobi iteration and Gauss-Seidel iteration for solving the matrix equations $oldsymbol{Ax}=oldsymbol{b}$. Iterative identification methods are based on the gradient search, least-squares search and Newton search principle. This paper studies the least squares based and gradient based iterative identification methods for CARMA systems and Box-Jenkins systems. The proposed methods can also be extended to other equation error type systems, output error type systems and nonlinear systems. Iterative methods usually apply system identification with finite data and their convergence analysis is very difficult and is a challenging research topic.