In this paper, stationary iterative methods for large-scale Toeplitz-type equations are investigated from a control theoretic point of view. We utilize spatially invariant structure of Toeplitz matrices, to avoid the curse of dimensionality arising in analysis and design of the convergence properties. Nonlinearities in the system are theoretically handled with the small gain theorem and stability analysis for Lur'e systems. This theory enables us to achieve the desired global convergence of the proposed numerical scheme. We also evaluate the efficacy of the proposed method through a numerical simulation comparison with the Broyden's method.