**7. Conclusions**

A novel method for solving the very important "matrix inversion" problem has been developed and validated in this paper. Consequently, it can also solve linear algebraic systems of equations. The concept developed is essentially an RNN-based analog computing machine.

We have compared this novel method with other dynamical systems methods from the relevant literature. It has been demonstrated that our novel method does theoretically have an exponential convergence rate towards the exact solution of the problem for any IVP (initial value problem). Also, the convergence rate of this method is much higher than those of other related competing concepts.

It is further possible to customize this novel model in order to enable its implementation on a cellular neural network processor machine. This has previously been done in our previous works.

For validation we have extensively compared our method with other relevant competing methods like gradient descent, Zhang, and Chen methods in one large simulation experiment. Hereby, we have considered the following scenarios: 1000 random matrixes of M are generated and applied in the different dynamical system models for matrix inversion. For each time-point, errors reached are summed-up using the formula *MX* − *I*.

It has been clear that while using our novel method we do visibly reach a significant speed-up even on one single CPU, this compared to the other methods. A use of more coefficients (i.e., higher values for n in model Equations (24) and (31)) will surely result in an even much higher convergence rate. On the other hand, however, if we use more coefficients, we would need more preparation time to create the templates and consume more computing resources for running the (our novel) RNN processor dynamical model. Therefore, it is recommended to reach a balance/tradeoff between convergence rate and number of required coefficients (i.e., value of n).

To finish, the last experiments have demonstrated that using the polynomial function for F in Equation (31) does lead to a clearly much higher convergence rate when compared to the other types of function. Further, the higher the polynomial order, the best.

**Author Contributions:** Conceptualization, V.T. and K.K.; Methodology, J.C.C. and K.K.; Software, V.T.; Validation, V.T., J.C.C. and K.K.; Formal Analysis, V.T.; Investigation, V.T.; Resources, V.T.; Data Curation, V.T.; Writing-Original Draft Preparation, V.T.; Writing-Review & Editing, J.C.C. and K.K; Visualization, V.T.; Supervision, J.C.C. and K.K.; Project Administration, K.K.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest
