## 6 Conclusions and Future Expansion  
The presented method in this work has a few advantages. First, it fits with rational functions, which are proven to be reliable at coping with many data behaviors. Second, the mathematics of finding the best fitting rational function is straight forward and is based primarily on solving a homogeneous linear system or finding the null space of a matrix, which can be performed efficiently in numerous ways [11]. Lastly, generating a null space base is efficiently computed via the \"Extraordinary SVD\" method as described in the work of [PERSON] and [PERSON] [12]. This will take away much of the burden in solving an overly determined, nonhomogeneous system \(\text{Ax}=\textbf{b}\) with optimization techniques; a task that is more challenging when the matrix A is badly scaled and conditioned, which is the case when trying to solve for the coefficients in \(f(s)\) via a nonhomogeneous system [1]. The latter limitation was the drive behind straying away from a direct approach when fitting with rational functions [1,2].  
In this paper, we proposed a robust method to fit data \((s,y_{s})\) with rational functions. We transformed the problem to a null space problem on scaled explanatory data. The presented method has a few advantages: First, we fit data with rational functions that are flexible in coping with different data behaviors. Second, the underlying null space problem is much easier to solve, as it entails computing a one-time SVD decomposition of a matrix that is not badly scaled due to controlling the terms \(s^{i}\) via scaling the independent variable \(s\). Third, the method provides high levels of efficiency and accuracy in the computations mainly because it attempts to solve for the exact solutions instead of iterative approximation schemes. As we showed through numerous examples, null space method succeeded to model many vital behaviors expressed as discrete measurements with high accuracy. As a potential future expansion of this work, we are interested in exploring prediction capabilities of the rational functions in many artificial intelligence algorithms in which cubic splines are used to smooth the data [24]. We are intrigued to explore whether cubic splines method replaced by our rational fit method would enhance prediction. We are also interested in deriving a numerical integration method to approximate \(\int_{a}^{b}f(x)dx\) based on fitting the integrand expressed discretely as \((x_{i},\ f(x_{i}))\).  
Conceptualization, T.M. and LH.; data curation, I.H.; formal analysis, T.M.; investigation, T.M.; methodology, T.M.; software, T.M.; validation T.M. and I.H.; writing--original draft preparation, T.M. and I.H.; writing--review and editing, T.M. and I.H.; visualization, T.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data used in this study are publicly available in various official sources, and all are already collected at Federal Reserve Economic Data, which is published by Federal Reserve Bank of St. Louis at [[https://fred.stlouisfed.org](https://fred.stlouisfed.org)]([https://fred.stlouisfed.org](https://fred.stlouisfed.org)), accessed on 3 September 2021.  
The authors declare no conflict of interest.