Numerical parameterization of stationary axisymmetric black holes in a theory agnostic framework

The pursuit of a comprehensive theory of gravity has led to the exploration of various alternative models, necessitating a model-independent framework. The Konoplya-Rezzolla-Zhidenko (KRZ) parameterization offers a robust method for approximating stationary axisymmetric black hole spacetimes, characterized by a rapidly converging continued-fraction expansion. However, while analytical metrics benefit from this approach, numerical metrics derived from complex gravitational theories remain presenting computational challenges. Bridging this gap, we propose a method for a numerical KRZ parameterization, tested and demonstrated on pseudo-numerical Kerr and Kerr-Sen spacetimes. Our approach involves constructing numerical grids to represent metric coefficients and using the grids for fitting the parameters up to an arbitrary order. We analyze the accuracy of our method across different orders of approximation, considering deviations in the metric functions and shadow images. In both Kerr and Kerr-Sen cases, we observe rapid convergence of errors with increasing orders of continued fractions, albeit with variations influenced by spin and charge. Our results underscore the potential of the proposed algorithm for parameterizing numerical metrics, offering a pathway for further investigations across diverse gravity theories.