AutoKnots: Adaptive knot allocation for spline interpolation

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. Interpolation methods, particularly spline-based approaches, play a critical role in this context. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods relying on manually configured knot distributions, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria, often requiring only a single parameter. The algorithm progressively improves interpolation by adaptively sampling the function-to-be-approximated, $f\left(x\right)$, in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute to the final approximation, ensuring efficiency. Although each resampling step involves recomputing the interpolation table, this process is highly optimized and computationally negligible compared to the cost of evaluating $f\left(x\right)$. For inherently fast functions, interpolation may not offer significant benefits. Precision tests on different functions demonstrate the algorithm’s efficacy, while a heuristic enhancement improves accuracy in flat regions. Integrated into the Numerical Cosmology library (NumCosmo), this algorithm has been extensively used and tested. NumCosmo includes rigorous unit tests that underscore its robustness and reliability. As a practical application, we compute the surface mass density $\Sigma \left(R\right)$ and average surface mass density $\overline{\Sigma }(<R)$ for Navarro–Frenk–White and Hernquist halo density profiles, offering analytical benchmarks. This adaptive algorithm provides a mature, user-friendly tool for interpolation challenges in computational astrophysics and cosmology.