# Fast Switch and Spline Scheme for Accurate Inversion of Nonlinear Functions: The New First Choice Solution to Kepler's Equation

## TL;DR

This paper introduces a fast, spline-based method for inverting monotonic functions, demonstrating superior speed and accuracy over traditional Newton-Raphson methods in solving equations like Kepler's and Lambert's W.

## Contribution

The authors develop a switching and spline interpolation scheme with an ultra-fast routine, providing highly accurate inverse function computations with significantly improved speed.

## Key findings

- The method achieves errors smaller than traditional spline analysis by up to 10^{-22}.
- It is 10^{-4} to 10^{-3} times faster than Newton-Raphson in tested cases.
- Outperforms Newton's method for Kepler's equation at high precision and large point sets.

## Abstract

Numerically obtaining the inverse of a function is a common task for many scientific problems, often solved using a Newton iteration method. Here we describe an alternative scheme, based on switching variables followed by spline interpolation, which can be applied to monotonic functions under very general conditions. To optimize the algorithm, we designed a specific ultra-fast spline routine. We also derive analytically the theoretical errors of the method and test it on examples that are of interest in physics. In particular, we compute the real branch of Lambert's $W(y)$ function, which is defined as the inverse of $x \exp(x)$, and we solve Kepler's equation. In all cases, our predictions for the theoretical errors are in excellent agreement with our numerical results, and are smaller than what could be expected from the general error analysis of spline interpolation by many orders of magnitude, namely by an astonishing $3\times 10^{-22}$ factor for the computation of $W$ in the range $W(y)\in [0,10]$, and by a factor $2\times 10^{-4}$ for Kepler's problem. In our tests, this scheme is much faster than Newton-Raphson method, by a factor in the range $10^{-4}$ to $10^{-3}$ for the execution time in the examples, when the values of the inverse function over an entire interval or for a large number of points are requested. For Kepler's equation and tolerance $10^{-6}$ rad, the algorithm outperforms Newton's method for all values of the number of points $N\ge 2$.