Fast Switch and Spline Scheme for Accurate Inversion of Nonlinear Functions: The New First Choice Solution to Kepler's Equation
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 ultrafast 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 NewtonRaphson 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 $N\ge 2$.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.02273
 arXiv:
 arXiv:1812.02273
 Bibcode:
 2018arXiv181202273T
 Keywords:

 Physics  Computational Physics;
 Astrophysics  Earth and Planetary Astrophysics;
 Physics  Space Physics
 EPrint:
 Applied Mathematics and Computation, Volume 364, 1 January 2020, 124677