Fast Switch and Spline Scheme for Accurate Inversion of Nonlinear Functions: The New First Choice Solution to Kepler’s Equation. (arXiv:1812.02273v1 [physics.comp-ph])

<a href="http://arxiv.org/find/physics/1/au:+Tommasini_D/0/1/0/all/0/1">Daniele Tommasini</a>, <a href="http://arxiv.org/find/physics/1/au:+Olivieri_D/0/1/0/all/0/1">David N. Olivieri</a>

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 spline routine

that is faster and more accurate than those that have been described in the

literature. 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 for the time evolution

of planets and spacecrafts. 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 $3times

10^{-22}$ factor for the computation of $W$ in the range $W(y)in [0,10]$. In

our tests, this scheme is much faster than the Newton-Raphson iteration 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 $Nge 2$.

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 spline routine

that is faster and more accurate than those that have been described in the

literature. 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 for the time evolution

of planets and spacecrafts. 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 $3times

10^{-22}$ factor for the computation of $W$ in the range $W(y)in [0,10]$. In

our tests, this scheme is much faster than the Newton-Raphson iteration 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 $Nge 2$.

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