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Good rotations

M. Henon, J-M. Petit

TL;DR

The paper addresses systematic drift caused by finite-precision roundoff in repeated fixed-angle rotations used in celestial mechanics. It proposes a constructive approach to choose representable sine and cosine values $(s,c)$ so that $c^2+s^2$ remains as close to unity as possible, using Diophantine and number-theoretic techniques to generate many accurate representations across single and double precision. The method substantially reduces the linear drift in radius and stabilizes energy errors in rotating-frame integrations, enabling reliable long-term simulations and improving the robustness of symplectic integrators in rotating frames. This 'good rotations' framework provides a practical tool for high-accuracy celestial mechanics computations where repeated rotations are essential.”

Abstract

Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in the representation of the sine s and cosine c of the angle theta. In a computer, one generally gets c^2 + s^2 <> 1, resulting in a mapping that is slightly contracting or expanding. In the present paper we present a method to find pairs of representable real numbers s and c such that c^2 + s^2 is as close to 1 as possible. We show that this results in a drastic decrease of the systematic error, making it negligible compared to the random error of other operations. We also verify that this approach gives good results in a realistic celestial mechanics integration.

Good rotations

TL;DR

The paper addresses systematic drift caused by finite-precision roundoff in repeated fixed-angle rotations used in celestial mechanics. It proposes a constructive approach to choose representable sine and cosine values so that remains as close to unity as possible, using Diophantine and number-theoretic techniques to generate many accurate representations across single and double precision. The method substantially reduces the linear drift in radius and stabilizes energy errors in rotating-frame integrations, enabling reliable long-term simulations and improving the robustness of symplectic integrators in rotating frames. This 'good rotations' framework provides a practical tool for high-accuracy celestial mechanics computations where repeated rotations are essential.”

Abstract

Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in the representation of the sine s and cosine c of the angle theta. In a computer, one generally gets c^2 + s^2 <> 1, resulting in a mapping that is slightly contracting or expanding. In the present paper we present a method to find pairs of representable real numbers s and c such that c^2 + s^2 is as close to 1 as possible. We show that this results in a drastic decrease of the systematic error, making it negligible compared to the random error of other operations. We also verify that this approach gives good results in a realistic celestial mechanics integration.

Paper Structure

This paper contains 11 sections, 3 theorems, 37 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Roundoff
  3. Some diophantine equations
  4. Single precision
  5. Double precision
  6. Solutions of $x ^ 2 + y ^ 2 = S$: general properties
  7. Solutions for a given $S$
  8. Numerical simulations
  9. Simple rotation
  10. Integration in a rotating frame
  11. Acknowledgements

Key Result

Theorem 1

The only solutions of (e:x2+y2-1) are ($x = \pm 2 ^ p$, $y = 0$) and ($x = 0$, $y = \pm 2 ^ p$).

Figures (3)

  • Figure 1: Roundoff errors as a function of the number of steps, in single precision. Dotted line: $\epsilon_0 =$ error due to the roundoff of $\cos \theta$ and $\sin \theta$, for an arbitrarily chosen $\theta$. Dashed line: $\epsilon_1 =$ error due to other roundoffs. Full line: $\epsilon_2 =$ errors due to the roundoff of $\cos \theta$ and $\sin \theta$, for a "good rotation" (Eq. (\ref{['e:x2+y2-3']})). Dash-dot line: $\epsilon_3 =$ same for Eq. (\ref{['e:x2+y2-4']}) with $k = 32$.
  • Figure 2: Relative square radius errors (absolute value) as a function of the number of steps. (a): $\theta = j / 512$, (b): $\theta = j \pi / 2000$, (c): solutions of equation (\ref{['e:x2+y2-6']}) with $n = 45$, (d): solutions of equation (\ref{['e:x2+y2-6']}) with $n = 51$.
  • Figure 3: Relative energy errors (absolute value) as a function of the number of steps. (a): "normal" angle $\theta = 0.0753$ (solid line) and "good" angles of approximately the same amplitude for $n = 45$ (dashed line) and $n = 51$ (dotted line). (b): same as (a), but for an normal angle of 0.00753, and corresponding good angles.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3