Are numerical theories irreplaceable? A computational complexity analysis

TL;DR

This paper demonstrates that for the Sitnikov three-body problem, achieving high-precision numerical solutions inherently requires exponential computational resources, making it impossible to combine the accuracy of numerical theories with the speed of analytical methods.

Contribution

It provides a formal complexity analysis showing that solving the Sitnikov problem with high precision cannot be done in polynomial time, highlighting fundamental computational limitations.

Findings

Numerical solutions require exponential precision in initial conditions.

No polynomial-time algorithm can solve the problem with arbitrary accuracy.

Chaotic behavior underpins the computational complexity result.

Abstract

It is widely known that numerically integrated orbits are more precise than analytical theories for celestial bodies. However, calculation of the positions of celestial bodies via numerical integration at time $t$ requires the amount of computer time proportional to $t$, while calculation by analytical series is usually asymptotically faster. The following question then arises: can the precision of numerical theories be combined with the computational speed of analytical ones? We give a negative answer to that question for a particular three-body problem known as Sitnikov problem. A formal problem statement is given for the the initial value problem (IVP) for a system of ordinary dynamical equations. The computational complexity of this problem is analyzed. The analysis is based on the result of Alexeyev (1968-1969) about the oscillatory solutions of the Sitnikov problem that have chaotic behavior. We prove that any algorithm calculating the state of the dynamical system in the Sitnikov problem needs to read the initial conditions with precision proportional to the required point in time (i.e. exponential in the length of the point's representation). That contradicts the existence of an algorithm that solves the IVP in polynomial time of the length of the input.