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Article overview
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Are numerical theories irreplaceable? A computational complexity analysis | | Nickolay Vasiliev
; Dmitry Pavlov
; | | Date: |
11 Sep 2017 | | 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. | | Source: | arXiv, 1709.3939 | | Services: | Forum | Review | PDF | Favorites | |
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