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

Are numerical theories irreplaceable? A computational complexity analysis