The complexity of initial-value problems is well studied for systems of equations of first order. In this paper, we study the
ε-complexity for initial-value problems for scalar equations of higher order. We consider two models of computation, the randomized model and the quantum model. We construct almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations. The analysis of these algorithms allows us to establish upper complexity bounds. We also show (almost) matching lower complexity bounds. The
ε-complexity in the randomized and quantum setting depends on the regularity of the right-hand side function, but is independent of the order of equation. Comparing the obtained bounds with results known in the deterministic case, we see that randomized algorithms give us a speed-up by
1/2, and quantum algorithms by
1 in the exponent. Hence, the speed-up does not depend on the order of equation, and is the same as for the systems of equations of first order. We also include results of some numerical experiments which confirm theoretical results.
We study possible advantages of randomized and quantum computing over deterministic computing for scalar initial-value problems for ordinary differential equations of order k. For systems of equations of the first order this question has been settled modulo some details in \cite{Kacewicz05}. A speed-up over deterministic computing shown in \cite{Kacewicz05} is related to the increased regularity of the solution with respect to that of the right-hand side function. For a scalar equation of order k (which can be transformed into a special system of the first order), the regularity of the solution is increased by k orders of magnitude. This leads to improved complexity bounds depending on k for linear information in the deterministic setting, see \cite{Szczesny05}. This may suggest that in the randomized and quantum settings a speed-up can also be achieved depending on k. We show in this paper that a speed-up dependent on k is not possible in the randomized and quantum settings. We establish lower complexity bounds, showing that the randomized and quantum complexities remain at the some level as for systems of the first order, no matter how large k is. Thus, the algorithms from \cite{Kacewicz05} remain (almost) optimal, even if we restrict ourselves to a subclass of systems arising from scalar equations of order k.
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.