Abstract
We deal with the problem, initiated in [8], of finding randomized and quantum complexity of initial-value problems. We showed in [8] that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the H\"older class of right-hand side functions with r continuous bounded partial derivatives, with r-th derivative being a H\"older function with exponent \rho, the \epsilon-complexity is shown to be O((1/\epsilon)^\{1/(r+\rho+1/3)\}) in the randomized setting, and O((1/\epsilon)^\{1/(r+\rho+1/2)\}) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from [8]. The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the s