On the complexity of solving initial value problems
Abstract
In this paper we prove that computing the solution of an initialvalue problem $\dot{y}=p(y)$ with initial condition $y(t_0)=y_0\in\R^d$ at time $t_0+T$ with precision $e^{\mu}$ where $p$ is a vector of polynomials can be done in time polynomial in the value of $T$, $\mu$ and $Y=\sup_{t_0\leqslant u\leqslant T}\infnorm{y(u)}$. Contrary to existing results, our algorithm works for any vector of polynomials $p$ over any bounded or unbounded domain and has a guaranteed complexity and precision. In particular we do not assume $p$ to be fixed, nor the solution to lie in a compact domain, nor we assume that $p$ has a Lipschitz constant.
 Publication:

arXiv eprints
 Pub Date:
 February 2012
 arXiv:
 arXiv:1202.4407
 Bibcode:
 2012arXiv1202.4407B
 Keywords:

 Computer Science  Numerical Analysis;
 Computer Science  Computational Complexity
 EPrint:
 8 pages (two columns per page), submitted to ISSAC'12 conference