On the complexity of solving polynomial initial value problems

Abstract : In this paper we prove that computing the solution of an initial-value 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.
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Olivier Bournez, Daniel Graça, Amaury Pouly. On the complexity of solving polynomial initial value problems. International Symposium on Symbolic and Algebraic Computation (ISSAC'12), 2012, France. ⟨hal-00760742⟩

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