Effective bounds for P-recursive sequences

We describe an algorithm that takes as input a complex sequence (_un)$\left(u_n\right)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound (_vn)$\left(v_n\right)$ such that |_un|≤_vn$\left|u_n\right|\le v_n$ for all n$n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of _un$u_n$. We discuss applications to the evaluation of power series with guaranteed precision.