  In this work we present the first practical
$\left(\frac{1}{e}-\epsilon\right)$-approximation algorithm to maximise a
general non-negative submodular function subject to a matroid constraint. Our
algorithm is based on combining the decreasing-threshold procedure of
Badanidiyuru and Vondrak (SODA 2014) with a smoother version of the measured
continuous greedy algorithm of Feldman et al. (FOCS 2011). This enables us to
obtain an algorithm that requires $O(\frac{nr^2}{\epsilon^4}
\big(\frac{a+b}{a}\big)^2 \log^2({\frac{n}{\epsilon}}))$ value oracle calls,
where $n$ is the cardinality of the ground set, $r$ is the matroid rank, and $
b, a \in \mathbb{R}^+$ are the absolute values of the minimum and maximum
marginal values that the function $f$ can take i.e.: $ -b \leq f_S(i) \leq a$,
for all $i\in E$ and $S\subseteq E$, (here, $E$ is the ground set). The
additional value oracle calls with respect to the work of Badanidiyuru and
Vondrak come from the greater spread in the sampling of the multilinear
extension that the possibility of negative marginal values introduce.
