Add contents of limitations section for Schmidt algorithm

main.tex

@@ -693,7 +693,38 @@ Goals:
 	\item noticeably slow (show benchmark)
 \end{itemize}
 
-\section{Refinement}
+The main downside of this algorithm is that many $r$,$c$ pairs which are tested
+end up not yielding any solutions for the problem.
+In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
+$\beta_{-}$:
+\begin{align*}
+	0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
+	0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
+\end{align*}
+In particular, it's the $\chern_1^{\beta_{-}}(v-u) \geq 0$ conditions which
+fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
+This condition is only checked within the internal loop.
+This, along with a conservative estimate for a bound on the $r$ values (as
+illustrated in example \ref{exmpl:recurring-first}) occasionally leads to slow
+computations.
+
+Here are some benchmarks to illustrate the performance benefits of the
+alternative algorithm which will later be described in this article [ref].
+
+\begin{center}
+\begin{tabular}{ |r|l|l| } 
+ \hline
+ Choice of $v$ on $\mathbb{P}^2$
+ & $(3, 2\ell, -2)$
+ & $(3, 2\ell, -\frac{15}{2})$ \\
+ \hline
+ Computation time for earlier [ref] program & \sim 20s & >1hr \\
+ Computation time for [ref] program  & \sim 50ms & \sim 50ms \\
+ \hline
+\end{tabular}
+\end{center}
+
+\section{Tighter Bounds}
 \label{sec:refinement}
 
 To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some