this solution as $E_n := u_{\alpha_{n}}(s_{n+1})$, which the new choice of $\alpha_{n+1}$ must rectify in order to achieve the condition $u_{\alpha_{n+1}}(s_{n+1}) = 0$. The first main estimate in the scheme is thus careful control of this error, $E_n$, throughout the iteration. Specifically, using backwards integration from $s_n$, we may obtain the decay estimate $$\begin{aligned}
|E_n| = |\mathcal{T}_n(\alpha_n)| \lesssim e^{- \gamma s_n}\,. \end{aligned}$$

Lower bounds on $\frac{{\ensuremath{\partial}}\mathcal{T}_n}{{\ensuremath{\partial}}\alpha}$ are achieved by differentiating the forward integration formula, in $\alpha$, as this formula importantly holds for all $\alpha$. A simple inspection shows that we may expect $\frac{{\ensuremath{\partial}}\mathcal{T}_n}{{\ensuremath{\partial}}\alpha} \sim e^{\frac s 2}$. Third, an upper bound of $\sup_{\alpha_\ast \in [\alpha_n, \alpha_{n+1}]} |\frac{{\ensuremath{\partial}}^2 \mathcal{T}_n}{{\ensuremath{\partial}}\alpha^2}|$ can also be computed by differentiating twice in $\alpha$.

Notational Conventions
----------------------

We now discuss some notational conventions that we will be using throughout the analysis. First, for a function $f = f(x, s)$, we use $\| f \|_\infty = \sup_{x} |f(s, x)|$, that is $L^\infty$ refers to in the $x$ variable only. Next, we define the bracket notation $\langle x \rangle := \sqrt{1 + x^2}$. Lastly, we will often use $A \lesssim B$ to mean $A \le CB$, where $C$ is a universal constant independent of $M, \eps, \kappa_0$. We will 