From d0d604e5fa0d06fc4dc899f3af9ede3ec9421b80 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 25 Oct 2021 14:21:53 +0200
Subject: Writing.

---
 ising_scaling.tex | 26 +++++++++++++++-----------
 1 file changed, 15 insertions(+), 11 deletions(-)

diff --git a/ising_scaling.tex b/ising_scaling.tex
index de49652..035c190 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -568,19 +568,19 @@ known numeric values, again weighted by their uncertainty.
 
 A Levenberg--Marquardt algorithm is performed on the cost function to find a
 parameter combination which minimizes it. As larger polynomial order in the
-series are fit, the truncations of $F$ and $h$ are extended to higher order so
-that the codimension of the fit is constant. A term is added to $F$ whenever a
-new coefficient of the high temperature series is added, and one is added to
-$h$ whenever a new coefficient of the low temperature series is added.
+series are fit, the truncations of $G$ and $g$ are extended to higher order so
+that the codimension of the fit is constant.
 
-We performed this procedure starting with $n=2$, or matching the scaling
+We performed this procedure starting at $n=2$, or matching the scaling
 function at the low and high temperature zero field points to quadratic order,
-through $n=9$. The resulting fit coefficients can be found in Table
+through $n=7$. The resulting fit coefficients can be found in Table
 \ref{tab:fits} without any sort of uncertainty, which is difficult to quantify
 directly due to the truncation of series. However, precise results exist for
 the value of the scaling function at the critical isotherm, or equivalently for
-the series coefficients of the scaling function $\mathcal F_0$, and the
-accuracy of the fit results can be checked against the known values here.
+the series coefficients of the scaling function $\mathcal F_0$. Since we do not
+use these coefficients to fix the unknown functions $G$ and $g$, the error in
+the approximate functions and their derivatives can be evaluated by comparison
+to their known values at the critical isotherm, or $\theta=1$.
 
 \begin{table}\label{tab:fits}
   \begin{tabular}{c|lll}
@@ -721,7 +721,7 @@ accuracy of the fit results can be checked against the known values here.
     set xlabel '$n$'
     set xrange [1.5:7.5]
 
-    set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$'
+    set ylabel '$|\Delta\mathcal F_0^{(m)}|$'
     set format y '$10^{%T}$'
     set logscale y
     set yrange [0.000002:0.003]
@@ -740,10 +740,14 @@ accuracy of the fit results can be checked against the known values here.
     The error in the $m$th derivative of the scaling function $\mathcal F_0$
     with respect to $\eta$ evaluated at $\eta=0$, as a function of the
     polynomial order $n$ at which the scaling function was fit.
-  }
+  } \label{fig:error}
 \end{figure}
 
-\subsection{Comparison}
+The difference between the numeric values the coefficients $\mathcal F_0^{(m)}$
+and those predicted by the iteratively fit scaling function are shown in
+Fig.~\ref{fig:error}. For the values for which we were able to make a fit, the
+error in the function and its first several derivatives appears to trend
+towards zero exponentially in the polynomial order $n$.
 
 \begin{figure}
   \begin{gnuplot}[terminal=epslatex]
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