From 0000fd8c04698cc1406e6a26dd20a7ce025d7450 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 19 Oct 2021 17:43:53 +0200
Subject: Writing.

---
 ising_scaling.tex | 48 +++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 43 insertions(+), 5 deletions(-)

(limited to 'ising_scaling.tex')

diff --git a/ising_scaling.tex b/ising_scaling.tex
index 2301e73..be5364d 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -491,11 +491,29 @@ This leaves as unknown variables the positions $\theta_0$ and
 $\theta_{\mathrm{YL}}$ of the abrupt transition and Yang--Lee edge singularity,
 the amplitude $A_\mathrm{YL}$ of the latter, and the unknown functions $F$ and
 $h$. We determine these approximately by iteration in the polynomial order at
-which the free energy and its derivative matches known results. Gradients can be computed with
-
-A Levenburg--Marquardt algorithm is performed
-
-\begin{table}
+which the free energy and its derivative matches known results. We write as a
+cost function the difference between the known series coefficients of the
+scaling functions $\mathcal F_\pm$ and the series coefficients of our
+parametric form evaluated at the same points, $\theta=0$ and $\theta=\theta_c$,
+weighted by the uncertainty in the value of the known coefficients or by a
+machine-precision cutoff, whichever is larger. A Levenburg--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.
+
+We performed this procedure starting with $n=2$, 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
+\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.
+
+\begin{table}\label{tab:fits}
   \begin{tabular}{c|ccc}
     $n$ & $\mathcal F_-^{(n)}$ & $\mathcal F_0^{(n)}$ & $\mathcal F_+^{(n)}$ \\\hline
     0   & 0                    & $-1.197733383797993$ & 0                    \\
@@ -619,6 +637,26 @@ A Levenburg--Marquardt algorithm is performed
   \end{tabular}
 \end{table}
 
+\begin{figure}
+  \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
+    dat = 'data/phi_comparison.dat'
+
+    set xlabel '$n$'
+    set ylabel '$|\mathcal F_0^{(n)}-|$'
+
+    set style data linespoints
+    set logscale y
+
+    plot \
+      dat using 1:2 title '0', \
+      dat using 1:3 title '1', \
+      dat using 1:4 title '2', \
+      dat using 1:5 title '3'
+  \end{gnuplot}
+  \caption{
+  }
+\end{figure}
+
 \begin{figure}
   \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
     dat9 = 'data/h_series_ours_9.dat'
-- 
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