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Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 18 |
1 files changed, 11 insertions, 7 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index d4ea507..de49652 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -562,13 +562,16 @@ 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 $\theta=\theta_0$, 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. +machine-precision cutoff, whichever is larger. We also add the difference +between the predictions for $A_\mathrm{YL}$ and $\xi_\mathrm{YL}$ and their +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. 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, @@ -721,6 +724,7 @@ accuracy of the fit results can be checked against the known values here. set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$' set format y '$10^{%T}$' set logscale y + set yrange [0.000002:0.003] set style data linespoints set key title '\raisebox{0.5em}{$m$}' bottom left |