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appendix-mesi_jacobian.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Appendix D - Jacobian Matrix of the MESI Equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Jacobian Matrix of the MESI Equation} \label{app:mesi_jacobian}
The Jacobian matrix ($\bm{J}$) is the matrix of all first-order partial derivatives of a vector-valued function ($\bm{f}$):
%
\begin{equation}
\bm{J} = \left[\frac{\partial \bm{f}}{\partial x_1} \ldots \frac{\partial \bm{f}}{\partial x_n}\right]
= \begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_m}{\partial x_1} & \dots & \frac{\partial f_m}{\partial x_n}
\end{bmatrix}
\end{equation}
%
and can be expressed component-wise as:
%
\begin{equation}
\bm{J_{ij}} = \frac{\partial f_i}{\partial x_j}
\end{equation}
%
The partial derivatives comprising the Jacobian matrix of the MESI equation (Equation \ref{eq:mesi}) for use in the Levenberg-Marquardt optimization algorithm are as follows:
% % dK/dB
% \begin{equation}
% \frac{\partial K^2}{\partial \beta}
% = \rho^2\tau_c^2\left(\frac{e^{-2T/\tau_c} + 2T/\tau_c - 1}{2T^2}\right)
% + 4\rho\tau_c^2(1 - \rho)\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right)
% + (1 - \rho)^2
% \end{equation}
\begin{align*}
\label{eq:mesi_jacobian}
\frac{\partial K^2}{\partial \beta}
&= \rho^2\tau_c^2\left(\frac{e^{-2T/\tau_c} + 2T/\tau_c - 1}{2T^2}\right)
+ 4\rho\tau_c^2(1 - \rho)\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right) \\
&+ (1 - \rho)^2
\\
\\
\\
\frac{\partial K^2}{\partial \rho}
&= \beta\rho\tau_c^2\left(\frac{e^{-2T/\tau_c} + 2T/\tau_c - 1}{T^2}\right)
+ 4\beta\tau_c^2(1 - \rho)\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right) \\
&- 4\beta\rho\tau_c^2\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right)
- 2\beta(1 - \rho)
\\
\\
\\
\frac{\partial K^2}{\partial \tau_c}
&= \beta\rho^2\left(\frac{e^{-2T/\tau_c} - 1}{T}\right)
+ \beta\rho^2\tau_c\left(\frac{e^{-2T/\tau_c} + 2T/\tau_c - 1}{T^2}\right) \\
&+ 4\beta\rho(1 - \rho)\left(\frac{e^{-T/\tau_c} - 1}{T}\right)
+ 8\beta\rho\tau_c(1 - \rho)\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right)
\\
\\
\\
\frac{\partial K^2}{\partial \nu} &= 1
\end{align*}
% % dK/dt
% \begin{equation}
% \begin{split}
% \frac{\partial K^2}{\partial \tau_c} =
% & \beta\rho^2\left(\frac{e^{-2T/\tau_c} - 1}{T}\right) +
% \beta\rho^2\tau_c\left(\frac{e^{-2T/\tau_c} + 2T/\tau_c - 1}{T^2}\right) \\ +
% & 4\beta\rho(1 - \rho)\left(\frac{e^{-T/\tau_c} - 1}{T}\right) +
% 8\beta\rho\tau_c(1 - \rho)\left(\frac{e^{-T/\tau_c} + T/\tau_c - 1}{T^2}\right)
% \end{split}
% \end{equation}
%
% % dK/dn
% \begin{equation}
% \frac{\partial K^2}{\partial \nu} = 1
% \end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% END Appendix D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%