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4thordereq_extended_from_Mu_Wang.tex
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4thordereq_extended_from_Mu_Wang.tex
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\documentclass[12pt,letter]{article}
\usepackage[top=30pt,bottom=30pt,left=30pt,right=30pt]{geometry}
\usepackage{hyperref}
\usepackage{amsmath}
\begin{document}
\thispagestyle{empty}
\begin{center}
Third order equations from pages 46 and 48, and derived 4th order equations, of
\url{www.autodiff.org/Docs/euroad/17th EuroAd Workshop - Mu Wang - Higher Order Reverse Mode With MPI.pdf}
ZTA, July 2016
\end{center}
\clearpage
\newpage
\section{Equations}
Given that
\begin{equation}
\frac{\hat{\partial} f_i}{\hat{\partial} v_c} =
\frac{\partial f_{i+1}}{\partial v_c} +
\left(\frac{\partial \phi_i}{\partial v_c} * \frac{\partial f_{i+1}}{\partial v_i}\right)
\end{equation}
Then
\begin{equation}
\begin{split}
\frac{\hat{\partial}}{\hat{\partial} v_b}\left[\frac{\hat{\partial} f_i}{\hat{\partial} v_c}\right]
&=
\frac{\partial^2 f_{i+1}}{\partial v_b \partial v_c} +
\left(\frac{\partial^2 \phi_i}{\partial v_b \partial v_c} * \frac{\partial f_{i+1}}{\partial v_i} \right) +
\left(\frac{\partial\phi_i}{\partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_b \partial v_i}\right)
\\
&+
\left(\frac{\partial\phi_i}{\partial v_b} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_c}\right) +
\left(\frac{\partial\phi_i}{\partial v_c} * \frac{\partial\phi_i}{\partial v_b} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right)
\end{split}
\end{equation}
It then follows that
\begin{equation}
\begin{split}
\frac{\hat{\partial}}{\hat{\partial} v_a}\left[\frac{\hat{\partial}}{\hat{\partial} v_b}\left(\frac{\hat{\partial} f_i}{\hat{\partial} v_c}\right)\right]
&=
\frac{\partial^3 f_{i+1}}{\partial v_a \partial v_b \partial v_c} +
\left(\frac{\partial^3 \phi_i}{\partial v_a \partial v_b \partial v_c} * \frac{\partial f_{i+1}}{\partial v_i}\right) +
\left(\frac{\partial^2 \phi_i}{\partial v_b \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_a \partial v_i}\right)
\\
&+
\left(\frac{\partial^2 \phi_i}{\partial v_a \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_b \partial v_i}\right) +
\left(\frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_b \partial v_i}\right)
\\
&+
\left(\frac{\partial^2 \phi_i}{\partial v_a \partial v_b} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_c}\right) +
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_i \partial v_c}\right)
\\
&+
\left\{\left[\left(\frac{\partial^2 \phi_i}{\partial v_a \partial v_c} * \frac{\partial \phi_i}{\partial v_b}\right) +
\left(\frac{\partial \phi_i}{\partial v_c} * \frac{\partial^2 \phi_i}{\partial v_a \partial v_b}\right)\right] * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right\}
\\
&+
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_i \partial v_i}\right)
\\
&+
\frac{\partial \phi_i}{\partial v_a} *
\left[\frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_c} +
\left(\frac{\partial^2 \phi_i}{\partial v_b \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right) +
\left(\frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_i}\right)
\right.
\\
&+
\left.
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_c}\right) +
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right)\right]
\end{split}
\end{equation}
Which leads to
\begin{equation}
\begin{split}
\frac{\hat{\partial}}{\hat{\partial} v_z}\left\{\frac{\hat{\partial}}{\hat{\partial} v_a}\left[
\frac{\hat{\partial}}{\hat{\partial} v_b}\left(\frac{\hat{\partial} f_i}{\hat{\partial} v_c}\right)\right]\right\}
&=
\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_a \partial v_b \partial v_c} +
\left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_a \partial v_b \partial v_c}\right)
\\
&+
\left(\frac{\partial^4 \phi_i}{\partial v_z \partial v_a \partial v_b \partial v_c} * \frac{\partial f_{i+1}}{\partial v_i}\right) + \left\{\frac{\partial^3 \phi_i}{\partial v_a \partial v_b \partial v_c} * \left[\frac{\partial^2 f_{i+1}}{\partial v_z \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right)\right]\right\}
\\
&+
\left(\frac{\partial^3 \phi_i}{\partial v_z \partial v_a \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_b \partial v_i}\right) + \left\{\frac{\partial^2 \phi_i}{\partial v_a \partial v_c} * \left[\frac{\partial^3 f_{i+1}}{\partial v_z \partial v_b \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_i}\right)\right]\right\}
\\
&+
\left(\frac{\partial^2 \phi_i}{\partial v_z \partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_b \partial v_i}\right) + \left\{\frac{\partial \phi_i}{\partial v_c} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_a \partial v_b \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_a \partial v_b \partial v_i}\right)\right]\right\}
\\
&+
\left(\frac{\partial^3 \phi_i}{\partial v_z \partial v_a \partial v_b} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_c}\right) + \left\{\frac{\partial^2 \phi_i}{\partial v_a \partial v_b} * \left[\frac{\partial^3 f_{i+1}}{\partial v_z \partial v_i \partial v_c} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_c}\right)\right]\right\}
\\
&+
\left(\frac{\partial^2 \phi_i}{\partial v_z \partial v_b} * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_i \partial v_c}\right) + \left\{\frac{\partial \phi_i}{\partial v_b} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_a \partial v_i \partial v_c} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_a \partial v_i \partial v_c}\right)\right]\right\}
\\
&+
\left\{\left[\left(\frac{\partial^3 \phi_i}{\partial v_z \partial v_a \partial v_b} * \frac{\partial \phi_i}{\partial v_c}\right) + \left(\frac{\partial^2 \phi_i}{\partial v_a \partial v_b} * \frac{\partial^2 \phi_i}{\partial v_z \partial v_c}\right)\right] * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right\}
\\
&+
\left\{\frac{\partial^2 \phi_i}{\partial v_a \partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \left[\frac{\partial^3 f_{i+1}}{\partial v_z \partial v_i \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right)\right]\right\}
\\
&+
\left\{\left[\left(\frac{\partial^2 \phi_i}{\partial v_z \partial v_b} * \frac{\partial^2 \phi_i}{\partial v_a \partial v_c}\right) + \left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^3 \phi_i}{\partial v_z \partial v_a \partial v_c}\right)\right] * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right\}
\\
&+
\left\{\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^2 \phi_i}{\partial v_a \partial v_c} * \left[\frac{\partial^3 f_{i+1}}{\partial v_z \partial v_i \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right)\right]\right\}
\\
&+
\left\{\left[\left(\frac{\partial^2 \phi_i}{\partial v_z \partial v_b} * \frac{\partial \phi_i}{\partial v_c}\right) + \left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^2 \phi_i}{\partial v_z \partial v_c}\right)\right] * \frac{\partial^3 f_{i+1}}{\partial v_a \partial v_i \partial v_i}\right\}
\\
&+
\left\{\frac{\partial \phi_i}{\partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_a \partial v_i \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_a \partial v_i \partial v_i}\right)\right]\right\}
\\
&+
\frac{\partial^2 \phi_i}{\partial v_z \partial v_a} *
\left[\frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_c} +
\left(\frac{\partial^2 \phi_i}{\partial v_b \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right) +
\left(\frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_i}\right)
\right.
\\
&+
\left.
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_c}\right) +
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right)\right]
\\
&+
\frac{\partial \phi_i}{\partial v_a} *
\left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_i \partial v_b \partial v_c} +
\left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_i \partial v_b \partial v_c}\right) +
\left(\frac{\partial^3 \phi_i}{\partial v_z \partial v_b \partial v_c} * \frac{\partial^2 f_{i+1}}{\partial v_i \partial v_i}\right)
\right.
\\
&+
\left.
\left\{\frac{\partial^2 \phi_i}{\partial v_b \partial v_c} * \left[\frac{\partial^3 f_{i+1}}{\partial v_z \partial v_i \partial v_i} +
\left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right)\right]\right\}
\right.
\\
&+
\left.
\left(\frac{\partial^2 \phi_i}{\partial v_z \partial v_c} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_b \partial v_i}\right) +
\left\{\frac{\partial \phi_i}{\partial v_c} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_i \partial v_b \partial v_i} + \left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_i \partial v_b \partial v_i}\right)\right]\right\}
\right.
\\
&+
\left.
\left(\frac{\partial^2 \phi_i}{\partial v_z\partial v_b} * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_c}\right) +
\left\{\frac{\partial \phi_i}{\partial v_b} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_i \partial v_i \partial v_c} +
\left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_i \partial v_i \partial v_c}\right)\right]\right\}
\right.
\\
&+
\left.
\left\{\left[\left(\frac{\partial^2 \phi_i}{\partial v_z\partial v_b} * \frac{\partial \phi_i}{\partial v_c}\right) +
\left(\frac{\partial \phi_i}{\partial v_b} * \frac{\partial^2 \phi_i}{\partial v_z \partial v_c}\right)\right] * \frac{\partial^3 f_{i+1}}{\partial v_i \partial v_i \partial v_i}\right\}
\right.
\\
&+
\left.
\left\{\frac{\partial \phi_i}{\partial v_b} * \frac{\partial \phi_i}{\partial v_c} * \left[\frac{\partial^4 f_{i+1}}{\partial v_z \partial v_i \partial v_i \partial v_i} +
\left(\frac{\partial \phi_i}{\partial v_z} * \frac{\partial^4 f_{i+1}}{\partial v_i \partial v_i \partial v_i \partial v_i}\right)\right]\right\}\right]
\end{split}
\end{equation}
\end{document}