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Exponential Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions $H$ and $G$. The zero order forward mode formula for the exponential is $$z^{(0)} = F ( x^{(0)} )$$ and for $j > 0$, $$z^{(j)} = x^{(j)} d^{(0)} + \frac{1}{j} \sum_{k=1}^{j} k x^{(k)} z^{(j-k)}$$ where $$d^{(0)} = \left\{ \begin{array}{ll} 0 & \R{if} \; F(x) = \R{exp}(x) \\ 1 & \R{if} \; F(x) = \R{expm1}(x) \end{array} \right.$$ For order $j = 0, 1, \ldots$ we note that $$\begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } ( d^{(0)} + z^{(0)} ) \end{array}$$ If $j > 0$, then for $k = 1 , \ldots , j$ $$\begin{array}{rcl} \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k z^{(j-k)} \\ \D{H}{ z^{(j-k)} } & = & \D{G}{ z^{(j-k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k x^{(k)} \end{array}$$
Input File: omh/appendix/theory/exp_reverse.omh