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tan_reverse

Headings-> Notation Eliminating Y(t) Positive Orders Z(t) Order Zero Z(t)

Tangent and Hyperbolic Tangent Reverse Mode Theory
Notation
We use the reverse theory standard math function definition for the functions $\mathit{H}$ and $\mathit{G}$ . In addition, we use the forward mode notation in tan_forward for $\mathit{X}(\mathit{t})$ , $\mathit{Y}(\mathit{t})$ and $\mathit{Z}(\mathit{t})$ .

Eliminating Y(t)
For $\mathit{j}>0$ , the forward mode coefficients are given by $${\mathit{y}}^{(\mathit{j}-1)}=\sum _{\mathit{k}=0}^{\mathit{j}-1}{\mathit{z}}^{(\mathit{k})}{\mathit{z}}^{(\mathit{j}-\mathit{k}-1)}$$ Fix $\mathit{j}>0$ and suppose that $\mathit{H}$ is the same as $\mathit{G}$ except that ${\mathit{y}}^{(\mathit{j}-1)}$ is replaced as a function of the Taylor coefficients for $\mathit{Z}(\mathit{t})$ . To be specific, for $\mathit{k}=0,\dots ,\mathit{j}-1$ , $$\begin{array}{ccc}\hfill \frac{\partial \mathit{H}}{\partial {\mathit{z}}^{(\mathit{k})}}& \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{k})}}+\frac{\partial \mathit{G}}{\partial {\mathit{y}}^{(\mathit{j}-1)}}\frac{\partial {\mathit{y}}^{(\mathit{j}-1)}}{\partial {\mathit{z}}^{(\mathit{k})}}\hfill \\ \hfill & \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{k})}}+\frac{\partial \mathit{G}}{\partial {\mathit{y}}^{(\mathit{j}-1)}}2{\mathit{z}}^{(\mathit{j}-\mathit{k}-1)}\hfill \end{array}$$
Positive Orders Z(t)
For order $\mathit{j}>0$ , suppose that $\mathit{H}$ is the same as $\mathit{G}$ except that ${\mathit{z}}^{(\mathit{j})}$ is expressed as a function of the coefficients for $\mathit{X}(\mathit{t})$ , and the lower order Taylor coefficients for $\mathit{Y}(\mathit{t})$ , $\mathit{Z}(\mathit{t})$ . $${\mathit{z}}^{(\mathit{j})}={\mathit{x}}^{(\mathit{j})}\pm \frac{1}{\mathit{j}}\sum _{\mathit{k}=1}^{\mathit{j}}\mathit{k}{\mathit{x}}^{(\mathit{k})}{\mathit{y}}^{(\mathit{j}-\mathit{k})}$$ For $\mathit{k}=1,\dots ,\mathit{j}$ , the partial of $\mathit{H}$ with respect to ${\mathit{x}}^{(\mathit{k})}$ is given by $$\begin{array}{ccc}\hfill \frac{\partial \mathit{H}}{\partial {\mathit{x}}^{(\mathit{k})}}& \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{x}}^{(\mathit{k})}}+\frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{j})}}\frac{\partial {\mathit{z}}^{(\mathit{j})}}{\partial {\mathit{x}}^{(\mathit{k})}}\hfill \\ \hfill & \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{x}}^{(\mathit{k})}}+\frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{j})}}\left[\mathrm{\delta }(\mathit{j}-\mathit{k})\pm \frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{j})}}\frac{\mathit{k}}{\mathit{j}}{\mathit{y}}^{(\mathit{j}-\mathit{k})}\right]\hfill \end{array}$$ where $\mathrm{\delta }(\mathit{j}-\mathit{k})$ is one if $\mathit{j}=\mathit{k}$ and zero otherwise. For $\mathit{k}=1,\dots ,\mathit{j}$ The partial of $\mathit{H}$ with respect to ${\mathit{y}}^{\mathit{j}-\mathit{k}}$ , is given by $$\begin{array}{ccc}\hfill \frac{\partial \mathit{H}}{\partial {\mathit{y}}^{(\mathit{j}-\mathit{k})}}& \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{y}}^{(\mathit{j}-\mathit{k})}}+\frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{j})}}\frac{\partial {\mathit{z}}^{(\mathit{j})}}{\partial {\mathit{y}}^{(\mathit{j}-\mathit{k})}}\hfill \\ \hfill & \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{y}}^{(\mathit{j}-\mathit{k})}}\pm \frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(\mathit{j})}}\frac{\mathit{k}}{\mathit{j}}{\mathit{x}}^{\mathit{k}}\hfill \end{array}$$
Order Zero Z(t)
The order zero coefficients for the tangent and hyperbolic tangent are $$\begin{array}{ccc}\hfill {\mathit{z}}^{(0)}& \hfill =\hfill & \{\begin{array}{c}\hfill \tan ({\mathit{x}}^{(0)})\hfill \\ \hfill \tanh ({\mathit{x}}^{(0)})\hfill \end{array}\hfill \end{array}$$ Suppose that $\mathit{H}$ is the same as $\mathit{G}$ except that ${\mathit{z}}^{(0)}$ is expressed as a function of the Taylor coefficients for $\mathit{X}(\mathit{t})$ . In this case, $$\begin{array}{ccc}\hfill \frac{\partial \mathit{H}}{\partial {\mathit{x}}^{(0)}}& \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{x}}^{(0)}}+\frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(0)}}\frac{\partial {\mathit{z}}^{(0)}}{\partial {\mathit{x}}^{(0)}}\hfill \\ \hfill & \hfill =\hfill & \frac{\partial \mathit{G}}{\partial {\mathit{x}}^{(0)}}+\frac{\partial \mathit{G}}{\partial {\mathit{z}}^{(0)}}(1\pm {\mathit{y}}^{(0)})\hfill \end{array}$$

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Input File: omh/tan_reverse.omh