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tan_reverse

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

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ Tangent and Hyperbolic Tangent Reverse Mode Theory
Notation
We use the reverse theory standard math function definition for the functions @(@ H @)@ and @(@ G @)@. In addition, we use the forward mode notation in tan_forward for @(@ X(t) @)@, @(@ Y(t) @)@ and @(@ Z(t) @)@.

Eliminating Y(t)
For @(@ j > 0 @)@, the forward mode coefficients are given by @[@ y^{(j-1)} = \sum_{k=0}^{j-1} z^{(k)} z^{(j-k-1)} @]@ Fix @(@ j > 0 @)@ and suppose that @(@ H @)@ is the same as @(@ G @)@ except that @(@ y^{(j-1)} @)@ is replaced as a function of the Taylor coefficients for @(@ Z(t) @)@. To be specific, for @(@ k = 0 , \ldots , j-1 @)@, @[@ \begin{array}{rcl} \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } \D{ y^{(j-1)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } 2 z^{(j-k-1)} \end{array} @]@

Positive Orders Z(t)
For order @(@ j > 0 @)@, suppose that @(@ H @)@ is the same as @(@ G @)@ except that @(@ z^{(j)} @)@ is expressed as a function of the coefficients for @(@ X(t) @)@, and the lower order Taylor coefficients for @(@ Y(t) @)@, @(@ Z(t) @)@. @[@ z^{(j)} = x^{(j)} \pm \frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)} @]@ For @(@ k = 1 , \ldots , j @)@, the partial of @(@ H @)@ with respect to @(@ x^{(k)} @)@ is given by @[@ \begin{array}{rcl} \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} } \\ & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \left[ \delta ( j - k ) \pm \frac{k}{j} y^{(j-k)} \right] \end{array} @]@ where @(@ \delta ( j - k ) @)@ is one if @(@ j = k @)@ and zero otherwise. For @(@ k = 1 , \ldots , j @)@ The partial of @(@ H @)@ with respect to @(@ y^{j-k} @)@, is given by @[@ \begin{array}{rcl} \D{H}{ y^{(j-k)} } & = & \D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} } \\ & = & \D{G}{ y^{(j-k)} } \pm \D{G}{ z^{(j)} }\frac{k}{j} x^{k} \end{array} @]@

Order Zero Z(t)
The order zero coefficients for the tangent and hyperbolic tangent are @[@ \begin{array}{rcl} z^{(0)} & = & \left\{ \begin{array}{c} \tan ( x^{(0)} ) \\ \tanh ( x^{(0)} ) \end{array} \right. \end{array} @]@ Suppose that @(@ H @)@ is the same as @(@ G @)@ except that @(@ z^{(0)} @)@ is expressed as a function of the Taylor coefficients for @(@ X(t) @)@. In this case, @[@ \begin{array}{rcl} \D{H}{ x^{(0)} } & = & \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } ( 1 \pm y^{(0)} ) \end{array} @]@

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Input File: omh/appendix/theory/tan_reverse.omh