Inverse Tangent and Hyperbolic Tangent Reverse Mode Theory

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atan_reverse

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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . Inverse Tangent and Hyperbolic Tangent Reverse Mode Theory We use the reverse theory standard math function definition for the functions

$H$ and

$G$ . In addition, we use the forward mode notation in atan_forward for

$B(t) = 1 \pm X(t) * X(t)$ We use

$b$ for the p-th order Taylor coefficient row vectors corresponding to

$B(t)$ and replace

$z^{(j)}$ by

$( z^{(j)} , b^{(j)} )$ in the definition for

$G$ and

$H$ . The zero order forward mode formulas for the atan function are

$\begin{array}{rcl} z^{(0)} & = & F ( x^{(0)} ) \\ b^{(0)} & = & 1 \pm x^{(0)} x^{(0)} \end{array}$ where

$F(x) = \R{atan} (x)$ for

$+$ and

$F(x) = \R{atanh} (x)$ for

$-$ . For orders

$j$ greater than zero we have

$\begin{array}{rcl} b^{(j)} & = & \pm \sum_{k=0}^j x^{(k)} x^{(j-k)} \\ z^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} } \left( j x^{(j)} - \sum_{k=1}^{j-1} k z^{(k)} b^{(j-k)} \right) \end{array}$ If

$j = 0$ , we note that

$F^{(1)} ( x^{(0)} ) = 1 / b^{(0)}$ and hence

$\begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} } \pm \D{G}{ b^{(j)} } 2 x^{(0)} \end{array}$ If

$j > 0$ , then for

$k = 1, \ldots , j-1$

$\begin{array}{rcl} \D{H}{ b^{(0)} } & = & \D{G}{ b^{(0)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(0)} } \\ & = & \D{G}{ b^{(0)} } - \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ b^{(0)} } \\ \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(j)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} } \pm \D{G}{ b^{(j)} } 2 x^{(0)} \\ \D{H}{ x^{(0)} } & = & \D{G}{ x^{(0)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(0)} } \pm \D{G}{ b^{(j)} } 2 x^{(j)} \\ \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(k)} } \\ & = & \D{G}{ x^{(k)} } \pm \D{G}{ b^{(j)} } 2 x^{(j-k)} \\ \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } - \D{G}{ z^{(j)} } \frac{k b^{(j-k)} }{ j b^{(0)} } \\ \D{H}{ b^{(j-k)} } & = & \D{G}{ b^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(j-k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(j-k)} } \\ & = & \D{G}{ b^{(j-k)} } - \D{G}{ z^{(j)} } \frac{k z^{(k)} }{ j b^{(0)} } \end{array}$

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