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atan_reverse

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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{eqnarray} z^{(0)} & = & F ( x^{(0)} ) \\ b^{(0)} & = & 1 \pm x^{(0)} x^{(0)} \end{eqnarray}

where \(F(x) = \R{atan} (x)\) for \(+\) and \(F(x) = \R{atanh} (x)\) for \(-\). For orders \(j\) greater than zero we have

\begin{eqnarray} 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{eqnarray}

If \(j = 0\), we note that \(F^{(1)} ( x^{(0)} ) = 1 / b^{(0)}\) and hence

\begin{eqnarray} \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{eqnarray}

If \(j > 0\), then for \(k = 1, \ldots , j-1\)

\begin{eqnarray} \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{eqnarray}