\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

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Exponential Function Forward Mode Theory

Derivatives

If \(F(x)\) is \(\R{exp} (x)\) or \(\R{expm1} (x)\) the corresponding derivative satisfies the equation

\[\begin{split}1 * F^{(1)} (x) - 1 * F (x) = d^{(0)} = \left\{ \begin{array}{ll} 0 & \R{if} \; F(x) = \R{exp}(x) \\ 1 & \R{if} \; F(x) = \R{expm1}(x) \end{array} \right.\end{split}\]

where the equation above defines \(d^{(0)}\). In the standard math function differential equation , \(A(x) = 1\), \(B(x) = 1\), and \(D(x) = d^{(0)}\). We use \(a\), \(b\), \(d\), and \(z\) to denote the Taylor coefficients for \(A [ X (t) ]\), \(B [ X (t) ]\), \(D [ X (t) ]\), and \(F [ X(t) ]\) respectively.

Taylor Coefficients Recursion

For orders \(j = 0 , 1, \ldots\),

\begin{eqnarray} z^{(0)} & = & F ( x^{(0)} ) \\ e^{(0)} & = & d^{(0)} + z^{(0)} \\ e^{(j+1)} & = & d^{(j+1)} + \sum_{k=0}^{j+1} a^{(j+1-k)} * z^{(k)} \\ & = & z^{(j+1)} \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} \right) \\ & = & x^{(j+1)} d^{(0)} + \frac{1}{j+1} \sum_{k=1}^{j+1} k x^{(k)} z^{(j+1-k)} \end{eqnarray}