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Logarithm Function Forward Mode Theory

Derivatives

If \(F(x)\) is \(\R{log} (x)\) or \(\R{log1p} (x)\) the corresponding derivative satisfies the equation

\[( \bar{b} + x ) * F^{(1)} (x) - 0 * F (x) = 1\]

where

\[\begin{split}\bar{b} = \left\{ \begin{array}{ll} 0 & \R{if} \; F(x) = \R{log}(x) \\ 1 & \R{if} \; F(x) = \R{log1p}(x) \end{array} \right.\end{split}\]

In the standard math function differential equation , \(A(x) = 0\), \(B(x) = \bar{b} + x\), and \(D(x) = 1\). 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^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 1 & {\rm if} \; j = 0 \\ 0 & {\rm otherwise} \end{array} \right. \\ 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) \\ & = & \frac{1}{j+1} \frac{1}{ \bar{b} + x^{(0)} } \left( (j+1) x^{(j+1) } - \sum_{k=1}^j k z^{(k)} x^{(j+1-k)} \right) \end{eqnarray}