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

sqrt_forward

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Square Root Function Forward Mode Theory

If \(F(x) = \sqrt{x}\)

\[F(x) * F^{(1)} (x) - 0 * F (x) = 1/2\]

and in the standard math function differential equation , \(A(x) = 0\), \(B(x) = F(x)\), and \(D(x) = 1/2\). 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. It now follows from the general Taylor Coefficients Recursion Formula that for \(j = 0 , 1, \ldots\),

\begin{eqnarray} z^{(0)} & = & \sqrt { x^{(0)} } \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 1/2 & {\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}{ z^{(0)} } \left( \frac{j+1}{2} x^{(j+1) } - \sum_{k=1}^j k z^{(k)} z^{(j+1-k)} \right) \end{eqnarray}