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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 $j = 0 , 1, \ldots$, $$\begin{array}{rcl} z^{(0)} & = & \sqrt { x^{(0)} } \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 1/2 & {\rm if} \; j = 0 \\ 0 & {\rm otherwise} \end{array} \right. \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} \right) \\ & = & \frac{1}{j+1} \frac{1}{ z^{(0)} } \left( \frac{j+1}{2} x^{(j+1) } - \sum_{k=1}^j k z^{(k)} z^{(j+1-k)} \right) \end{array}$$
Input File: omh/appendix/theory/sqrt_forward.omh