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Error Function Reverse Mode Theory

Notation
We use the reverse theory standard math function definition for the functions $H$ and $G$.

Positive Orders Z(t)
For order $j > 0$, suppose that $H$ is the same as $G$. $$z^{(j)} = \frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)}$$ For $k = 1 , \ldots , j$, the partial of $H$ with respect to $x^{(k)}$ is given by $$\D{H}{ x^{(k)} } = \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} } = \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{k}{j} y^{(j-k)}$$ For $k = 1 , \ldots , j$ The partial of $H$ with respect to $y^{j-k}$, is given by $$\D{H}{ y^{(j-k)} } = \D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} } = \D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \frac{k}{j} x^{k}$$

Order Zero Z(t)
The $z^{(0)}$ coefficient is expressed as a function of the Taylor coefficients for $X(t)$ and $Y(t)$ as follows: In this case, $$\D{H}{ x^{(0)} } = \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} } = \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } y^{(0)}$$
Input File: omh/appendix/theory/erf_reverse.omh