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erf_reverse

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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,

\[\D{H}{ x^{(0)} } = \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} } = \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } y^{(0)}\]