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@(@\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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
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/theory/erf_reverse.omh