Error Function Reverse Mode Theory
Keywords: error function reverse mode theory erf notation positive orders z(t) order zero

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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 ) = 1 j ∑ k = 1 j k x ( k ) y ( j - k ) For k = 1 , … , j , the partial of H with respect to x ( k ) is given by ∂ H ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) k j y ( j - k ) For k = 1 , … , j The partial of H with respect to y j - k , is given by ∂ H ∂ y ( j - k ) = ∂ G ∂ y ( j - k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ y ( j - k ) = ∂ G ∂ y ( j - k ) + ∂ G ∂ z ( j ) 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, ∂ H ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) + ∂ G ∂ z ( 0 ) ∂ z ( 0 ) ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) + ∂ G ∂ z ( 0 ) y ( 0 )

Input File: omh/theory/erf_reverse.omh