Exponential Function Reverse Mode Theory
Keywords: exponential function reverse mode theory exp expm1

| Prev | Next |  |  |  |  |  | exp_reverse | Headings

Exponential Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions H and G . The zero order forward mode formula for the exponential is z ( 0 ) = F ( x ( 0 ) ) and for j > 0 , z ( j ) = x ( j ) d ( 0 ) + 1 j ∑ k = 1 j k x ( k ) z ( j - k ) where d ( 0 ) = { 0 if F ( x ) = exp ( x ) 1 if F ( x ) = expm 1 ( x ) For order j = 0 , 1 , … we note that ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ( d ( 0 ) + z ( 0 ) ) If j > 0 , then for k = 1 , … , j ∂ H ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) 1 j k z ( j - k ) ∂ H ∂ z ( j - k ) = ∂ G ∂ z ( j - k ) + ∂ G ∂ z ( j ) 1 j k x ( k )

Input File: omh/theory/exp_reverse.omh