Square Root Function Reverse Mode Theory
Keywords: square root function reverse mode theory sqrt

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Square Root Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions H and G . The forward mode formulas for the square root function are z ( j ) = x ( 0 ) for the case j = 0 , and for j > 0 , z ( j ) = 1 j 1 z ( 0 ) (j 2 x ( j ) - ∑ ℓ = 1 j -1 ℓ z ( ℓ ) z ( j - ℓ ) ) If j = 0 , we have the relation ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( 0 ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) 1 2 z ( 0 ) If j > 0 , then for k = 1 , … , j -1 ∂ H ∂ z ( 0 ) = ∂ G ∂ z ( 0 ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ z ( 0 ) = ∂ G ∂ z ( 0 ) - ∂ G ∂ z ( j ) z ( j ) z ( 0 ) ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) 1 2 z ( 0 ) ∂ H ∂ z ( k ) = ∂ G ∂ z ( k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ z ( k ) = ∂ G ∂ z ( k ) - ∂ G ∂ z ( j ) z ( j - k ) z ( 0 )

Input File: omh/theory/sqrt_reverse.omh