Logarithm Function Forward Mode Theory
Keywords: logarithm function forward mode theory log1p derivatives taylor coefficients recursion

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Logarithm Function Forward Mode Theory
Derivatives
If F ( x ) is log ( x ) or log 1 p ( x ) the corresponding derivative satisfies the equation ( b ¯ + x ) * F ( 1 ) ( x ) - 0 * F ( x ) = 1 where b ¯ = { 0 if F ( x ) = log ( x ) 1 if F ( x ) = log 1 p ( x ) In the standard math function differential equation , A ( x ) = 0 , B ( x ) = b ¯ + x , and D ( x ) = 1 . We use a , b , d , and z to denote the Taylor coefficients for A [ X ( t ) ] , B [ X ( t ) ] , D [ X ( t ) ] , and F [ X ( t ) ] respectively.
Taylor Coefficients Recursion
For orders j = 0 , 1 , … , z ( 0 ) = F ( x ( 0 ) ) e ( j ) = d ( j ) + ∑ k = 0 j a ( j - k ) * z ( k ) = { 1 if j = 0 0 otherwise z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∑ k = 1 j + 1 k x ( k ) e ( j + 1 - k ) - ∑ k = 1 j k z ( k ) b ( j + 1 - k ) ) = 1 j + 1 1 b ¯ + x ( 0 ) (( j + 1 ) x ( j + 1 ) - ∑ k = 1 j k z ( k ) x ( j + 1 - k ) )

Input File: omh/theory/log_forward.omh