First Order Forward Mode: Derivative Values
Keywords: first order forward mode: derivative values one syntax purpose f x0 x1 vector example special case

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First Order Forward Mode: Derivative Values
Syntax
`y1 = f.Forward(1, x1)`
Purpose
We use F : B n → B m to denote the AD function corresponding to `f` . The result of the syntax above is y 1 = F ( 1 ) ( x 0 ) * x 1 where F ( 1 ) ( x 0 ) is the Jacobian of F evaluated at `x0` .
f
The object `f` has prototype `
ADFun<Base> f
` Note that the ADFun object `f` is not `const`. Before this call to `Forward`, the value returned by `
f.size_order()
` must be greater than or equal one. After this call it will be will be two (see size_order ).
x0
The vector `x0` in the formula y 1 = F ( 1 ) ( x 0 ) * x 1 corresponds to the previous call to forward_zero using this ADFun object `f` ; i.e., `
f.Forward(0, x0)
` If there is no previous call with the first argument zero, the value of the independent variables during the recording of the AD sequence of operations is used for `x0` .
x1
The argument `x1` has prototype `
const Vector& x1
` (see Vector below) and its size must be equal to `n` , the dimension of the domain space for `f` .
Vector
The type `Vector` must be a SimpleVector class with elements of type `Base` . The routine CheckSimpleVector will generate an error message if this is not the case.
Example
The file forward.cpp contains an example and test of this operation. It returns true if it succeeds and false otherwise.
Special Case
This is special case of forward_order where Y ( t ) = F [ X ( t ) ] X ( t ) = x ( 0 ) t 0 + x ( 1 ) * t 1 + ⋯ , + x ( q ) * t q + o ( t q ) Y ( t ) = y ( 0 ) t 0 + y ( 1 ) * t 1 + ⋯ , + y ( q ) * t q + o ( t q ) and o ( t q ) * t - q → 0 as t → 0 . For this special case, q = 1 , x ( 0 ) `= x0` , x ( 1 ) `= x1` , X ( t ) = x ( 0 ) + x ( 1 ) t , and y ( 0 ) + y ( 1 ) t = F [ x ( 0 ) + x ( 1 ) t ] + o ( t ) Taking the derivative with respect to t , at t = 0 , we obtain y ( 1 ) = F ( 1 ) [ x ( 0 ) ] x ( 1 ) which agrees with the specifications for `y1` in the purpose above.

Input File: omh/forward/forward_one.omh