Atomic Forward Mode
Keywords: atomic forward mode callback virtual syntax purpose implementation p q vx vy tx ty ok discussion examples define use

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Atomic Forward Mode
Syntax
`ok = afun.forward(p, q, vx, vy, tx, ty)`
Purpose
This virtual function is used by atomic_afun to evaluate function values. It is also used buy forward to compute function vales and derivatives.
Implementation
This virtual function must be defined by the atomic_user class. It can just return `ok == false` (and not compute anything) for values of `q > 0` that are greater than those used by your forward mode calculations.
p
The argument `p` has prototype `
size_t p
` It specifies the lowest order Taylor coefficient that we are evaluating. During calls to atomic_afun , `p == 0` .
q
The argument `q` has prototype `
size_t q
` It specifies the highest order Taylor coefficient that we are evaluating. During calls to atomic_afun , `q == 0` .
vx
The `forward` argument `vx` has prototype `
const CppAD::vector<bool>& vx
` The case `vx.size() > 0` only occurs while evaluating a call to atomic_afun . In this case, `p == q == 0` , `vx.size() == n` , and for j = 0 , … , n -1 , `vx[j]` is true if and only if `ax[j]` is a variable in the corresponding call to `
afun(ax, ay, id)
` If `vx.size() == 0` , then `vy.size() == 0` and neither of these vectors should be used.
vy
The `forward` argument `vy` has prototype `
CppAD::vector<bool>& vy
` If `vy.size() == 0` , it should not be used. Otherwise, `q == 0` and `vy.size() == m` . The input values of the elements of `vy` are not specified (must not matter). Upon return, for j = 0 , … , m -1 , `vy[i]` is true if and only if `ay[i]` is a variable (CppAD uses `vy` to reduce the necessary computations).
tx
The argument `tx` has prototype `
const CppAD::vector<Base>& tx
` and `tx.size() == (q+1)*n` . For j = 0 , … , n -1 and k = 0 , … , q , we use the Taylor coefficient notation x j k = tx [ j * ( q + 1 ) + k ] X j ( t ) = x j 0 + x j 1 t 1 + ⋯ + x j q t q Note that superscripts represent an index for x j k and an exponent for t k . Also note that the Taylor coefficients for X ( t ) correspond to the derivatives of X ( t ) at t = 0 in the following way: x j k = 1 k ! X j ( k ) ( 0 )
ty
The argument `ty` has prototype `
CppAD::vector<Base>& ty
` and `tx.size() == (q+1)*m` . Upon return, For i = 0 , … , m -1 and k = 0 , … , q , Y i ( t ) = f i [ X ( t ) ] Y i ( t ) = y i 0 + y i 1 t 1 + ⋯ + y i q t q + o ( t q ) ty [ i * ( q + 1 ) + k ] = y i k where o ( t q ) / t q → 0 as t → 0 . Note that superscripts represent an index for y j k and an exponent for t k . Also note that the Taylor coefficients for Y ( t ) correspond to the derivatives of Y ( t ) at t = 0 in the following way: y j k = 1 k ! Y j ( k ) ( 0 ) If p > 0 , for i = 0 , … , m -1 and k = 0 , … , p -1 , the input of `ty` satisfies ty [ i * ( q + 1 ) + k ] = y i k and hence the corresponding elements need not be recalculated.
ok
If the required results are calculated, `ok` should be true. Otherwise, it should be false.
Discussion
For example, suppose that `q == 2` , and you know how to compute the function f ( x ) , its first derivative f ( 1 ) ( x ) , and it component wise Hessian f i ( 2 ) ( x ) . Then you can compute `ty` using the following formulas: y i 0 = Y ( 0 ) = f i ( x 0 ) y i 1 = Y ( 1 ) ( 0 ) = f i ( 1 ) ( x 0 ) X ( 1 ) ( 0 ) = f i ( 1 ) ( x 0 ) x 1 y i 2 = 1 2 ! Y ( 2 ) ( 0 ) = 1 2 X ( 1 ) ( 0 ) T f i ( 2 ) ( x 0 ) X ( 1 ) ( 0 ) + 1 2 f i ( 1 ) ( x 0 ) X ( 2 ) ( 0 ) = 1 2 ( x 1 ) T f i ( 2 ) ( x 0 ) x 1 + f i ( 1 ) ( x 0 ) x 2 For i = 0 , … , m -1 , and k = 0 , 1 , 2 , ty [ i * ( q + 1 ) + k ] = y i k
Examples
Define forward
The following files contain example atomic `forward` functions: get_started.cpp , norm_sq.cpp , reciprocal.cpp , tangent.cpp , matrix_mul.hpp .
Use forward
The following are links to user atomic function forward uses: get_started.cpp , norm_sq.cpp , reciprocal.cpp , tangent.cpp , mat_mul.cpp .

Input File: cppad/local/atomic_base.hpp