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Second Order Reverse Mode

Syntax
dw = f.Reverse(2, w)

Purpose
We use $F : B^n \rightarrow B^m$ to denote the AD function corresponding to f . Reverse mode computes the derivative of the Forward mode Taylor coefficients with respect to the domain variable $x$.

x^(k)
For $k = 0, 1$, the vector $x^{(k)} \in B^n$ is defined as the value of x_k in the previous call (counting this call) of the form       f.Forward(k, x_k)  If there is no previous call with $k = 0$, $x^{(0)}$ is the value of the independent variables when the corresponding AD of Base operation sequence was recorded.

W
The functions $W_0 : B^n \rightarrow B$ and $W_1 : B^n \rightarrow B$ are defined by $$\begin{array}{rcl} W_0 ( u ) & = & w_0 * F_0 ( u ) + \cdots + w_{m-1} * F_{m-1} (u) \\ W_1 ( u ) & = & w_0 * F_0^{(1)} ( u ) * x^{(1)} + \cdots + w_{m-1} * F_{m-1}^{(1)} (u) * x^{(1)} \end{array}$$ This operation computes the derivatives $$\begin{array}{rcl} W_0^{(1)} (u) & = & w_0 * F_0^{(1)} ( u ) + \cdots + w_{m-1} * F_{m-1}^{(1)} (u) \\ W_1^{(1)} (u) & = & w_0 * \left( x^{(1)} \right)^\R{T} * F_0^{(2)} ( u ) + \cdots + w_{m-1} * \left( x^{(1)} \right)^\R{T} F_{m-1}^{(2)} (u) \end{array}$$ at $u = x^{(0)}$.

f
The object f has prototype       const ADFun<Base> f  Before this call to Reverse, the value returned by       f.size_order()  must be greater than or equal two (see size_order ).

w
The argument w has prototype       const Vector &w  (see Vector below) and its size must be equal to m , the dimension of the range space for f .

dw
The result dw has prototype       Vector dw  (see Vector below). It contains both the derivative $W^{(1)} (x)$ and the derivative $U^{(1)} (x)$. The size of dw is equal to $n \times 2$, where $n$ is the dimension of the domain space for f .

First Order Partials
For $j = 0 , \ldots , n - 1$, $$dw [ j * 2 + 0 ] = \D{ W_0 }{ u_j } \left( x^{(0)} \right) = w_0 * \D{ F_0 }{ u_j } \left( x^{(0)} \right) + \cdots + w_{m-1} * \D{ F_{m-1} }{ u_j } \left( x^{(0)} \right)$$ This part of dw contains the same values as are returned by reverse_one .

Second Order Partials
For $j = 0 , \ldots , n - 1$, $$dw [ j * 2 + 1 ] = \D{ W_1 }{ u_j } \left( x^{(0)} \right) = \sum_{\ell=0}^{n-1} x_\ell^{(1)} \left[ w_0 * \DD{ F_0 }{ u_\ell }{ u_j } \left( x^{(0)} \right) + \cdots + w_{m-1} * \DD{ F_{m-1} }{ u_\ell }{ u_j } \left( x^{(0)} \right) \right]$$

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.

Hessian Times Direction
Suppose that $w$ is the i-th elementary vector. It follows that for $j = 0, \ldots, n-1$ $$\begin{array}{rcl} dw[ j * 2 + 1 ] & = & w_i \sum_{\ell=0}^{n-1} \DD{F_i}{ u_j }{ u_\ell } \left( x^{(0)} \right) x_\ell^{(1)} \\ & = & \left[ F_i^{(2)} \left( x^{(0)} \right) * x^{(1)} \right]_j \end{array}$$ Thus the vector $( dw[1], dw[3], \ldots , dw[ n * q - 1 ] )$ is equal to the Hessian of $F_i (x)$ times the direction $x^{(1)}$. In the special case where $x^{(1)}$ is the l-th elementary vector , $$dw[ j * 2 + 1 ] = \DD{ F_i }{ x_j }{ x_\ell } \left( x^{(0)} \right)$$

Example
The files reverse_two.cpp and hes_times_dir.cpp contain a examples and tests of reverse mode calculations. They return true if they succeed and false otherwise.
Input File: omh/reverse/reverse_two.omh