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Up-> CppAD ADFun Reverse reverse_two

Headings-> Syntax Purpose x^(k) W f w dw ---..First Order Partials ---..Second Order Partials Vector Hessian Times Direction Example

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation . Second Order Reverse Mode
Syntax
dw = f.Reverse(2, w)

Purpose
We use @(@ F : \B{R}^n \rightarrow \B{R}^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{R}^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{R}^n \rightarrow \B{R} @)@ and @(@ W_1 : \B{R}^n \rightarrow \B{R} @)@ 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.

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Input File: omh/reverse/reverse_two.omh