\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

reverse_one

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First Order Reverse Mode

Syntax

dw = f . Reverse (1, w )

Purpose

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the AD Function corresponding to f . The function \(W : \B{R}^n \rightarrow \B{R}\) is defined by

\[W(x) = w_0 * F_0 ( x ) + \cdots + w_{m-1} * F_{m-1} (x)\]

The result of this operation is the derivative \(dw = W^{(1)} (x)\); i.e.,

\[dw = w_0 * F_0^{(1)} ( x ) + \cdots + w_{m-1} * F_{m-1}^{(1)} (x)\]

Note that if \(w\) is the i-th Elementary Vector , \(dw = F_i^{(1)} (x)\).

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 one (see size_order ).

x

The vector x in expression for dw above corresponds to the previous call to forward_zero using this ADFun object f ; i.e.,

f . Forward (0, x )

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 x .

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) and its value is the derivative \(W^{(1)} (x)\). The size of dw is 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 reverse_one.cpp contains an example and test of this operation.