First Order Reverse Mode

Loading [MathJax]/extensions/TeX/newcommand.js

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . 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.

Input File: omh/reverse/reverse_one.omh