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

subgraph_reverse

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Reverse Mode Using Subgraphs

Syntax

f . subgraph_reverse ( select_domain )

f . subgraph_reverse ( q , ell , col , dw )

f . clear_subgraph ()

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 \(x\).

Notation

We use the reverse mode Notation with the following change: the vector w^(k) is defined

\[\begin{split}w_i^{(k)} = \left\{ \begin{array}{ll} 1 & {\rm if} \; k = q-1 \; \R{and} \; i = \ell \\ 0 & {\rm otherwise} \end{array} \right.\end{split}\]

BaseVector

The type BaseVector must be a SimpleVector class with elements of type Base . The routine CheckSimpleVector will generate an error message if this is not the case.

BoolVector

The type BoolVector is a SimpleVector class with elements of type bool .

SizeVector

The type SizeVector is a SimpleVector class with elements of type size_t .

select_domain

The argument select_domain has prototype

const BoolVector & select_domain

It has size \(n\) and specifies which independent variables to include in future subgraph_reverse calculations. If select_domain [ j ] is false, it is assumed that \(u^{(k)}_j = 0\) for \(k > 0\); i.e., the j-th component of the Taylor coefficient for \(x\), with order greater that zero, are zero; see u^(k) .

q

The argument q has prototype

size_t q

and specifies the number of Taylor coefficient orders to be differentiated.

ell

The argument ell has prototype

size_t ell

and specifies the dependent variable index that we are computing the derivatives for; i.e. \(\ell\). This index can only be used once per, and after, a call that selects the independent variables using select_domain .

col

This argument col has prototype

SizeVector col

The input size and value of its elements do not matter. The col . resize member function is used to change its size to the number the number of possible non-zero derivative components. For each c ,

      select_domain [ col [ c ] ] == true

      col [ c +1] >= col [ c ]

and the derivative with respect to the j-th independent variable is possibly non-zero where j = col [ c ] .

dw

The argument dw has prototype

Vector dw

Its input size and value does not matter. Upon return, it is a vector with size \(n \times q\). For \(c = 0 , \ldots , %col%.size()-1\), and \(k = 0, \ldots , q-1\),

\[dw[ j * q + k ] = W^{(1)} ( x )_{j,k}\]

is the derivative of the specified Taylor coefficients w.r.t the j-th independent variable where j = col [ c ] . Note that this corresponds to the reverse_any convention when w has size m * q .

clear_subgraph

Calling this routine will free memory that holds information between calls to subgraph calculations so that it does not need to be recalculated. (This memory is automatically freed when f is deleted.) You cannot free this memory between calls that select the domain and corresponding calls that compute reverse mode derivatives. Some of this information is also used by subgraph_sparsity .

Example

The file subgraph_reverse.cpp contains an example and test of this operation.