Reverse Mode Jacobian Sparsity Patterns

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

f.rev_jac_sparsity(

    pattern_in, transpose, dependency, internal_bool, pattern_out

)

Purpose
We use

$F : \B{R}^n \rightarrow \B{R}^m$ to denote the AD function corresponding to the operation sequence stored in f . Fix

$R \in \B{R}^{\ell \times m}$ and define the function

$J(x) = R * F^{(1)} ( x )$ Given the sparsity pattern for

$R$ , rev_jac_sparsity computes a sparsity pattern for

$J(x)$ .

x
Note that the sparsity pattern

$J(x)$ corresponds to the operation sequence stored in f and does not depend on the argument x . (The operation sequence may contain CondExp and VecAD operations.)

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

f
The object f has prototype



    ADFun<Base> f

pattern_in
The argument pattern_in has prototype



    const sparse_rc<SizeVector>& pattern_in

see sparse_rc . If transpose it is false (true), pattern_in is a sparsity pattern for

$R$ (

$R^\R{T}$ ).

transpose
This argument has prototype



    bool transpose

See pattern_in above and pattern_out below.

dependency
This argument has prototype



    bool dependency

see pattern_out below.

internal_bool
If this is true, calculations are done with sets represented by a vector of boolean values. Otherwise, a vector of sets of integers is used.

pattern_out
This argument has prototype



    sparse_rc<SizeVector>& pattern_out

This input value of pattern_out does not matter. If transpose it is false (true), upon return pattern_out is a sparsity pattern for

$J(x)$ (

$J(x)^\R{T}$ ). If dependency is true, pattern_out is a dependency pattern instead of sparsity pattern.

Sparsity for Entire Jacobian
Suppose that

$R$ is the

$m \times m$ identity matrix. In this case, pattern_out is a sparsity pattern for

$F^{(1)} ( x )$ (

$F^{(1)} (x)^\R{T}$ ) if transpose is false (true).

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

Input File: include/cppad/core/rev_jac_sparsity.hpp