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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 .
Subgraph Dependency Sparsity Patterns

Syntax
f.subgraph_sparsity(
    select_domain, select_range, transpose, pattern_out
)

See Also
clear_subgraph .

Notation
We use @(@ F : \B{R}^n \rightarrow \B{R}^m @)@ to denote the AD function corresponding to the operation sequence stored in f .

Method
This routine uses a subgraph technique. To be specific, for each dependent variable, it creates a subgraph of the operation sequence containing the variables that affect the dependent variable. This avoids the overhead of performing set operations that is inherent in other methods for computing sparsity patterns.

Atomic Function
The sparsity calculation for atomic functions in the f operation sequence are not efficient. To be specific, each atomic function is treated as if all of its outputs depend on all of its inputs. This may be improved upon in the future; see the subgraph sparsity wish list item.

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.

f
The object f has prototype
    ADFun<Base> f
select_domain
The argument select_domain has prototype
    const BoolVector& select_domain
 It has size @(@ n @)@ and specifies which independent variables to include in the calculation. If not all the independent variables are included in the calculation, a forward pass on the operation sequence is used to determine which nodes may be in the subgraphs.

select_range
The argument select_range has prototype
    const BoolVector& select_range
 It has size @(@ m @)@ and specifies which components of the range to include in the calculation. A subgraph is built for each dependent variable and the selected set of independent variables.

transpose
This argument has prototype
    bool transpose
 If transpose it is false (true), upon return pattern_out is a sparsity pattern for @(@ J(x) @)@ (@(@ J(x)^\R{T} @)@) defined below.

pattern_out
This argument has prototype
    sparse_rc<SizeVector>& pattern_out
 This input value of pattern_out does not matter. Upon return pattern_out is a dependency pattern for @(@ F(x) @)@. The pattern has @(@ m @)@ rows, @(@ n @)@ columns. If select_domain[j] is true, select_range[i] is true, and @(@ F_i (x) @)@ depends on @(@ x_j @)@, then the pair @(@ (i, j) @)@ is in pattern_out . Not that this is also a sparsity pattern for the Jacobian @[@ J(x) = R F^{(1)} (x) D @]@ where @(@ D @)@ (@(@ R @)@) is the diagonal matrix corresponding to select_domain ( select_range ).

Example
The file subgraph_sparsity.cpp contains an example and test of this operation.
Input File: include/cppad/core/subgraph_sparsity.hpp