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Subgraph Dependency Sparsity Patterns


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

This routine uses a subgraph technique. To be specific, for each dependent variable, it a subgraph of the operation sequence to determine which independent variables affect it. This avoids to 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 atomic functions wish list item.

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

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

The object f has prototype

The argument select_domain has prototype
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.

The argument select_range has prototype
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.

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

This argument has prototype
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 ).

The file subgraph_sparsity.cpp contains an example and test of this operation. It returns true if it succeeds and false otherwise.
Input File: cppad/core/subgraph_sparsity.hpp