lines 8-136 of file: include/cppad/core/subgraph_sparsity.hpp

{xrst_begin subgraph_sparsity}
{xrst_spell
   subgraphs
}

Subgraph Dependency Sparsity Patterns
#####################################

Syntax
******

| *f* . ``subgraph_sparsity`` (
| |tab| *select_domain* , *select_range* , *transpose* , *pattern_out*
| )

See Also
********
:ref:`subgraph_reverse@clear_subgraph` .

Notation
********
We use :math:`F : \B{R}^n \rightarrow \B{R}^m` to denote the
:ref:`glossary@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
:ref:`atomic functions<atomic_two_afun-name>` 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
:ref:`subgraph sparsity<wish_list@Subgraph@Sparsity>`
wish list item.

BoolVector
**********
The type *BoolVector* is a :ref:`SimpleVector-name` class with
:ref:`elements of type<SimpleVector@Elements of Specified Type>`
``bool`` .

SizeVector
**********
The type *SizeVector* is a :ref:`SimpleVector-name` class with
:ref:`elements of type<SimpleVector@Elements of Specified 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 :math:`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 :math:`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
:math:`J(x)` (:math:`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
:ref:`dependency.cpp@Dependency Pattern`
for :math:`F(x)`.
The pattern has :math:`m` rows, :math:`n` columns.
If *select_domain* [ *j* ] is true,
*select_range* [ *i* ] is true, and
:math:`F_i (x)` depends on :math:`x_j`,
then the pair :math:`(i, j)` is in *pattern_out* .
Not that this is also a sparsity pattern for the Jacobian

.. math::

   J(x) = R F^{(1)} (x) D

where :math:`D` (:math:`R`) is the diagonal matrix corresponding
to *select_domain* ( *select_range* ).

Example
*******
{xrst_toc_hidden
   example/sparse/subgraph_sparsity.cpp
}
The file
:ref:`subgraph_sparsity.cpp-name`
contains an example and test of this operation.

{xrst_end subgraph_sparsity}