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Forward Mode Jacobian Sparsity Patterns

Syntax
f.for_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 $R \in \B{R}^{n \times \ell}$ and define the function $$J(x) = F^{(1)} ( x ) * R$$ Given the sparsity pattern for $R$, for_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  The ADFun object f is not const. After a call to for_jac_sparsity, a sparsity pattern for each of the variables in the operation sequence is held in f for possible later use during reverse Hessian sparsity calculations.

size_forward_bool
After for_jac_sparsity, if k is a size_t object,       k = f.size_forward_bool()  sets k to the amount of memory (in unsigned character units) used to store the boolean vector sparsity patterns. If internal_bool if false, k will be zero. Otherwise it will be non-zero. If you do not need this information for RevSparseHes calculations, it can be deleted (and the corresponding memory freed) using       f.size_forward_bool(0)  after which f.size_forward_bool() will return zero.

size_forward_set
After for_jac_sparsity, if k is a size_t object,       k = f.size_forward_set()  sets k to the amount of memory (in unsigned character units) used to store the vector of sets sparsity patterns. If internal_bool if true, k will be zero. Otherwise it will be non-zero. If you do not need this information for future rev_hes_sparsity calculations, it can be deleted (and the corresponding memory freed) using       f.size_forward_set(0)  after which f.size_forward_set() will return zero.

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 $n \times n$ 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 for_jac_sparsity.cpp contains an example and test of this operation. It returns true if it succeeds and false otherwise.