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Hessian Sparsity Pattern: Forward Mode

Syntax
h = f.ForSparseHes(r, s) 
Purpose
We use $F : \B{R}^n \rightarrow \B{R}^m$ to denote the AD function corresponding to f . we define $$\begin{array}{rcl} H(x) & = & \partial_x \left[ \partial_u S \cdot F[ x + R \cdot u ] \right]_{u=0} \\ & = & R^\R{T} \cdot (S \cdot F)^{(2)} ( x ) \cdot R \end{array}$$ Where $R \in \B{R}^{n \times n}$ is a diagonal matrix and $S \in \B{R}^{1 \times m}$ is a row vector. Given a sparsity pattern for the diagonal of $R$ and the vector $S$, ForSparseHes returns a sparsity pattern for the $H(x)$.

f
The object f has prototype       const ADFun<Base> f 
x
If the operation sequence in f is independent of the independent variables in $x \in B^n$, the sparsity pattern is valid for all values of (even if it has CondExp or VecAD operations).

r
The argument r has prototype       const VectorSet& r  (see VectorSet below) If it has elements of type bool, its size is $n$. If it has elements of type std::set<size_t>, its size is one and all the elements of s[0] are between zero and $n - 1$. It specifies a sparsity pattern for the diagonal of $R$. The fewer non-zero elements in this sparsity pattern, the faster the calculation should be and the more sparse $H(x)$ should be.

s
The argument s has prototype       const VectorSet& s  (see VectorSet below) If it has elements of type bool, its size is $m$. If it has elements of type std::set<size_t>, its size is one and all the elements of s[0] are between zero and $m - 1$. It specifies a sparsity pattern for the vector S . The fewer non-zero elements in this sparsity pattern, the faster the calculation should be and the more sparse $H(x)$ should be.

h
The result h has prototype       VectorSet& h  (see VectorSet below). If h has elements of type bool, its size is $n * n$. If it has elements of type std::set<size_t>, its size is $n$ and all the set elements are between zero and n-1 inclusive. It specifies a sparsity pattern for the matrix $H(x)$.

VectorSet
The type VectorSet must be a SimpleVector class with elements of type bool or std::set<size_t>; see sparsity pattern for a discussion of the difference. The type of the elements of VectorSet must be the same as the type of the elements of r .

Algorithm
See Algorithm II in Computing sparse Hessians with automatic differentiation by Andrea Walther. Note that s provides the information so that 'dead ends' are not included in the sparsity pattern.

Example
The file for_sparse_hes.cpp contains an example and test of this operation. It returns true if it succeeds and false otherwise.