cpp_sparsity#

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Hessian Sparsity Patterns#

Syntax#

f.for_hes_sparsity ( select_domain , select_range , pattern_out )
f.rev_hes_sparsity ( select_domain , select_range , pattern_out )

Purpose#

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the function corresponding to the operation sequence stored in f . Fix a diagonal matrix \(D \in \B{R}^{n \times n}\), fix a vector \(r \in \B{R}^m\), and define

\[H(x) = D (r^\R{T} F)^{(2)} ( x ) D\]

Given a sparsity pattern for \(D\) and \(r\), these routines compute a sparsity pattern for \(H(x)\).

x#

Note that a sparsity pattern for \(H(x)\) corresponds to the operation sequence stored in f and does not depend on the argument x .

f#

The object f has prototype

      d_fun f

select_domain#

The argument select_domain has prototype

      const vec_bool& select_domain

It has size n and is a sparsity pattern for the diagonal of \(D\); i.e., select_domain [ j ] is true if and only if \(D_{j,j}\) is possibly non-zero.

select_range#

The argument select_range has prototype

      const vec_bool& select_range

It has size m and is a sparsity pattern for the vector \(r\); i.e., select_range [ i ] is true if and only if \(r_i\) is possibly non-zero.

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 sparsity pattern for \(J(x)\).

Sparsity for Component Wise Hessian#

Suppose that \(D\) is the identity matrix, and only the i-th component of r is possibly non-zero. In this case, pattern_out is a sparsity pattern for \(F_i^{(2)} ( x )\).

Example#

c++