rev_hes_sparsity¶

Reverse Mode Hessian Sparsity Patterns¶

Syntax¶

f . rev_hes_sparsity (
      select_range , transpose , 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 \(R \in \B{R}^{n \times \ell}\), \(s \in \B{R}^m\) and define the function

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

Given a Sparsity Pattern for \(R\) and for the vector \(s\), rev_hes_sparsity computes a sparsity pattern for \(H(x)\).

x¶

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

BoolVector¶

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

SizeVector¶

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

f¶

The object f has prototype

ADFun < Base > f

R¶

The sparsity pattern for the matrix \(R\) is specified by pattern_in in the previous call

      f . for_jac_sparsity (
            pattern_in , transpose , dependency , internal_bool , pattern_out
)

select_range¶

The argument select_range has prototype

const BoolVector & select_range

It has size \(m\) and specifies which components of the vector \(s\) are non-zero; i.e., select_range [ i ] is true if and only if \(s_i\) is possibly non-zero.

transpose¶

This argument has prototype

bool transpose

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. This must be the same as in the previous call to f . for_jac_sparsity .

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 \(H(x)\) (\(H(x)^\R{T}\)).

Sparsity for Entire Hessian¶

Suppose that \(R\) is the \(n \times n\) identity matrix. In this case, pattern_out is a sparsity pattern for \((s^\R{T} F) F^{(2)} ( x )\).

Example¶

The file rev_hes_sparsity.cpp contains an example and test of this operation.