\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

RevSparseHes

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Hessian Sparsity Pattern: Reverse Mode

Syntax

h = f . RevSparseHes ( q , s )

h = f . RevSparseHes ( q , s , transpose )

Purpose

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the AD Function corresponding to f . For a fixed matrix \(R \in \B{R}^{n \times q}\) and a fixed vector \(S \in \B{R}^{1 \times m}\), we define

\begin{eqnarray} H(x) & = & \partial_x \left[ \partial_u S * F[ x + R * u ] \right]_{u=0} \\ & = & R^\R{T} * (S * F)^{(2)} ( x ) \\ H(x)^\R{T} & = & (S * F)^{(2)} ( x ) * R \end{eqnarray}

Given a Sparsity Pattern for the matrix \(R\) and the vector \(S\), RevSparseHes 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 \(x \in \B{R}^n\), the sparsity pattern is valid for all values of (even if it has CondExp or VecAD operations).

q

The argument q has prototype

size_t q

It specifies the number of columns in \(R \in \B{R}^{n \times q}\) and the number of rows in \(H(x) \in \B{R}^{q \times n}\). It must be the same value as in the previous ForSparseJac call

f . ForSparseJac ( q , r , r_transpose )

Note that if r_transpose is true, r in the call above corresponding to \(R^\R{T} \in \B{R}^{q \times n}\)

transpose

The argument transpose has prototype

bool transpose

The default value false is used when transpose is not present.

r

The matrix \(R\) is specified by the previous call

f . ForSparseJac ( q , r , transpose )

see r . The type of the elements of SetVector must be the same as the type of the elements of r .

s

The argument s has prototype

const SetVector & s

(see SetVector 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 .

h

The result h has prototype

SetVector & h

(see SetVector below).

transpose false

If h has elements of type bool , its size is \(q * n\). If it has elements of type std::set<size_t> , its size is \(q\) and all the set elements are between zero and n -1 inclusive. It specifies a Sparsity Pattern for the matrix \(H(x)\).

transpose true

If h has elements of type bool , its size is \(n * q\). If it has elements of type std::set<size_t> , its size is \(n\) and all the set elements are between zero and q -1 inclusive. It specifies a Sparsity Pattern for the matrix \(H(x)^\R{T}\).

SetVector

The type SetVector 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 SetVector must be the same as the type of the elements of r .

Entire Sparsity Pattern

Suppose that \(q = n\) and \(R \in \B{R}^{n \times n}\) is the \(n \times n\) identity matrix. Further suppose that the \(S\) is the k-th Elementary Vector ; i.e.

\[\begin{split}S_j = \left\{ \begin{array}{ll} 1 & {\rm if} \; j = k \\ 0 & {\rm otherwise} \end{array} \right.\end{split}\]

In this case, the corresponding value h is a sparsity pattern for the Hessian matrix \(F_k^{(2)} (x) \in \B{R}^{n \times n}\).

Example

The file rev_sparse_hes.cpp contains an example and test of this operation. The file sparsity_sub.cpp contains an example and test of using RevSparseHes to compute the sparsity pattern for a subset of the Hessian.