RevSparseJac

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Jacobian Sparsity Pattern: Reverse Mode

Syntax

s = f . RevSparseJac ( q , r )
s = f . RevSparseJac ( q , r , transpose , dependency )

Purpose

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the AD Function corresponding to f . For a fixed matrix \(R \in \B{R}^{q \times m}\), the Jacobian of \(R * F( x )\) with respect to \(x\) is

\[S(x) = R * F^{(1)} ( x )\]

Given a Sparsity Pattern for \(R\), RevSparseJac returns a sparsity pattern for the \(S(x)\).

f

The object f has prototype

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 rows in \(R \in \B{R}^{q \times m}\) and the Jacobian \(S(x) \in \B{R}^{q \times n}\).

transpose

The argument transpose has prototype

bool transpose

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

dependency

The argument dependency has prototype

bool dependency

If dependency is true, the Dependency Pattern (instead of sparsity pattern) is computed.

r

The argument s has prototype

const SetVector & r

see SetVector below.

transpose false

If r has elements of type bool , its size is \(q * m\). If it has elements of type std::set<size_t> , its size is q and all its set elements are between zero and \(m - 1\). It specifies a Sparsity Pattern for the matrix \(R \in \B{R}^{q \times m}\).

transpose true

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

s

The return value s has prototype

SetVector s

see SetVector below.

transpose false

If it 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 its set elements are between zero and \(n - 1\). It specifies a Sparsity Pattern for the matrix \(S(x) \in {q \times n}\).

transpose true

If it 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 its set elements are between zero and \(q - 1\). It specifies a Sparsity Pattern for the matrix \(S(x)^\R{T} \in {n \times q}\).

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.

Entire Sparsity Pattern

Suppose that \(q = m\) and \(R\) is the \(m \times m\) identity matrix. In this case, the corresponding value for s is a sparsity pattern for the Jacobian \(S(x) = F^{(1)} ( x )\).

Example

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