Forward Mode Jacobian Sparsity Patterns

@(@\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}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation . Forward Mode Jacobian Sparsity Patterns
Syntax

f.for_jac_sparsity(

    pattern_in, transpose, dependency, 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} @)@ and define the function @[@ J(x) = F^{(1)} ( x ) * R @]@ Given the sparsity pattern for @(@ R @)@, for_jac_sparsity computes a sparsity pattern for @(@ J(x) @)@.

x
Note that the sparsity pattern @(@ J(x) @)@ corresponds to the operation sequence stored in f and does not depend on the argument x . (The operation sequence may contain CondExp and VecAD operations.)

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

f
The object f has prototype



    ADFun<Base> f

The ADFun object f is not const. After a call to for_jac_sparsity, a sparsity pattern for each of the variables in the operation sequence is held in f for possible later use during reverse Hessian sparsity calculations.

size_forward_bool
After for_jac_sparsity, if k is a size_t object,



    k = f.size_forward_bool()

sets k to the amount of memory (in unsigned character units) used to store the boolean vector sparsity patterns. If internal_bool if false, k will be zero. Otherwise it will be non-zero. If you do not need this information for RevSparseHes calculations, it can be deleted (and the corresponding memory freed) using



    f.size_forward_bool(0)

after which f.size_forward_bool() will return zero.

size_forward_set
After for_jac_sparsity, if k is a size_t object,



    k = f.size_forward_set()

sets k to the amount of memory (in unsigned character units) used to store the vector of sets sparsity patterns. If internal_bool if true, k will be zero. Otherwise it will be non-zero. If you do not need this information for future rev_hes_sparsity calculations, it can be deleted (and the corresponding memory freed) using



    f.size_forward_set(0)

after which f.size_forward_set() will return zero.

pattern_in
The argument pattern_in has prototype



    const sparse_rc<SizeVector>& pattern_in

see sparse_rc . If transpose it is false (true), pattern_in is a sparsity pattern for @(@ R @)@ (@(@ R^\R{T} @)@).

transpose
This argument has prototype



    bool transpose

See pattern_in above and pattern_out below.

dependency
This argument has prototype



    bool dependency

see pattern_out below.

internal_bool
This argument has prototype



    bool 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.

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 @(@ J(x) @)@ (@(@ J(x)^\R{T} @)@). If dependency is true, pattern_out is a dependency pattern instead of sparsity pattern.

Sparsity for Entire Jacobian
Suppose that @(@ R @)@ is the @(@ n \times n @)@ identity matrix. In this case, pattern_out is a sparsity pattern for @(@ F^{(1)} ( x ) @)@ ( @(@ F^{(1)} (x)^\R{T} @)@ ) if transpose is false (true).

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

Input File: include/cppad/core/for_jac_sparsity.hpp