Evaluate a Function That Has a Sparse Jacobian

@(@\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 . Evaluate a Function That Has a Sparse Jacobian
Syntax

# include <cppad/speed/sparse_jac_fun.hpp>

sparse_jac_fun(m, n, x, row, col, p, fp)

Purpose
This routine evaluates @(@ f(x) @)@ and @(@ f^{(1)} (x) @)@ where the Jacobian @(@ f^{(1)} (x) @)@ is sparse. The function @(@ f : \B{R}^n \rightarrow \B{R}^m @)@ only depends on the size and contents of the index vectors row and col . The non-zero entries in the Jacobian of this function have one of the following forms: @[@ \D{ f[row[k]]}{x[col[k]]} @]@ for some @(@ k @)@ between zero and @(@ K-1 @)@. All the other terms of the Jacobian are zero.

Inclusion
The template function sparse_jac_fun is defined in the CppAD namespace by including the file cppad/speed/sparse_jac_fun.hpp (relative to the CppAD distribution directory).

Float
The type Float must be a NumericType . In addition, if y and z are Float objects,



    y = exp(z)

must set the y equal the exponential of z , i.e., the derivative of y with respect to z is equal to y .

FloatVector
The type FloatVector is any SimpleVector , or it can be a raw pointer, with elements of type Float .

n
The argument n has prototype



    size_t n

It specifies the dimension for the domain space for @(@ f(x) @)@.

m
The argument m has prototype



    size_t m

It specifies the dimension for the range space for @(@ f(x) @)@.

x
The argument x has prototype



    const FloatVector& x

It contains the argument value for which the function, or its derivative, is being evaluated. We use @(@ n @)@ to denote the size of the vector x .

row
The argument row has prototype



     const CppAD::vector<size_t>& row

It specifies indices in the range of @(@ f(x) @)@ for non-zero components of the Jacobian (see purpose above). The value @(@ K @)@ is defined by K = row.size() . All the elements of row must be between zero and m-1 .

col
The argument col has prototype



     const CppAD::vector<size_t>& col

and its size must be @(@ K @)@; i.e., the same as row . It specifies the component of @(@ x @)@ for the non-zero Jacobian terms. All the elements of col must be between zero and n-1 .

p
The argument p has prototype



    size_t p

It is either zero or one and specifies the order of the derivative of @(@ f @)@ that is being evaluated, i.e., @(@ f^{(p)} (x) @)@ is evaluated.

fp
The argument fp has prototype



    FloatVector& fp

If p = 0 , it size is m otherwise its size is K . The input value of the elements of fp does not matter.

Function
If p is zero, fp has size @(@ m @)@ and (fp[0], ... , fp[m-1]) is the value of @(@ f(x) @)@.

Jacobian
If p is one, fp has size K and for @(@ k = 0 , \ldots , K-1 @)@, @[@ \D{f[ \R{row}[i] ]}{x[ \R{col}[j] ]} = fp [k] @]@

Example
The file sparse_jac_fun.cpp contains an example and test of sparse_jac_fun.hpp.

Source Code
The file sparse_jac_fun.hpp contains the source code for this template function.

Input File: include/cppad/speed/sparse_jac_fun.hpp