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

sparse_jac_fun

View page source

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.