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@(@\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 Hessian

Syntax
# include <cppad/speed/sparse_hes_fun.hpp>
 sparse_hes_fun(n, x, row, col, p, fp)

Purpose
This routine evaluates @(@ f(x) @)@, @(@ f^{(1)} (x) @)@, or @(@ f^{(2)} (x) @)@ where the Hessian @(@ f^{(2)} (x) @)@ is sparse. The function @(@ f : \B{R}^n \rightarrow \B{R} @)@ only depends on the size and contents of the index vectors row and col . The non-zero entries in the Hessian of this function have one of the following forms: @[@ \DD{f}{x[row[k]]}{x[row[k]]} \; , \; \DD{f}{x[row[k]]}{x[col[k]]} \; , \; \DD{f}{x[col[k]]}{x[row[k]]} \; , \; \DD{f}{x[col[k]]}{x[col[k]]} @]@ for some @(@ k @)@ between zero and @(@ K-1 @)@. All the other terms of the Hessian are zero.

Inclusion
The template function sparse_hes_fun is defined in the CppAD namespace by including the file cppad/speed/sparse_hes_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) @)@.

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 one of the first index of @(@ x @)@ for each non-zero Hessian term (see purpose above). All the elements of row must be between zero and n-1 . The value @(@ K @)@ is defined by K = row.size() .

col
The argument col has prototype
     const CppAD::vector<size_t>& col
 and its size must be @(@ K @)@; i.e., the same as for col . It specifies the second index of @(@ x @)@ for the non-zero Hessian terms. All the elements of col must be between zero and n-1 . There are no duplicated entries requested, to be specific, if k1 != k2 then
    ( row[k1] , col[k1] ) != ( row[k2] , col[k2] )
p
The argument p has prototype
    size_t p
 It is either zero or two 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
 The input value of the elements of fp does not matter.

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

Hessian
If p is two, fp has size K and for @(@ k = 0 , \ldots , K-1 @)@, @[@ \DD{f}{ x[ \R{row}[k] ] }{ x[ \R{col}[k] ]} = fp [k] @]@

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

Source Code
The file sparse_hes_fun.hpp contains the source code for this template function.
Input File: include/cppad/speed/sparse_hes_fun.hpp