Link to Speed Test Sparse Hessian

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link_sparse_hessian

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . Link to Speed Test Sparse Hessian
Syntax

ok = link_sparse_hessian(

        size, repeat, row, col, x, hessian, n_color

)

Prototype


extern bool link_sparse_hessian(
    size_t                           size      ,
    size_t                           repeat    ,
    const CppAD::vector<size_t>&     row       ,
    const CppAD::vector<size_t>&     col       ,
    CppAD::vector<double>&           x         ,
    CppAD::vector<double>&           hessian   ,
    size_t&                          n_color
);

Namespace
The function link_sparse_hessian is in the global namespace, not the CppAD namespace.

Method
Given a row index vector

$row$ and a second column vector

$col$ , the corresponding function

$f : \B{R}^n \rightarrow \B{R}$ is defined by sparse_hes_fun . 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.

Sparsity Pattern
The combination of row and col determine the sparsity pattern for the Hessian that is computed. The calculation of this sparsity pattern, if necessary to compute the Hessian, is intended to be part of the timing for this test.

size
The argument size , referred to as

$n$ below, is the dimension of the domain space for

$f(x)$ .

repeat
The argument repeat is the number of times to repeat the test (with a different value for x corresponding to each repetition).

x
The size of x is

$n$ ; i.e., x.size() == size . The input value of the elements of x does not matter. On output, it has been set to the argument value for which the function, or its derivative, is being evaluated. The value of this vector need not change with each repetition.

row
The size of the vector row defines the value

$K$ . The input value of its elements does not matter. On output, all the elements of row are between zero and

$n-1$ .

col
The argument col is a vector with size

$K$ . The input value of its elements does not matter. On output, all the elements of col are between zero and

$n-1$ .

Row Major
The indices row and col are in row major order; i.e., for each k < row.size()-2



    row[k] <= row[k+1]

and if row[k] == row[k+1] then



    col[k] < col[k+1]

Lower Triangular
Only the lower triangle of the Hessian is included.



    col[k] <= row[k]

.

hessian
The size of hessian is K . The input value of its elements does not matter. The output value of its elements is the Hessian of the function

$f(x)$ . To be more specific, for

$k = 0 , \ldots , K-1$ ,

$\DD{f}{ x[ \R{row}[k] ] }{ x[ \R{col}[k] ]} = \R{hessian} [k]$

n_color
The input value of n_color does not matter. On output, it is the value n_sweep corresponding to the evaluation of hessian . This is also the number of colors corresponding to the coloring method , which can be set to colpack , and is otherwise cppad.

double
In the case where package is double, only the first element of hessian is used and it is actually the value of

$f(x)$ (derivatives are not computed).

Input File: speed/src/link_sparse_hessian.hpp