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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 .
Atomic Linear ODE Forward Type Calculation: Example Implementation

Purpose
The for_type routine overrides the virtual functions used by the atomic_four base; see for_type .

Notation
We use the notation: call_id r pattern transpose nnz , row , col , x , n , A(x) , b(x) , y(x) , m , vk(x) , and the following additional notation:

T(s)
We use @(@ \R{T} ( s ) @)@ to denote the ad_type of a scalar value @(@ s @)@. There are four possible ad_types : identical_zero, constant, dynamic, and variable in that order.

Theory
This routine must calculate the following value for @(@ i = 0, \ldots, m-1 @)@; see m : @[@ \R{T} [ y_i (x) ] = \max_k \R{T} [ v_i^k (x) ] @]@ The type @(@ \R{T} [ v_i^0 (x) ] = \R{T}[ b_i (x) ] @)@. This is easy to calculate given the type of the components of x ; see b(x) . Furthermore, for @(@ k > 0 @)@ @[@ v_i^k (x) = \frac{r}{k} \sum_{j=0}^{m-1} A_{i,j} (x) v_j^{k-1} (x) @]@ @[@ \R{T} [ v_i^k (x) ] = \max_j \R{T} [ A_{i,j} (x) v_j^{k-1} (x) ] @]@ @[@ \R{T} [ A_{i,j} (x) v_j^k (x) ] = \left\{ \begin{array}{ll} \R{identical\_zero} & \R{if} A_{i,j} (x) \W{\R{or}} v_j^{k-1} (x) \W{\R{is}} \R{identical\_zero} \\ \max\{ \R{T} [ A_{i,j} (x) ] \W{,} \R{T} [ v_j^{k-1} (x) ] \} & \R{otherwise} \end{array} \right. @]@ If @(@ A_{i,j} (x) @)@ is not in the sparsity pattern , it is identically zero. Furthermore we are allowing for the case where @(@ A_{i,j} (x) @)@ is in the pattern and it is identically zero; i.e., the sparsity pattern is not efficient as it could be. The type @(@ \R{T} [ A_{i,j} (x) ] @)@ for components in the sparsity pattern is easy to calculate given the type of the components of x ; see A(x) . Suppose @(@ \ell @)@ is such that for all @(@ i @)@ @[@ \R{T} [ v_i^\ell (x) ] \leq \max_{k < \ell} \R{T} [ v_i^k (x) ] @]@ It follows that @[@ \R{T} [ v_j^{\ell+1} (x) ] = \max_j \R{T} [ A_{i,j} (x) v_j^\ell (x) ] @]@ @[@ \R{T} [ v_j^{\ell+1} (x) ] \leq \max_{k < \ell} \max_j \R{T} [ A_{i,j} (x) v_j^k (x) ] @]@ @[@ \R{T} [ v_j^{\ell+1} (x) ] \leq \max_{k < \ell} \R{T} [ v_i^k (x) ] @]@ From this it is clear that @[@ \R{T} [ y_i (x) ] = \max_{k < \ell} \R{T} [ v_i^k (x) ] @]@

Source 

# include <cppad/example/atomic_four/lin_ode/lin_ode.hpp>

namespace CppAD { // BEGIN_CPPAD_NAMESPACE
//
// for_type override
template <class Base>
bool atomic_lin_ode<Base>::for_type(
    size_t                                     call_id     ,
    const CppAD::vector<CppAD::ad_type_enum>&  type_x      ,
    CppAD::vector<CppAD::ad_type_enum>&        type_y      )
{
    // pattern, transpose, nnz
    Base      r;
    Base      step;
    sparse_rc pattern;
    bool      transpose;
    get(call_id, step, r, pattern, transpose);
    size_t nnz = pattern.nnz();
    //
    // m
    size_t m     = type_y.size();
    CPPAD_ASSERT_UNKNOWN( pattern.nr() == m );
    CPPAD_ASSERT_UNKNOWN( pattern.nc() == m );
    //
    // type_x
    CPPAD_ASSERT_UNKNOWN( type_x.size() == nnz + m );
    //
    // type_y[i] = type_b[i]
    // type_y[i] = max T[ v_i^k (x) ] for k = 0
    for(size_t i = 0; i < m; ++i)
        type_y[i] = type_x[nnz + i];
    //
    // change
    // Did type_y change during the previous iteration of the while loop
    bool change = true;
    while(change)
    {   change = false;
        // we use k = 1, 2, ... to denote the pass through this loop
        // type_y[i] = max q < k T[ v_i^q (x) ]
        //
        for(size_t p = 0; p < nnz; ++p)
        {   size_t i = pattern.row()[p];
            size_t j = pattern.col()[p];
            if( transpose )
                std::swap(i, j);
            //
            // type_y[i], change
            if( type_x[p] != identical_zero_enum )
            {   // A_ij (x) is not identically zero
                if( type_y[j] > type_y[i] )
                {   change = true;
                    type_y[i] = type_y[j];
                 }
            }
            //
            // type_y[i], change
            if( type_y[j] != identical_zero_enum )
            {   // There is a q < k such that v_j^q (x) not identically zero
                if( type_x[p] > type_y[i] )
                {   change = true;
                    type_y[i] = type_x[p];
                }
            }
        }
    }
    return true;
}
} // END_CPPAD_NAMESPACE
Input File: include/cppad/example/atomic_four/lin_ode/for_type.hpp