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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 .
Forward Mode Jacobian Sparsity: Example and Test

# include <set>
# include <cppad/cppad.hpp>

namespace { // -------------------------------------------------------------
// define the template function BoolCases<Vector>
template <class Vector>  // vector class, elements of type bool
bool BoolCases(void)
{   bool ok = true;
    using CppAD::AD;

    // domain space vector
    size_t n = 2;
    CPPAD_TESTVECTOR(AD<double>) X(n);
    X[0] = 0.;
    X[1] = 1.;

    // declare independent variables and start recording
    CppAD::Independent(X);

    // range space vector
    size_t m = 3;
    CPPAD_TESTVECTOR(AD<double>) Y(m);
    Y[0] = X[0];
    Y[1] = X[0] * X[1];
    Y[2] = X[1];

    // create f: X -> Y and stop tape recording
    CppAD::ADFun<double> f(X, Y);

    // sparsity pattern for the identity matrix
    Vector r(n * n);
    size_t i, j;
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
            r[ i * n + j ] = (i == j);
    }

    // sparsity pattern for F'(x)
    Vector s(m * n);
    s = f.ForSparseJac(n, r);

    // check values
    ok &= (s[ 0 * n + 0 ] == true);  // Y[0] does     depend on X[0]
    ok &= (s[ 0 * n + 1 ] == false); // Y[0] does not depend on X[1]
    ok &= (s[ 1 * n + 0 ] == true);  // Y[1] does     depend on X[0]
    ok &= (s[ 1 * n + 1 ] == true);  // Y[1] does     depend on X[1]
    ok &= (s[ 2 * n + 0 ] == false); // Y[2] does not depend on X[0]
    ok &= (s[ 2 * n + 1 ] == true);  // Y[2] does     depend on X[1]

    // check that values are stored
    ok &= (f.size_forward_bool() > 0);
    ok &= (f.size_forward_set() == 0);

    // sparsity pattern for F'(x)^T, note R is the identity, so R^T = R
    bool transpose = true;
    Vector st(n * m);
    st = f.ForSparseJac(n, r, transpose);

    // check values
    ok &= (st[ 0 * m + 0 ] == true);  // Y[0] does     depend on X[0]
    ok &= (st[ 1 * m + 0 ] == false); // Y[0] does not depend on X[1]
    ok &= (st[ 0 * m + 1 ] == true);  // Y[1] does     depend on X[0]
    ok &= (st[ 1 * m + 1 ] == true);  // Y[1] does     depend on X[1]
    ok &= (st[ 0 * m + 2 ] == false); // Y[2] does not depend on X[0]
    ok &= (st[ 1 * m + 2 ] == true);  // Y[2] does     depend on X[1]

    // check that values are stored
    ok &= (f.size_forward_bool() > 0);
    ok &= (f.size_forward_set() == 0);

    // free values from forward calculation
    f.size_forward_bool(0);
    ok &= (f.size_forward_bool() == 0);

    return ok;
}
// define the template function SetCases<Vector>
template <class Vector>  // vector class, elements of type std::set<size_t>
bool SetCases(void)
{   bool ok = true;
    using CppAD::AD;

    // domain space vector
    size_t n = 2;
    CPPAD_TESTVECTOR(AD<double>) X(n);
    X[0] = 0.;
    X[1] = 1.;

    // declare independent variables and start recording
    CppAD::Independent(X);

    // range space vector
    size_t m = 3;
    CPPAD_TESTVECTOR(AD<double>) Y(m);
    Y[0] = X[0];
    Y[1] = X[0] * X[1];
    Y[2] = X[1];

    // create f: X -> Y and stop tape recording
    CppAD::ADFun<double> f(X, Y);

    // sparsity pattern for the identity matrix
    Vector r(n);
    size_t i;
    for(i = 0; i < n; i++)
    {   assert( r[i].empty() );
        r[i].insert(i);
    }

    // sparsity pattern for F'(x)
    Vector s(m);
    s = f.ForSparseJac(n, r);

    // an interator to a standard set
    std::set<size_t>::iterator itr;
    bool found;

    // Y[0] does     depend on X[0]
    found = s[0].find(0) != s[0].end();  ok &= ( found == true );
    // Y[0] does not depend on X[1]
    found = s[0].find(1) != s[0].end();  ok &= ( found == false );
    // Y[1] does     depend on X[0]
    found = s[1].find(0) != s[1].end();  ok &= ( found == true );
    // Y[1] does     depend on X[1]
    found = s[1].find(1) != s[1].end();  ok &= ( found == true );
    // Y[2] does not depend on X[0]
    found = s[2].find(0) != s[2].end();  ok &= ( found == false );
    // Y[2] does     depend on X[1]
    found = s[2].find(1) != s[2].end();  ok &= ( found == true );

    // check that values are stored
    ok &= (f.size_forward_set() > 0);
    ok &= (f.size_forward_bool() == 0);


    // sparsity pattern for F'(x)^T
    bool transpose = true;
    Vector st(n);
    st = f.ForSparseJac(n, r, transpose);

    // Y[0] does     depend on X[0]
    found = st[0].find(0) != st[0].end();  ok &= ( found == true );
    // Y[0] does not depend on X[1]
    found = st[1].find(0) != st[1].end();  ok &= ( found == false );
    // Y[1] does     depend on X[0]
    found = st[0].find(1) != st[0].end();  ok &= ( found == true );
    // Y[1] does     depend on X[1]
    found = st[1].find(1) != st[1].end();  ok &= ( found == true );
    // Y[2] does not depend on X[0]
    found = st[0].find(2) != st[0].end();  ok &= ( found == false );
    // Y[2] does     depend on X[1]
    found = st[1].find(2) != st[1].end();  ok &= ( found == true );

    // check that values are stored
    ok &= (f.size_forward_set() > 0);
    ok &= (f.size_forward_bool() == 0);

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool ForSparseJac(void)
{   bool ok = true;
    // Run with Vector equal to four different cases
    // all of which are Simple Vectors with elements of type bool.
    ok &= BoolCases< CppAD::vectorBool     >();
    ok &= BoolCases< CppAD::vector  <bool> >();
    ok &= BoolCases< std::vector    <bool> >();
    ok &= BoolCases< std::valarray  <bool> >();

    // Run with Vector equal to two different cases both of which are
    // Simple Vectors with elements of type std::set<size_t>
    typedef std::set<size_t> set;
    ok &= SetCases< CppAD::vector  <set> >();
    // ok &= SetCases< std::vector    <set> >();

    // Do not use valarray because its element access in the const case
    // returns a copy instead of a reference
    // ok &= SetCases< std::valarray  <set> >();

    return ok;
}

Input File: example/sparse/for_sparse_jac.cpp