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Computing Dependency: Example and Test
Discussion
The partial of an dependent variable with respect to an independent variable might always be zero even though the dependent variable depends on the value of the dependent variable. Consider the following case f ( x ) = sign ( x ) = { + 1 if x > 0 0 if x = 0 -1 if x < 0 In this case the value of f ( x ) depends on the value of x but CppAD always returns zero for the derivative of the sign function.
Dependency Pattern
If the `i`-th dependent variables depends on the value of the `j`-th independent variable, the corresponding entry in the dependency pattern is non-zero (true). Otherwise it is zero (false). CppAD uses sparsity patterns to represent dependency patterns.
Computation
The `dependency` argument to for_jac_sparsity and RevSparseJac is a flag that signals that the dependency pattern (instead of the sparsity pattern) is computed. `

    # include <cppad/cppad.hpp>
    namespace {
         double heavyside(const double& x)
         {     if( x <= 0.0 )
                   return 0.0;
              return 1.0;
         }
         CPPAD_DISCRETE_FUNCTION(double, heavyside)
    }

    bool dependency(void)
    {     bool ok = true;
         using CppAD::AD;
         using CppAD::NearEqual;
         typedef CPPAD_TESTVECTOR(size_t)     SizeVector;
         typedef CppAD::sparse_rc<SizeVector> sparsity;

         // VecAD object for use later
         CppAD::VecAD<double> vec_ad(2);
         vec_ad[0] = 0.0;
         vec_ad[1] = 1.0;

         // domain space vector
         size_t n  = 5;
         CPPAD_TESTVECTOR(AD<double>) ax(n);
         for(size_t j = 0; j < n; j++)
              ax[j] = AD<double>(j + 1);

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

         // some AD constants
         AD<double> azero(0.0), aone(1.0);

         // range space vector
         size_t m  = n;
         size_t m1 = n - 1;
         CPPAD_TESTVECTOR(AD<double>) ay(m);
         // Note that ay[m1 - j] depends on ax[j]
         ay[m1 - 0] = sign( ax[0] );
         ay[m1 - 1] = CondExpLe( ax[1], azero, azero, aone);
         ay[m1 - 2] = CondExpLe( azero, ax[2], azero, aone);
         ay[m1 - 3] = heavyside( ax[3] );
         ay[m1 - 4] = vec_ad[ ax[4] - AD<double>(4.0) ];

         // create f: x -> y and stop tape recording
         CppAD::ADFun<double> f(ax, ay);

         // sparsity pattern for n by n identity matrix
         size_t nr  = n;
         size_t nc  = n;
         size_t nnz = n;
         sparsity pattern_in(nr, nc, nnz);
         for(size_t k = 0; k < nnz; k++)
         {     size_t r = k;
              size_t c = k;
              pattern_in.set(k, r, c);
         }

         // compute dependency pattern
         bool transpose     = false;
         bool dependency    = true;  // would transpose dependency pattern
         bool internal_bool = true;  // does not affect result
         sparsity pattern_out;
         f.for_jac_sparsity(
              pattern_in, transpose, dependency, internal_bool, pattern_out
         );
         const SizeVector& row( pattern_out.row() );
         const SizeVector& col( pattern_out.col() );
         SizeVector col_major = pattern_out.col_major();

         // check result
         ok &= pattern_out.nr()  == n;
         ok &= pattern_out.nc()  == n;
         ok &= pattern_out.nnz() == n;
         for(size_t k = 0; k < n; k++)
         {     ok &= row[ col_major[k] ] == m1 - k;
              ok &= col[ col_major[k] ] == k;
         }
         // -----------------------------------------------------------
         // RevSparseJac and set dependency
         CppAD::vector<    std::set<size_t> > eye_set(m), depend_set(m);
         for(size_t i = 0; i < m; i++)
         {     ok &= eye_set[i].empty();
              eye_set[i].insert(i);
         }
         depend_set = f.RevSparseJac(n, eye_set, transpose, dependency);
         for(size_t i = 0; i < m; i++)
         {     std::set<size_t> check;
              check.insert(m1 - i);
              ok &= depend_set[i] == check;
         }
         dependency = false;
         depend_set = f.RevSparseJac(n, eye_set, transpose, dependency);
         for(size_t i = 0; i < m; i++)
              ok &= depend_set[i].empty();
         return ok;
    }

`

Input File: example/sparse/dependency.cpp