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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}} }@)@
Reverse Mode Hessian Sparsity: Example and Test

# include <cppad/cppad.hpp>
namespace { // -------------------------------------------------------------

// expected sparsity pattern
bool check_f0[] = {
     false, false, false,  // partials w.r.t x0 and (x0, x1, x2)
     false, false, false,  // partials w.r.t x1 and (x0, x1, x2)
     false, false, true    // partials w.r.t x2 and (x0, x1, x2)
};
bool check_f1[] = {
     false,  true, false,  // partials w.r.t x0 and (x0, x1, x2)
     true,  false, false,  // partials w.r.t x1 and (x0, x1, x2)
     false, false, false   // partials w.r.t x2 and (x0, x1, x2)
};

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

     // domain space vector
     size_t n = 3;
     CPPAD_TESTVECTOR(AD<double>) ax(n);
     ax[0] = 0.;
     ax[1] = 1.;
     ax[2] = 2.;

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

     // range space vector
     size_t m = 2;
     CPPAD_TESTVECTOR(AD<double>) ay(m);
     ay[0] = sin( ax[2] );
     ay[1] = ax[0] * ax[1];

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

     // 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);
     }

     // compute sparsity pattern for J(x) = F^{(1)} (x)
     f.ForSparseJac(n, r);

     // compute sparsity pattern for H(x) = F_0^{(2)} (x)
     Vector s(m);
     for(i = 0; i < m; i++)
          s[i] = false;
     s[0] = true;
     Vector h(n * n);
     h    = f.RevSparseHes(n, s);

     // check values
     for(i = 0; i < n; i++)
          for(j = 0; j < n; j++)
               ok &= (h[ i * n + j ] == check_f0[ i * n + j ] );

     // compute sparsity pattern for H(x) = F_1^{(2)} (x)
     for(i = 0; i < m; i++)
          s[i] = false;
     s[1] = true;
     h    = f.RevSparseHes(n, s);

     // check values
     for(i = 0; i < n; i++)
          for(j = 0; j < n; j++)
               ok &= (h[ i * n + j ] == check_f1[ i * n + j ] );

     // call that transposed the result
     bool transpose = true;
     h    = f.RevSparseHes(n, s, transpose);

     // This h is symmetric, because R is symmetric, not really testing here
     for(i = 0; i < n; i++)
          for(j = 0; j < n; j++)
               ok &= (h[ j * n + i ] == check_f1[ i * n + j ] );

     return ok;
}
// define the template function SetCases<Vector> in empty namespace
template <typename 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 = 3;
     CPPAD_TESTVECTOR(AD<double>) ax(n);
     ax[0] = 0.;
     ax[1] = 1.;
     ax[2] = 2.;

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

     // range space vector
     size_t m = 2;
     CPPAD_TESTVECTOR(AD<double>) ay(m);
     ay[0] = sin( ax[2] );
     ay[1] = ax[0] * ax[1];

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

     // 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);
     }

     // compute sparsity pattern for J(x) = F^{(1)} (x)
     f.ForSparseJac(n, r);

     // compute sparsity pattern for H(x) = F_0^{(2)} (x)
     Vector s(1);
     assert( s[0].empty() );
     s[0].insert(0);
     Vector h(n);
     h    = f.RevSparseHes(n, s);

     // check values
     std::set<size_t>::iterator itr;
     size_t j;
     for(i = 0; i < n; i++)
     {     for(j = 0; j < n; j++)
          {     bool found = h[i].find(j) != h[i].end();
               ok        &= (found == check_f0[i * n + j]);
          }
     }

     // compute sparsity pattern for H(x) = F_1^{(2)} (x)
     s[0].clear();
     assert( s[0].empty() );
     s[0].insert(1);
     h    = f.RevSparseHes(n, s);

     // check values
     for(i = 0; i < n; i++)
     {     for(j = 0; j < n; j++)
          {     bool found = h[i].find(j) != h[i].end();
               ok        &= (found == check_f1[i * n + j]);
          }
     }

     // call that transposed the result
     bool transpose = true;
     h    = f.RevSparseHes(n, s, transpose);

     // This h is symmetric, because R is symmetric, not really testing here
     for(i = 0; i < n; i++)
     {     for(j = 0; j < n; j++)
          {     bool found = h[j].find(i) != h[j].end();
               ok        &= (found == check_f1[i * n + j]);
          }
     }

     return ok;
}
} // End empty namespace

# include <vector>
# include <valarray>
bool rev_sparse_hes(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::vector  <bool> >();
     ok &= BoolCases< CppAD::vectorBool     >();
     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/rev_sparse_hes.cpp