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lines 7-406 of file: example/atomic_three/norm_sq.cpp
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{xrst_begin atomic_three_norm_sq.cpp}

Atomic Euclidean Norm Squared: Example and Test
###############################################

Function
********
This example demonstrates using :ref:`atomic_three-name`
to define the operation
:math:`g : \B{R}^n \rightarrow \B{R}^m` where
:math:`n = 2`, :math:`m = 1`, where

.. math::

   g(x) =  x_0^2 + x_1^2

Start Class Definition
**********************
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{xrst_code cpp} */
# include <cppad/cppad.hpp>
namespace {           // isolate items below to this file
using CppAD::vector;  // abbreivate CppAD::vector as vector
//
class atomic_norm_sq : public CppAD::atomic_three<double> {
/* {xrst_code}
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Constructor
***********
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{xrst_code cpp} */
public:
   atomic_norm_sq(const std::string& name) :
   CppAD::atomic_three<double>(name)
   { }
private:
/* {xrst_code}
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for_type
********
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{xrst_code cpp} */
   // calculate type_y
   bool for_type(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      vector<CppAD::ad_type_enum>&        type_y      ) override
   {  assert( parameter_x.size() == type_x.size() );
      bool ok = type_x.size() == 2; // n
      ok     &= type_y.size() == 1; // m
      if( ! ok )
         return false;
      type_y[0] = std::max(type_x[0], type_x[1]);
      return true;
   }
/* {xrst_code}
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forward
*******
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{xrst_code cpp} */
   // forward mode routine called by CppAD
   bool forward(
      const vector<double>&              parameter_x ,
      const vector<CppAD::ad_type_enum>& type_x      ,
      size_t                             need_y      ,
      size_t                             p           ,
      size_t                             q           ,
      const vector<double>&              tx          ,
      vector<double>&                    ty          ) override
   {
# ifndef NDEBUG
      size_t n = tx.size() / (q+1);
      size_t m = ty.size() / (q+1);
# endif
      assert( type_x.size() == n );
      assert( n == 2 );
      assert( m == 1 );
      assert( p <= q );

      // return flag
      bool ok = q <= 1;

      // Order zero forward mode must always be implemented.
      // y^0 = g( x^0 )
      double x_00 = tx[ 0*(q+1) + 0];        // x_0^0
      double x_10 = tx[ 1*(q+1) + 0];        // x_10
      double g = x_00 * x_00 + x_10 * x_10;  // g( x^0 )
      if( p <= 0 )
         ty[0] = g;   // y_0^0
      if( q <= 0 )
         return ok;

      // Order one forward mode.
      // This case needed if first order forward mode is used.
      // y^1 = g'( x^0 ) x^1
      double x_01 = tx[ 0*(q+1) + 1];   // x_0^1
      double x_11 = tx[ 1*(q+1) + 1];   // x_1^1
      double gp_0 = 2.0 * x_00;         // partial f w.r.t x_0^0
      double gp_1 = 2.0 * x_10;         // partial f w.r.t x_1^0
      if( p <= 1 )
         ty[1] = gp_0 * x_01 + gp_1 * x_11; // g'( x^0 ) * x^1
      if( q <= 1 )
         return ok;

      // Assume we are not using forward mode with order > 1
      assert( ! ok );
      return ok;
   }
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reverse
*******
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{xrst_code cpp} */
   // reverse mode routine called by CppAD
   bool reverse(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      size_t                              q           ,
      const vector<double>&               tx          ,
      const vector<double>&               ty          ,
      vector<double>&                     px          ,
      const vector<double>&               py          ) override
   {
# ifndef NDEBUG
      size_t n = tx.size() / (q+1);
      size_t m = ty.size() / (q+1);
# endif
      assert( px.size() == tx.size() );
      assert( py.size() == ty.size() );
      assert( n == 2 );
      assert( m == 1 );
      bool ok = q <= 1;

      double gp_0, gp_1;
      switch(q)
      {  case 0:
         // This case needed if first order reverse mode is used
         // F ( {x} ) = g( x^0 ) = y^0
         gp_0  =  2.0 * tx[0];  // partial F w.r.t. x_0^0
         gp_1  =  2.0 * tx[1];  // partial F w.r.t. x_0^1
         px[0] = py[0] * gp_0;; // partial G w.r.t. x_0^0
         px[1] = py[0] * gp_1;; // partial G w.r.t. x_0^1
         assert(ok);
         break;

         default:
         // Assume we are not using reverse with order > 1 (q > 0)
         assert(!ok);
      }
      return ok;
   }
/* {xrst_code}
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jac_sparsity
************
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{xrst_code cpp} */
   // Jacobian sparsity routine called by CppAD
   bool jac_sparsity(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      bool                                dependency  ,
      const vector<bool>&                 select_x    ,
      const vector<bool>&                 select_y    ,
      CppAD::sparse_rc< vector<size_t> >& pattern_out ) override
   {  size_t n = select_x.size();
      size_t m = select_y.size();
      assert( n == 2 );
      assert( m == 1 );
      assert( parameter_x.size() == select_x.size() );
      //
      // count number non-zeros
      size_t nnz = 0;
      if( select_y[0] )
      {  if( select_x[0] )
            ++nnz;
         if( select_x[1] )
            ++nnz;
      }
      // sparsity pattern
      pattern_out.resize(m, n, nnz);
      size_t k = 0;
      if( select_y[0] )
      {  if( select_x[0] )
            pattern_out.set(k++, 0, 0);
         if( select_x[1] )
            pattern_out.set(k++, 0, 1);
      }
      return true;
   }
/* {xrst_code}
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hes_sparsity
************
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{xrst_code cpp} */
   // Hessian sparsity routine called by CppAD
   bool hes_sparsity(
      const vector<double>&               parameter_x ,
      const vector<CppAD::ad_type_enum>&  type_x      ,
      const vector<bool>&                 select_x    ,
      const vector<bool>&                 select_y    ,
      CppAD::sparse_rc< vector<size_t> >& pattern_out ) override
   {  size_t n = select_x.size();
      assert( n == 2 );
      assert( select_y.size() == 1 ); // m
      assert( parameter_x.size() == select_x.size() );
      //
      // count number non-zeros
      size_t nnz = 0;
      if( select_y[0] )
      {  if( select_x[0] )
            ++nnz;
         if( select_x[1] )
            ++nnz;
      }
      // sparsity pattern
      pattern_out.resize(n, n, nnz);
      size_t k = 0;
      if( select_y[0] )
      {  if( select_x[0] )
            pattern_out.set(k++, 0, 0);
         if( select_x[1] )
            pattern_out.set(k++, 1, 1);
      }
      return true;
   }
/* {xrst_code}
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End Class Definition
********************
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{xrst_code cpp} */
}; // End of atomic_norm_sq class
}  // End empty namespace

/* {xrst_code}
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Use Atomic Function
*******************
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{xrst_code cpp} */
bool norm_sq(void)
{  bool ok = true;
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps = 10. * CppAD::numeric_limits<double>::epsilon();
/* {xrst_code}
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Constructor
===========
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{xrst_code cpp} */
   // --------------------------------------------------------------------
   // Create the atomic reciprocal object
   atomic_norm_sq afun("atomic_norm_sq");
/* {xrst_code}
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Recording
=========
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{xrst_code cpp} */
   // Create the function f(x) = g(x)
   //
   // domain space vector
   size_t  n  = 2;
   double  x0 = 0.25;
   double  x1 = 0.75;
   vector< AD<double> > ax(n);
   ax[0]      = x0;
   ax[1]      = x1;

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

   // range space vector
   size_t m = 1;
   vector< AD<double> > ay(m);

   // call atomic function and store norm_sq(x) in au[0]
   afun(ax, ay);        // y_0 = x_0 * x_0 + x_1 * x_1

   // create g: x -> y and stop tape recording
   CppAD::ADFun<double> f;
   f.Dependent (ax, ay);
/* {xrst_code}
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forward
=======
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{xrst_code cpp} */
   // check function value
   double check = x0 * x0 + x1 * x1;
   ok &= NearEqual( Value(ay[0]) , check,  eps, eps);

   // check zero order forward mode
   size_t q;
   vector<double> x_q(n), y_q(m);
   q      = 0;
   x_q[0] = x0;
   x_q[1] = x1;
   y_q    = f.Forward(q, x_q);
   ok &= NearEqual(y_q[0] , check,  eps, eps);

   // check first order forward mode
   q      = 1;
   x_q[0] = 0.3;
   x_q[1] = 0.7;
   y_q    = f.Forward(q, x_q);
   check  = 2.0 * x0 * x_q[0] + 2.0 * x1 * x_q[1];
   ok &= NearEqual(y_q[0] , check,  eps, eps);

/* {xrst_code}
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reverse
=======
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{xrst_code cpp} */
   // first order reverse mode
   q     = 1;
   vector<double> w(m), dw(n * q);
   w[0]  = 1.;
   dw    = f.Reverse(q, w);
   check = 2.0 * x0;
   ok &= NearEqual(dw[0] , check,  eps, eps);
   check = 2.0 * x1;
   ok &= NearEqual(dw[1] , check,  eps, eps);
/* {xrst_code}
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rev_jac_sparsity
================
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{xrst_code cpp} */
   // reverse mode Jacobian sparstiy pattern
   CppAD::sparse_rc< CPPAD_TESTVECTOR(size_t) > pattern_in, pattern_out;
   pattern_in.resize(m, m, m);
   for(size_t i = 0; i < m; ++i)
      pattern_in.set(i, i, i);
   bool transpose     = false;
   bool dependency    = false;
   bool internal_bool = false;
   f.rev_jac_sparsity(
      pattern_in, transpose, dependency, internal_bool, pattern_out
   );
   CPPAD_TESTVECTOR(size_t) row_major  = pattern_out.row_major();
   //
   // first element in row major order is (0, 0)
   size_t k = 0;
   size_t r = pattern_out.row()[ row_major[k] ];
   size_t c = pattern_out.col()[ row_major[k] ];
   ok      &= r == 0 && c == 0;
   //
   // second element in row major order is (0, 1)
   ++k;
   r        = pattern_out.row()[ row_major[k] ];
   c        = pattern_out.col()[ row_major[k] ];
   ok      &= r == 0 && c == 1;
   //
   // k + 1 should be number of values in sparsity pattern
   ok      &= k + 1 == pattern_out.nnz();
/* {xrst_code}
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for_hes_sparsity
================
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{xrst_code cpp} */
   // forward mode Hessian sparsity pattern
   CPPAD_TESTVECTOR(bool) select_x(n), select_y(m);
   for(size_t j = 0; j < n; ++j)
      select_x[j] = true;
   for(size_t i = 0; i < m; ++i)
      select_y[i] = true;
   f.for_hes_sparsity(
      select_x, select_y, internal_bool, pattern_out
   );
   CPPAD_TESTVECTOR(size_t) order  = pattern_out.row_major();
   //
   // first element in row major order is (0, 0)
   k   = 0;
   r   = pattern_out.row()[ order[k] ];
   c   = pattern_out.col()[ order[k] ];
   ok &= r == 0 && c == 0;
   //
   // second element in row major order is (1, 1)
   ++k;
   r   = pattern_out.row()[ order[k] ];
   c   = pattern_out.col()[ order[k] ];
   ok &= r == 1 && c == 1;
   //
   // k + 1 should be number of values in sparsity pattern
   ok &= k + 1 == pattern_out.nnz();
   //
   return ok;
}
/* {xrst_code}
{xrst_spell_on}

{xrst_end atomic_three_norm_sq.cpp}