// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-24 Bradley M. Bell
// ----------------------------------------------------------------------------

/*
{xrst_begin chkpoint_two_ode.cpp}
{xrst_spell
  kutta
  runge
}

Checkpointing an ODE Solver: Example and Test
#############################################

Purpose
*******
In this example we :ref:`checkpoint<chkpoint_two-name>` one step of an ODE solver.

Problem
*******
We consider the initial value problem with parameter :math:`x` defined by,
:math:`z(0, x) = z_0 (x)`,

.. math::

   \partial_t z(t, x ) = h [ x , z(t, x) ]

Note that if :math:`t` needs to be in the equation, one can define
the first component of :math:`z(t, x)` to be equal to :math:`t`.

ODE Solver
**********
For this example, we consider the Fourth order Runge-Kutta ODE solver.
Given an approximation solution at time :math:`t_k` denoted by
:math:`\tilde{z}_k (x)`, and :math:`\Delta t = t_{k+1} - t_k`,
it defines the approximation solution :math:`\tilde{z}_{k+1} (x)`
at time :math:`t_{k+1}` by

.. math::
   :nowrap:

   \begin{eqnarray}
   h_1 & =  & h [ x , \tilde{z}_k (x) ]
   \\
   h_2 & =  & h [ x , \tilde{z}_k (x) + \Delta t \; h_1 / 2 ]
   \\
   h_3 & =  & h [ x , \tilde{z}_k (x) + \Delta t \; h_2 / 2 ]
   \\
   h_4 & =  & h [ x , \tilde{z}_k (x) + \Delta t \; h_3 ]
   \\
   \tilde{z}_{k+1} (x) & = &
      \tilde{z}_k (x) + \Delta t \; ( h_1 +  2 h_2 + 2 h_3 + h_4 ) / 6
   \end{eqnarray}

If :math:`\tilde{z}_k (x) = z_k (x)`,
:math:`\tilde{z}_{k+1} (x) = z_{k+1} (x) + O( \Delta t^5 )`.
Other ODE solvers can use a similar method to the one used below.

ODE
***
For this example the ODE is defined by
:math:`z(0, x) = 0` and

.. math::

   h[ x, z(t, x) ] =
   \left( \begin{array}{c}
         x_0                     \\
         x_1 z_0 (t, x)          \\
         \vdots                  \\
         x_{n-1} z_{n-2} (t, x)
   \end{array} \right)
   =
   \left( \begin{array}{c}
         \partial_t z_0 (t , x)      \\
         \partial_t z_1 (t , x)      \\
         \vdots                      \\
         \partial_t z_{n-1} (t , x)
   \end{array} \right)

Solution
********
The solution of the ODE for this example,
which is used to check the results,
can be calculated by
starting with the first row and then using the solution
for the first row to solve the second and so on.
Doing this we obtain

.. math::

   z(t, x) =
   \left( \begin{array}{c}
      x_0 t                  \\
      x_1 x_0 t^2 / 2        \\
      \vdots                 \\
      x_{n-1} x_{n-2} \ldots x_0 t^n / n !
   \end{array} \right)

Source
******
{xrst_literal
   // BEGIN C++
   // END C++
}

{xrst_end chkpoint_two_ode.cpp}
*/
// BEGIN C++

# include <cppad/cppad.hpp>

namespace {
   using CppAD::AD;
   typedef AD<double>                     a1double;
   typedef AD<a1double>                   a2double;
   //
   typedef CPPAD_TESTVECTOR(   double )   a0vector;
   typedef CPPAD_TESTVECTOR( a1double )   a1vector;
   typedef CPPAD_TESTVECTOR( a2double )   a2vector;
   //
   // set once by main and kept that way
   double delta_t_ = std::numeric_limits<double>::quiet_NaN();
   size_t n_       = 0;
   //
   // The function h( x , y)
   template <class FloatVector>
   FloatVector h(const FloatVector& x, const FloatVector& y)
   {  assert( size_t( x.size() ) == n_ );
      assert( size_t( y.size() ) == n_ );
      FloatVector result(n_);
      result[0] = x[0];
      for(size_t i = 1; i < n_; i++)
         result[i] = x[i] * y[i-1];
      return result;
   }

   // The 4-th Order Runge-Kutta Step
   template <class FloatVector>
   FloatVector Runge4(const FloatVector& x, const FloatVector& z0
   )
   {  assert( size_t( x.size() ) == n_ );
      assert( size_t( z0.size() ) == n_ );
      //
      typedef typename FloatVector::value_type Float;
      //
      Float  dt = Float(delta_t_);
      size_t m  = z0.size();
      //
      FloatVector h1(m), h2(m), h3(m), h4(m), result(m);
      h1 = h( x, z0 );
      //
      for(size_t i = 0; i < m; i++)
         h2[i] = z0[i] + dt * h1[i] / 2.0;
      h2 = h( x, h2 );
      //
      for(size_t i = 0; i < m; i++)
         h3[i] = z0[i] + dt * h2[i] / 2.0;
      h3 = h( x, h3 );
      //
      for(size_t i = 0; i < m; i++)
         h4[i] = z0[i] + dt * h3[i];
      h4 = h( x, h4 );
      //
      for(size_t i = 0; i < m; i++)
      {  Float dz = dt * ( h1[i] + 2.0*h2[i] + 2.0*h3[i] + h4[i] ) / 6.0;
         result[i] = z0[i] + dz;
      }
      return result;
   }

   // pack x and z into an axz vector
   template <class FloatVector>
   void pack(
      FloatVector&         axz      ,
      const FloatVector&   x        ,
      const FloatVector&   z        )
   {  assert( size_t( axz.size() ) == n_ + n_ );
      assert( size_t( x.size()        ) == n_      );
      assert( size_t( z.size()        ) == n_      );
      //
      size_t offset = 0;
      for(size_t i = 0; i < n_; i++)
         axz[offset + i] = x[i];
      offset += n_;
      for(size_t i = 0; i < n_; i++)
         axz[offset + i] = z[i];
   }

   // unpack an axz vector
   template <class FloatVector>
   void unpack(
      const FloatVector&         axz      ,
      FloatVector&               x        ,
      FloatVector&               z        )
   {  assert( size_t( axz.size() ) == n_ + n_ );
      assert( size_t( x.size()        ) == n_      );
      assert( size_t( z.size()        ) == n_      );
      //
      size_t offset = 0;
      for(size_t i = 0; i < n_; i++)
         x[i] = axz[offset + i];
      offset += n_;
      for(size_t i = 0; i < n_; i++)
         z[i] = axz[offset + i];
   }

   // Algorithm that z(t, x)
   void ode_algo(const a1vector& axz_in, a1vector& axz_out)
   {  assert( size_t( axz_in.size()  ) == n_ + n_ );
      assert( size_t( axz_out.size() ) == n_ + n_ );
      //
      // initial ode information
      a1vector x(n_), z0(n_);
      unpack(axz_in, x, z0);
      //
      // advance z(t, x)
      a1vector z1 = Runge4(x, z0);
      //
      // final ode information
      pack(axz_out, x, z1);
      //
      return;
   }
}
//
bool ode(void)
{  bool ok = true;
   using CppAD::NearEqual;
   double eps = std::numeric_limits<double>::epsilon();
   //
   // number of terms in the differential equation
   n_ = 6;
   //
   // step size for the differentiail equation
   size_t n_step = 10;
   double T      = 1.0;
   delta_t_ = T / double(n_step);
   //
   // set parameter value and initial value of the ode
   a1vector ax(n_), az0(n_);
   for(size_t i = 0; i < n_; i++)
   {  ax[i]  = a1double(i + 1);
      az0[i] = a1double(0);
   }
   //
   // pack ode information input vector
   //
   // function corresponding to one step of the ode Algorithm
   a1vector axz_in(2 * n_), axz_out(2 * n_);
   pack(axz_in, ax, az0);
   CppAD::Independent(axz_in);
   ode_algo(axz_in, axz_out);
   CppAD::ADFun<double> ode_fun(axz_in, axz_out);
   //
   // create checkpoint version of the algorithm
   bool internal_bool    = false;
   bool use_hes_sparsity = false;
   bool use_base2ad      = false;
   bool use_in_parallel  = false;
   CppAD::chkpoint_two<double> ode_check(ode_fun, "ode",
      internal_bool, use_hes_sparsity, use_base2ad, use_in_parallel
   );
   //
   // set the independent variables for recording
   CppAD::Independent( ax );
   //
   // repack to get dependence on ax
   pack(axz_in, ax, az0);
   //
   // Now run the checkpoint algorithm n_step times
   for(size_t k = 0; k < n_step; k++)
   {  ode_check(axz_in, axz_out);
      axz_in = axz_out;
   }
   //
   // Unpack the results (must use ax1 so do not overwrite ax)
   a1vector ax1(n_), az1(n_);
   unpack(axz_out, ax1, az1);
   //
   // We could record a complicated function of x and z(T, x) in f,
   // but make this example simpler we record x -> z(T, x).
   CppAD::ADFun<double> f(ax, az1);
   //
   // check function values
   a0vector x(n_), z1(n_);
   for(size_t j = 0; j < n_; j++)
      x[j] = double(j + 1);
   z1 = f.Forward(0, x);
   //
   // separate calculation of z(t, x)
   a0vector check_z1(n_);
   check_z1[0] = x[0] * T;
   for(size_t i = 1; i < n_; i++)
      check_z1[i] = x[i] * T * check_z1[i-1] / double(i+1);
   //
   // expected accuracy for each component of of z(t, x)
   a0vector acc(n_);
   for(size_t i = 0; i < n_; i++)
   {  if( i < 4 )
      {  // Runge-Kutta methos is exact for this case
         acc[i] = 10. * eps;
      }
      else
      {  acc[i] = 1.0;
         for(size_t k = 0; k < 5; k++)
               acc[i] *= x[k] * delta_t_;
      }
   }
   // check z1(T, x)
   for(size_t i = 0; i < n_; i++)
      ok &= NearEqual(z1[i] , check_z1[i], acc[i], acc[i]);
   //
   // Now use f to compute a derivative. For this 'simple' example it is
   // the derivative of z_{n-1} (T, x) respect to x of the
   a0vector w(n_), dw(n_);
   for(size_t i = 0; i < n_; i++)
   {  w[i] = 0.0;
      if( i == n_ - 1 )
         w[i] = 1.0;
   }
   dw = f.Reverse(1, w);
   for(size_t j = 0; j < n_; j++)
   {  double check = z1[n_ - 1] / x[j];
      ok &= NearEqual(dw[j] , check, 100.*eps, 100.*eps);
   }
   //
   return ok;
}
// END C++