boost/numeric/odeint/stepper/runge_kutta_dopri5.hpp
/* [auto_generated] boost/numeric/odeint/stepper/runge_kutta_dopri5.hpp [begin_description] Implementation of the Dormand-Prince 5(4) method. This stepper can also be used with the dense-output controlled stepper. [end_description] Copyright 2010-2013 Karsten Ahnert Copyright 2010-2013 Mario Mulansky Copyright 2012 Christoph Koke Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) */ #ifndef BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED #define BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED #include <boost/numeric/odeint/util/bind.hpp> #include <boost/numeric/odeint/stepper/base/explicit_error_stepper_fsal_base.hpp> #include <boost/numeric/odeint/algebra/range_algebra.hpp> #include <boost/numeric/odeint/algebra/default_operations.hpp> #include <boost/numeric/odeint/algebra/algebra_dispatcher.hpp> #include <boost/numeric/odeint/algebra/operations_dispatcher.hpp> #include <boost/numeric/odeint/stepper/stepper_categories.hpp> #include <boost/numeric/odeint/util/state_wrapper.hpp> #include <boost/numeric/odeint/util/is_resizeable.hpp> #include <boost/numeric/odeint/util/resizer.hpp> #include <boost/numeric/odeint/util/same_instance.hpp> namespace boost { namespace numeric { namespace odeint { template< class State , class Value = double , class Deriv = State , class Time = Value , class Algebra = typename algebra_dispatcher< State >::algebra_type , class Operations = typename operations_dispatcher< State >::operations_type , class Resizer = initially_resizer > class runge_kutta_dopri5 #ifndef DOXYGEN_SKIP : public explicit_error_stepper_fsal_base< runge_kutta_dopri5< State , Value , Deriv , Time , Algebra , Operations , Resizer > , 5 , 5 , 4 , State , Value , Deriv , Time , Algebra , Operations , Resizer > #else : public explicit_error_stepper_fsal_base #endif { public : #ifndef DOXYGEN_SKIP typedef explicit_error_stepper_fsal_base< runge_kutta_dopri5< State , Value , Deriv , Time , Algebra , Operations , Resizer > , 5 , 5 , 4 , State , Value , Deriv , Time , Algebra , Operations , Resizer > stepper_base_type; #else typedef explicit_error_stepper_fsal_base< runge_kutta_dopri5< ... > , ... > stepper_base_type; #endif typedef typename stepper_base_type::state_type state_type; typedef typename stepper_base_type::value_type value_type; typedef typename stepper_base_type::deriv_type deriv_type; typedef typename stepper_base_type::time_type time_type; typedef typename stepper_base_type::algebra_type algebra_type; typedef typename stepper_base_type::operations_type operations_type; typedef typename stepper_base_type::resizer_type resizer_type; #ifndef DOXYGEN_SKIP typedef typename stepper_base_type::stepper_type stepper_type; typedef typename stepper_base_type::wrapped_state_type wrapped_state_type; typedef typename stepper_base_type::wrapped_deriv_type wrapped_deriv_type; #endif // DOXYGEN_SKIP runge_kutta_dopri5( const algebra_type &algebra = algebra_type() ) : stepper_base_type( algebra ) { } template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut > void do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , StateOut &out , DerivOut &dxdt_out , time_type dt ) { const value_type a2 = static_cast<value_type> ( 1 ) / static_cast<value_type>( 5 ); const value_type a3 = static_cast<value_type> ( 3 ) / static_cast<value_type> ( 10 ); const value_type a4 = static_cast<value_type> ( 4 ) / static_cast<value_type> ( 5 ); const value_type a5 = static_cast<value_type> ( 8 )/static_cast<value_type> ( 9 ); const value_type b21 = static_cast<value_type> ( 1 ) / static_cast<value_type> ( 5 ); const value_type b31 = static_cast<value_type> ( 3 ) / static_cast<value_type>( 40 ); const value_type b32 = static_cast<value_type> ( 9 ) / static_cast<value_type>( 40 ); const value_type b41 = static_cast<value_type> ( 44 ) / static_cast<value_type> ( 45 ); const value_type b42 = static_cast<value_type> ( -56 ) / static_cast<value_type> ( 15 ); const value_type b43 = static_cast<value_type> ( 32 ) / static_cast<value_type> ( 9 ); const value_type b51 = static_cast<value_type> ( 19372 ) / static_cast<value_type>( 6561 ); const value_type b52 = static_cast<value_type> ( -25360 ) / static_cast<value_type> ( 2187 ); const value_type b53 = static_cast<value_type> ( 64448 ) / static_cast<value_type>( 6561 ); const value_type b54 = static_cast<value_type> ( -212 ) / static_cast<value_type>( 729 ); const value_type b61 = static_cast<value_type> ( 9017 ) / static_cast<value_type>( 3168 ); const value_type b62 = static_cast<value_type> ( -355 ) / static_cast<value_type>( 33 ); const value_type b63 = static_cast<value_type> ( 46732 ) / static_cast<value_type>( 5247 ); const value_type b64 = static_cast<value_type> ( 49 ) / static_cast<value_type>( 176 ); const value_type b65 = static_cast<value_type> ( -5103 ) / static_cast<value_type>( 18656 ); const value_type c1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); const value_type c3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); const value_type c4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); const value_type c5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); const value_type c6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); typename odeint::unwrap_reference< System >::type &sys = system; m_k_x_tmp_resizer.adjust_size( in , detail::bind( &stepper_type::template resize_k_x_tmp_impl<StateIn> , detail::ref( *this ) , detail::_1 ) ); //m_x_tmp = x + dt*b21*dxdt stepper_base_type::m_algebra.for_each3( m_x_tmp.m_v , in , dxdt_in , typename operations_type::template scale_sum2< value_type , time_type >( 1.0 , dt*b21 ) ); sys( m_x_tmp.m_v , m_k2.m_v , t + dt*a2 ); // m_x_tmp = x + dt*b31*dxdt + dt*b32*m_k2 stepper_base_type::m_algebra.for_each4( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , typename operations_type::template scale_sum3< value_type , time_type , time_type >( 1.0 , dt*b31 , dt*b32 )); sys( m_x_tmp.m_v , m_k3.m_v , t + dt*a3 ); // m_x_tmp = x + dt * (b41*dxdt + b42*m_k2 + b43*m_k3) stepper_base_type::m_algebra.for_each5( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , m_k3.m_v , typename operations_type::template scale_sum4< value_type , time_type , time_type , time_type >( 1.0 , dt*b41 , dt*b42 , dt*b43 )); sys( m_x_tmp.m_v, m_k4.m_v , t + dt*a4 ); stepper_base_type::m_algebra.for_each6( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , m_k3.m_v , m_k4.m_v , typename operations_type::template scale_sum5< value_type , time_type , time_type , time_type , time_type >( 1.0 , dt*b51 , dt*b52 , dt*b53 , dt*b54 )); sys( m_x_tmp.m_v , m_k5.m_v , t + dt*a5 ); stepper_base_type::m_algebra.for_each7( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , m_k3.m_v , m_k4.m_v , m_k5.m_v , typename operations_type::template scale_sum6< value_type , time_type , time_type , time_type , time_type , time_type >( 1.0 , dt*b61 , dt*b62 , dt*b63 , dt*b64 , dt*b65 )); sys( m_x_tmp.m_v , m_k6.m_v , t + dt ); stepper_base_type::m_algebra.for_each7( out , in , dxdt_in , m_k3.m_v , m_k4.m_v , m_k5.m_v , m_k6.m_v , typename operations_type::template scale_sum6< value_type , time_type , time_type , time_type , time_type , time_type >( 1.0 , dt*c1 , dt*c3 , dt*c4 , dt*c5 , dt*c6 )); // the new derivative sys( out , dxdt_out , t + dt ); } template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut , class Err > void do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , StateOut &out , DerivOut &dxdt_out , time_type dt , Err &xerr ) { const value_type c1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); const value_type c3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); const value_type c4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); const value_type c5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); const value_type c6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); const value_type dc1 = c1 - static_cast<value_type> ( 5179 ) / static_cast<value_type>( 57600 ); const value_type dc3 = c3 - static_cast<value_type> ( 7571 ) / static_cast<value_type>( 16695 ); const value_type dc4 = c4 - static_cast<value_type> ( 393 ) / static_cast<value_type>( 640 ); const value_type dc5 = c5 - static_cast<value_type> ( -92097 ) / static_cast<value_type>( 339200 ); const value_type dc6 = c6 - static_cast<value_type> ( 187 ) / static_cast<value_type>( 2100 ); const value_type dc7 = static_cast<value_type>( -1 ) / static_cast<value_type> ( 40 ); /* ToDo: copy only if &dxdt_in == &dxdt_out ? */ if( same_instance( dxdt_in , dxdt_out ) ) { m_dxdt_tmp_resizer.adjust_size( in , detail::bind( &stepper_type::template resize_dxdt_tmp_impl<StateIn> , detail::ref( *this ) , detail::_1 ) ); boost::numeric::odeint::copy( dxdt_in , m_dxdt_tmp.m_v ); do_step_impl( system , in , dxdt_in , t , out , dxdt_out , dt ); //error estimate stepper_base_type::m_algebra.for_each7( xerr , m_dxdt_tmp.m_v , m_k3.m_v , m_k4.m_v , m_k5.m_v , m_k6.m_v , dxdt_out , typename operations_type::template scale_sum6< time_type , time_type , time_type , time_type , time_type , time_type >( dt*dc1 , dt*dc3 , dt*dc4 , dt*dc5 , dt*dc6 , dt*dc7 ) ); } else { do_step_impl( system , in , dxdt_in , t , out , dxdt_out , dt ); //error estimate stepper_base_type::m_algebra.for_each7( xerr , dxdt_in , m_k3.m_v , m_k4.m_v , m_k5.m_v , m_k6.m_v , dxdt_out , typename operations_type::template scale_sum6< time_type , time_type , time_type , time_type , time_type , time_type >( dt*dc1 , dt*dc3 , dt*dc4 , dt*dc5 , dt*dc6 , dt*dc7 ) ); } } /* * Calculates Dense-Output for Dopri5 * * See Hairer, Norsett, Wanner: Solving Ordinary Differential Equations, Nonstiff Problems. I, p.191/192 * * y(t+theta) = y(t) + h * sum_i^7 b_i(theta) * k_i * * A = theta^2 * ( 3 - 2 theta ) * B = theta^2 * ( theta - 1 ) * C = theta^2 * ( theta - 1 )^2 * D = theta * ( theta - 1 )^2 * * b_1( theta ) = A * b_1 - C * X1( theta ) + D * b_2( theta ) = 0 * b_3( theta ) = A * b_3 + C * X3( theta ) * b_4( theta ) = A * b_4 - C * X4( theta ) * b_5( theta ) = A * b_5 + C * X5( theta ) * b_6( theta ) = A * b_6 - C * X6( theta ) * b_7( theta ) = B + C * X7( theta ) * * An alternative Method is described in: * * www-m2.ma.tum.de/homepages/simeon/numerik3/kap3.ps */ template< class StateOut , class StateIn1 , class DerivIn1 , class StateIn2 , class DerivIn2 > void calc_state( time_type t , StateOut &x , const StateIn1 &x_old , const DerivIn1 &deriv_old , time_type t_old , const StateIn2 & /* x_new */ , const DerivIn2 &deriv_new , time_type t_new ) const { const value_type b1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); const value_type b3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); const value_type b4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); const value_type b5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); const value_type b6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); const time_type dt = ( t_new - t_old ); const value_type theta = ( t - t_old ) / dt; const value_type X1 = static_cast< value_type >( 5 ) * ( static_cast< value_type >( 2558722523LL ) - static_cast< value_type >( 31403016 ) * theta ) / static_cast< value_type >( 11282082432LL ); const value_type X3 = static_cast< value_type >( 100 ) * ( static_cast< value_type >( 882725551 ) - static_cast< value_type >( 15701508 ) * theta ) / static_cast< value_type >( 32700410799LL ); const value_type X4 = static_cast< value_type >( 25 ) * ( static_cast< value_type >( 443332067 ) - static_cast< value_type >( 31403016 ) * theta ) / static_cast< value_type >( 1880347072LL ) ; const value_type X5 = static_cast< value_type >( 32805 ) * ( static_cast< value_type >( 23143187 ) - static_cast< value_type >( 3489224 ) * theta ) / static_cast< value_type >( 199316789632LL ); const value_type X6 = static_cast< value_type >( 55 ) * ( static_cast< value_type >( 29972135 ) - static_cast< value_type >( 7076736 ) * theta ) / static_cast< value_type >( 822651844 ); const value_type X7 = static_cast< value_type >( 10 ) * ( static_cast< value_type >( 7414447 ) - static_cast< value_type >( 829305 ) * theta ) / static_cast< value_type >( 29380423 ); const value_type theta_m_1 = theta - static_cast< value_type >( 1 ); const value_type theta_sq = theta * theta; const value_type A = theta_sq * ( static_cast< value_type >( 3 ) - static_cast< value_type >( 2 ) * theta ); const value_type B = theta_sq * theta_m_1; const value_type C = theta_sq * theta_m_1 * theta_m_1; const value_type D = theta * theta_m_1 * theta_m_1; const value_type b1_theta = A * b1 - C * X1 + D; const value_type b3_theta = A * b3 + C * X3; const value_type b4_theta = A * b4 - C * X4; const value_type b5_theta = A * b5 + C * X5; const value_type b6_theta = A * b6 - C * X6; const value_type b7_theta = B + C * X7; // const state_type &k1 = *m_old_deriv; // const state_type &k3 = dopri5().m_k3; // const state_type &k4 = dopri5().m_k4; // const state_type &k5 = dopri5().m_k5; // const state_type &k6 = dopri5().m_k6; // const state_type &k7 = *m_current_deriv; stepper_base_type::m_algebra.for_each8( x , x_old , deriv_old , m_k3.m_v , m_k4.m_v , m_k5.m_v , m_k6.m_v , deriv_new , typename operations_type::template scale_sum7< value_type , time_type , time_type , time_type , time_type , time_type , time_type >( 1.0 , dt * b1_theta , dt * b3_theta , dt * b4_theta , dt * b5_theta , dt * b6_theta , dt * b7_theta ) ); } template< class StateIn > void adjust_size( const StateIn &x ) { resize_k_x_tmp_impl( x ); resize_dxdt_tmp_impl( x ); stepper_base_type::adjust_size( x ); } private: template< class StateIn > bool resize_k_x_tmp_impl( const StateIn &x ) { bool resized = false; resized |= adjust_size_by_resizeability( m_x_tmp , x , typename is_resizeable<state_type>::type() ); resized |= adjust_size_by_resizeability( m_k2 , x , typename is_resizeable<deriv_type>::type() ); resized |= adjust_size_by_resizeability( m_k3 , x , typename is_resizeable<deriv_type>::type() ); resized |= adjust_size_by_resizeability( m_k4 , x , typename is_resizeable<deriv_type>::type() ); resized |= adjust_size_by_resizeability( m_k5 , x , typename is_resizeable<deriv_type>::type() ); resized |= adjust_size_by_resizeability( m_k6 , x , typename is_resizeable<deriv_type>::type() ); return resized; } template< class StateIn > bool resize_dxdt_tmp_impl( const StateIn &x ) { return adjust_size_by_resizeability( m_dxdt_tmp , x , typename is_resizeable<deriv_type>::type() ); } wrapped_state_type m_x_tmp; wrapped_deriv_type m_k2 , m_k3 , m_k4 , m_k5 , m_k6 ; wrapped_deriv_type m_dxdt_tmp; resizer_type m_k_x_tmp_resizer; resizer_type m_dxdt_tmp_resizer; }; /************* DOXYGEN ************/ /** * \class runge_kutta_dopri5 * \brief The Runge-Kutta Dormand-Prince 5 method. * * The Runge-Kutta Dormand-Prince 5 method is a very popular method for solving ODEs, see * <a href=""></a>. * The method is explicit and fulfills the Error Stepper concept. Step size control * is provided but continuous output is available which make this method favourable for many applications. * * This class derives from explicit_error_stepper_fsal_base and inherits its interface via CRTP (current recurring * template pattern). The method possesses the FSAL (first-same-as-last) property. See * explicit_error_stepper_fsal_base for more details. * * \tparam State The state type. * \tparam Value The value type. * \tparam Deriv The type representing the time derivative of the state. * \tparam Time The time representing the independent variable - the time. * \tparam Algebra The algebra type. * \tparam Operations The operations type. * \tparam Resizer The resizer policy type. */ /** * \fn runge_kutta_dopri5::runge_kutta_dopri5( const algebra_type &algebra ) * \brief Constructs the runge_kutta_dopri5 class. This constructor can be used as a default * constructor if the algebra has a default constructor. * \param algebra A copy of algebra is made and stored inside explicit_stepper_base. */ /** * \fn runge_kutta_dopri5::do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , StateOut &out , DerivOut &dxdt_out , time_type dt ) * \brief This method performs one step. The derivative `dxdt_in` of `in` at the time `t` is passed to the * method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Furthermore, * the derivative is update out-of-place, hence the input is assumed to be in `dxdt_in` and the output in * `dxdt_out`. * Access to this step functionality is provided by explicit_error_stepper_fsal_base and * `do_step_impl` should not be called directly. * * \param system The system function to solve, hence the r.h.s. of the ODE. It must fulfill the * Simple System concept. * \param in The state of the ODE which should be solved. in is not modified in this method * \param dxdt_in The derivative of x at t. dxdt_in is not modified by this method * \param t The value of the time, at which the step should be performed. * \param out The result of the step is written in out. * \param dxdt_out The result of the new derivative at time t+dt. * \param dt The step size. */ /** * \fn runge_kutta_dopri5::do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , StateOut &out , DerivOut &dxdt_out , time_type dt , Err &xerr ) * \brief This method performs one step. The derivative `dxdt_in` of `in` at the time `t` is passed to the * method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Furthermore, * the derivative is update out-of-place, hence the input is assumed to be in `dxdt_in` and the output in * `dxdt_out`. * Access to this step functionality is provided by explicit_error_stepper_fsal_base and * `do_step_impl` should not be called directly. * An estimation of the error is calculated. * * \param system The system function to solve, hence the r.h.s. of the ODE. It must fulfill the * Simple System concept. * \param in The state of the ODE which should be solved. in is not modified in this method * \param dxdt_in The derivative of x at t. dxdt_in is not modified by this method * \param t The value of the time, at which the step should be performed. * \param out The result of the step is written in out. * \param dxdt_out The result of the new derivative at time t+dt. * \param dt The step size. * \param xerr An estimation of the error. */ /** * \fn runge_kutta_dopri5::calc_state( time_type t , StateOut &x , const StateIn1 &x_old , const DerivIn1 &deriv_old , time_type t_old , const StateIn2 & , const DerivIn2 &deriv_new , time_type t_new ) const * \brief This method is used for continuous output and it calculates the state `x` at a time `t` from the * knowledge of two states `old_state` and `current_state` at time points `t_old` and `t_new`. It also uses * internal variables to calculate the result. Hence this method must be called after two successful `do_step` * calls. */ /** * \fn runge_kutta_dopri5::adjust_size( const StateIn &x ) * \brief Adjust the size of all temporaries in the stepper manually. * \param x A state from which the size of the temporaries to be resized is deduced. */ } // odeint } // numeric } // boost #endif // BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED