| /* |
| [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 |