1 /* 2 [auto_generated] 3 boost/numeric/odeint/stepper/runge_kutta_dopri5.hpp 4 5 [begin_description] 6 Implementation of the Dormand-Prince 5(4) method. This stepper can also be used with the dense-output controlled stepper. 7 [end_description] 8 9 Copyright 2010-2013 Karsten Ahnert 10 Copyright 2010-2013 Mario Mulansky 11 Copyright 2012 Christoph Koke 12 13 Distributed under the Boost Software License, Version 1.0. 14 (See accompanying file LICENSE_1_0.txt or 15 copy at http://www.boost.org/LICENSE_1_0.txt) 16 */ 17 18 19 #ifndef BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED 20 #define BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED 21 22 23 #include <boost/numeric/odeint/util/bind.hpp> 24 25 #include <boost/numeric/odeint/stepper/base/explicit_error_stepper_fsal_base.hpp> 26 #include <boost/numeric/odeint/algebra/range_algebra.hpp> 27 #include <boost/numeric/odeint/algebra/default_operations.hpp> 28 #include <boost/numeric/odeint/algebra/algebra_dispatcher.hpp> 29 #include <boost/numeric/odeint/algebra/operations_dispatcher.hpp> 30 #include <boost/numeric/odeint/stepper/stepper_categories.hpp> 31 32 #include <boost/numeric/odeint/util/state_wrapper.hpp> 33 #include <boost/numeric/odeint/util/is_resizeable.hpp> 34 #include <boost/numeric/odeint/util/resizer.hpp> 35 #include <boost/numeric/odeint/util/same_instance.hpp> 36 37 namespace boost { 38 namespace numeric { 39 namespace odeint { 40 41 42 43 template< 44 class State , 45 class Value = double , 46 class Deriv = State , 47 class Time = Value , 48 class Algebra = typename algebra_dispatcher< State >::algebra_type , 49 class Operations = typename operations_dispatcher< State >::operations_type , 50 class Resizer = initially_resizer 51 > 52 class runge_kutta_dopri5 53 #ifndef DOXYGEN_SKIP 54 : public explicit_error_stepper_fsal_base< 55 runge_kutta_dopri5< State , Value , Deriv , Time , Algebra , Operations , Resizer > , 56 5 , 5 , 4 , State , Value , Deriv , Time , Algebra , Operations , Resizer > 57 #else 58 : public explicit_error_stepper_fsal_base 59 #endif 60 { 61 62 public : 63 64 #ifndef DOXYGEN_SKIP 65 typedef explicit_error_stepper_fsal_base< 66 runge_kutta_dopri5< State , Value , Deriv , Time , Algebra , Operations , Resizer > , 67 5 , 5 , 4 , State , Value , Deriv , Time , Algebra , Operations , Resizer > stepper_base_type; 68 #else 69 typedef explicit_error_stepper_fsal_base< runge_kutta_dopri5< ... > , ... > stepper_base_type; 70 #endif 71 72 typedef typename stepper_base_type::state_type state_type; 73 typedef typename stepper_base_type::value_type value_type; 74 typedef typename stepper_base_type::deriv_type deriv_type; 75 typedef typename stepper_base_type::time_type time_type; 76 typedef typename stepper_base_type::algebra_type algebra_type; 77 typedef typename stepper_base_type::operations_type operations_type; 78 typedef typename stepper_base_type::resizer_type resizer_type; 79 80 #ifndef DOXYGEN_SKIP 81 typedef typename stepper_base_type::stepper_type stepper_type; 82 typedef typename stepper_base_type::wrapped_state_type wrapped_state_type; 83 typedef typename stepper_base_type::wrapped_deriv_type wrapped_deriv_type; 84 #endif // DOXYGEN_SKIP 85 86 runge_kutta_dopri5(const algebra_type & algebra=algebra_type ())87 runge_kutta_dopri5( const algebra_type &algebra = algebra_type() ) : stepper_base_type( algebra ) 88 { } 89 90 91 template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut > do_step_impl(System system,const StateIn & in,const DerivIn & dxdt_in,time_type t,StateOut & out,DerivOut & dxdt_out,time_type dt)92 void do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , 93 StateOut &out , DerivOut &dxdt_out , time_type dt ) 94 { 95 const value_type a2 = static_cast<value_type> ( 1 ) / static_cast<value_type>( 5 ); 96 const value_type a3 = static_cast<value_type> ( 3 ) / static_cast<value_type> ( 10 ); 97 const value_type a4 = static_cast<value_type> ( 4 ) / static_cast<value_type> ( 5 ); 98 const value_type a5 = static_cast<value_type> ( 8 )/static_cast<value_type> ( 9 ); 99 100 const value_type b21 = static_cast<value_type> ( 1 ) / static_cast<value_type> ( 5 ); 101 102 const value_type b31 = static_cast<value_type> ( 3 ) / static_cast<value_type>( 40 ); 103 const value_type b32 = static_cast<value_type> ( 9 ) / static_cast<value_type>( 40 ); 104 105 const value_type b41 = static_cast<value_type> ( 44 ) / static_cast<value_type> ( 45 ); 106 const value_type b42 = static_cast<value_type> ( -56 ) / static_cast<value_type> ( 15 ); 107 const value_type b43 = static_cast<value_type> ( 32 ) / static_cast<value_type> ( 9 ); 108 109 const value_type b51 = static_cast<value_type> ( 19372 ) / static_cast<value_type>( 6561 ); 110 const value_type b52 = static_cast<value_type> ( -25360 ) / static_cast<value_type> ( 2187 ); 111 const value_type b53 = static_cast<value_type> ( 64448 ) / static_cast<value_type>( 6561 ); 112 const value_type b54 = static_cast<value_type> ( -212 ) / static_cast<value_type>( 729 ); 113 114 const value_type b61 = static_cast<value_type> ( 9017 ) / static_cast<value_type>( 3168 ); 115 const value_type b62 = static_cast<value_type> ( -355 ) / static_cast<value_type>( 33 ); 116 const value_type b63 = static_cast<value_type> ( 46732 ) / static_cast<value_type>( 5247 ); 117 const value_type b64 = static_cast<value_type> ( 49 ) / static_cast<value_type>( 176 ); 118 const value_type b65 = static_cast<value_type> ( -5103 ) / static_cast<value_type>( 18656 ); 119 120 const value_type c1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); 121 const value_type c3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); 122 const value_type c4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); 123 const value_type c5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); 124 const value_type c6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); 125 126 typename odeint::unwrap_reference< System >::type &sys = system; 127 128 m_k_x_tmp_resizer.adjust_size( in , detail::bind( &stepper_type::template resize_k_x_tmp_impl<StateIn> , detail::ref( *this ) , detail::_1 ) ); 129 130 //m_x_tmp = x + dt*b21*dxdt 131 stepper_base_type::m_algebra.for_each3( m_x_tmp.m_v , in , dxdt_in , 132 typename operations_type::template scale_sum2< value_type , time_type >( 1.0 , dt*b21 ) ); 133 134 sys( m_x_tmp.m_v , m_k2.m_v , t + dt*a2 ); 135 // m_x_tmp = x + dt*b31*dxdt + dt*b32*m_k2 136 stepper_base_type::m_algebra.for_each4( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , 137 typename operations_type::template scale_sum3< value_type , time_type , time_type >( 1.0 , dt*b31 , dt*b32 )); 138 139 sys( m_x_tmp.m_v , m_k3.m_v , t + dt*a3 ); 140 // m_x_tmp = x + dt * (b41*dxdt + b42*m_k2 + b43*m_k3) 141 stepper_base_type::m_algebra.for_each5( m_x_tmp.m_v , in , dxdt_in , m_k2.m_v , m_k3.m_v , 142 typename operations_type::template scale_sum4< value_type , time_type , time_type , time_type >( 1.0 , dt*b41 , dt*b42 , dt*b43 )); 143 144 sys( m_x_tmp.m_v, m_k4.m_v , t + dt*a4 ); 145 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 , 146 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 )); 147 148 sys( m_x_tmp.m_v , m_k5.m_v , t + dt*a5 ); 149 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 , 150 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 )); 151 152 sys( m_x_tmp.m_v , m_k6.m_v , t + dt ); 153 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 , 154 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 )); 155 156 // the new derivative 157 sys( out , dxdt_out , t + dt ); 158 } 159 160 161 162 template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut , class Err > 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)163 void do_step_impl( System system , const StateIn &in , const DerivIn &dxdt_in , time_type t , 164 StateOut &out , DerivOut &dxdt_out , time_type dt , Err &xerr ) 165 { 166 const value_type c1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); 167 const value_type c3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); 168 const value_type c4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); 169 const value_type c5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); 170 const value_type c6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); 171 172 const value_type dc1 = c1 - static_cast<value_type> ( 5179 ) / static_cast<value_type>( 57600 ); 173 const value_type dc3 = c3 - static_cast<value_type> ( 7571 ) / static_cast<value_type>( 16695 ); 174 const value_type dc4 = c4 - static_cast<value_type> ( 393 ) / static_cast<value_type>( 640 ); 175 const value_type dc5 = c5 - static_cast<value_type> ( -92097 ) / static_cast<value_type>( 339200 ); 176 const value_type dc6 = c6 - static_cast<value_type> ( 187 ) / static_cast<value_type>( 2100 ); 177 const value_type dc7 = static_cast<value_type>( -1 ) / static_cast<value_type> ( 40 ); 178 179 /* ToDo: copy only if &dxdt_in == &dxdt_out ? */ 180 if( same_instance( dxdt_in , dxdt_out ) ) 181 { 182 m_dxdt_tmp_resizer.adjust_size( in , detail::bind( &stepper_type::template resize_dxdt_tmp_impl<StateIn> , detail::ref( *this ) , detail::_1 ) ); 183 boost::numeric::odeint::copy( dxdt_in , m_dxdt_tmp.m_v ); 184 do_step_impl( system , in , dxdt_in , t , out , dxdt_out , dt ); 185 //error estimate 186 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 , 187 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 ) ); 188 189 } 190 else 191 { 192 do_step_impl( system , in , dxdt_in , t , out , dxdt_out , dt ); 193 //error estimate 194 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 , 195 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 ) ); 196 197 } 198 199 } 200 201 202 /* 203 * Calculates Dense-Output for Dopri5 204 * 205 * See Hairer, Norsett, Wanner: Solving Ordinary Differential Equations, Nonstiff Problems. I, p.191/192 206 * 207 * y(t+theta) = y(t) + h * sum_i^7 b_i(theta) * k_i 208 * 209 * A = theta^2 * ( 3 - 2 theta ) 210 * B = theta^2 * ( theta - 1 ) 211 * C = theta^2 * ( theta - 1 )^2 212 * D = theta * ( theta - 1 )^2 213 * 214 * b_1( theta ) = A * b_1 - C * X1( theta ) + D 215 * b_2( theta ) = 0 216 * b_3( theta ) = A * b_3 + C * X3( theta ) 217 * b_4( theta ) = A * b_4 - C * X4( theta ) 218 * b_5( theta ) = A * b_5 + C * X5( theta ) 219 * b_6( theta ) = A * b_6 - C * X6( theta ) 220 * b_7( theta ) = B + C * X7( theta ) 221 * 222 * An alternative Method is described in: 223 * 224 * www-m2.ma.tum.de/homepages/simeon/numerik3/kap3.ps 225 */ 226 template< class StateOut , class StateIn1 , class DerivIn1 , class StateIn2 , class DerivIn2 > 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) const227 void calc_state( time_type t , StateOut &x , 228 const StateIn1 &x_old , const DerivIn1 &deriv_old , time_type t_old , 229 const StateIn2 & /* x_new */ , const DerivIn2 &deriv_new , time_type t_new ) const 230 { 231 const value_type b1 = static_cast<value_type> ( 35 ) / static_cast<value_type>( 384 ); 232 const value_type b3 = static_cast<value_type> ( 500 ) / static_cast<value_type>( 1113 ); 233 const value_type b4 = static_cast<value_type> ( 125 ) / static_cast<value_type>( 192 ); 234 const value_type b5 = static_cast<value_type> ( -2187 ) / static_cast<value_type>( 6784 ); 235 const value_type b6 = static_cast<value_type> ( 11 ) / static_cast<value_type>( 84 ); 236 237 const time_type dt = ( t_new - t_old ); 238 const value_type theta = ( t - t_old ) / dt; 239 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 ); 240 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 ); 241 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 ) ; 242 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 ); 243 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 ); 244 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 ); 245 246 const value_type theta_m_1 = theta - static_cast< value_type >( 1 ); 247 const value_type theta_sq = theta * theta; 248 const value_type A = theta_sq * ( static_cast< value_type >( 3 ) - static_cast< value_type >( 2 ) * theta ); 249 const value_type B = theta_sq * theta_m_1; 250 const value_type C = theta_sq * theta_m_1 * theta_m_1; 251 const value_type D = theta * theta_m_1 * theta_m_1; 252 253 const value_type b1_theta = A * b1 - C * X1 + D; 254 const value_type b3_theta = A * b3 + C * X3; 255 const value_type b4_theta = A * b4 - C * X4; 256 const value_type b5_theta = A * b5 + C * X5; 257 const value_type b6_theta = A * b6 - C * X6; 258 const value_type b7_theta = B + C * X7; 259 260 // const state_type &k1 = *m_old_deriv; 261 // const state_type &k3 = dopri5().m_k3; 262 // const state_type &k4 = dopri5().m_k4; 263 // const state_type &k5 = dopri5().m_k5; 264 // const state_type &k6 = dopri5().m_k6; 265 // const state_type &k7 = *m_current_deriv; 266 267 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 , 268 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 ) ); 269 } 270 271 272 template< class StateIn > adjust_size(const StateIn & x)273 void adjust_size( const StateIn &x ) 274 { 275 resize_k_x_tmp_impl( x ); 276 resize_dxdt_tmp_impl( x ); 277 stepper_base_type::adjust_size( x ); 278 } 279 280 281 private: 282 283 template< class StateIn > resize_k_x_tmp_impl(const StateIn & x)284 bool resize_k_x_tmp_impl( const StateIn &x ) 285 { 286 bool resized = false; 287 resized |= adjust_size_by_resizeability( m_x_tmp , x , typename is_resizeable<state_type>::type() ); 288 resized |= adjust_size_by_resizeability( m_k2 , x , typename is_resizeable<deriv_type>::type() ); 289 resized |= adjust_size_by_resizeability( m_k3 , x , typename is_resizeable<deriv_type>::type() ); 290 resized |= adjust_size_by_resizeability( m_k4 , x , typename is_resizeable<deriv_type>::type() ); 291 resized |= adjust_size_by_resizeability( m_k5 , x , typename is_resizeable<deriv_type>::type() ); 292 resized |= adjust_size_by_resizeability( m_k6 , x , typename is_resizeable<deriv_type>::type() ); 293 return resized; 294 } 295 296 template< class StateIn > resize_dxdt_tmp_impl(const StateIn & x)297 bool resize_dxdt_tmp_impl( const StateIn &x ) 298 { 299 return adjust_size_by_resizeability( m_dxdt_tmp , x , typename is_resizeable<deriv_type>::type() ); 300 } 301 302 303 304 wrapped_state_type m_x_tmp; 305 wrapped_deriv_type m_k2 , m_k3 , m_k4 , m_k5 , m_k6 ; 306 wrapped_deriv_type m_dxdt_tmp; 307 resizer_type m_k_x_tmp_resizer; 308 resizer_type m_dxdt_tmp_resizer; 309 }; 310 311 312 313 /************* DOXYGEN ************/ 314 /** 315 * \class runge_kutta_dopri5 316 * \brief The Runge-Kutta Dormand-Prince 5 method. 317 * 318 * The Runge-Kutta Dormand-Prince 5 method is a very popular method for solving ODEs, see 319 * <a href=""></a>. 320 * The method is explicit and fulfills the Error Stepper concept. Step size control 321 * is provided but continuous output is available which make this method favourable for many applications. 322 * 323 * This class derives from explicit_error_stepper_fsal_base and inherits its interface via CRTP (current recurring 324 * template pattern). The method possesses the FSAL (first-same-as-last) property. See 325 * explicit_error_stepper_fsal_base for more details. 326 * 327 * \tparam State The state type. 328 * \tparam Value The value type. 329 * \tparam Deriv The type representing the time derivative of the state. 330 * \tparam Time The time representing the independent variable - the time. 331 * \tparam Algebra The algebra type. 332 * \tparam Operations The operations type. 333 * \tparam Resizer The resizer policy type. 334 */ 335 336 337 /** 338 * \fn runge_kutta_dopri5::runge_kutta_dopri5( const algebra_type &algebra ) 339 * \brief Constructs the runge_kutta_dopri5 class. This constructor can be used as a default 340 * constructor if the algebra has a default constructor. 341 * \param algebra A copy of algebra is made and stored inside explicit_stepper_base. 342 */ 343 344 /** 345 * \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 ) 346 * \brief This method performs one step. The derivative `dxdt_in` of `in` at the time `t` is passed to the 347 * method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Furthermore, 348 * the derivative is update out-of-place, hence the input is assumed to be in `dxdt_in` and the output in 349 * `dxdt_out`. 350 * Access to this step functionality is provided by explicit_error_stepper_fsal_base and 351 * `do_step_impl` should not be called directly. 352 * 353 * \param system The system function to solve, hence the r.h.s. of the ODE. It must fulfill the 354 * Simple System concept. 355 * \param in The state of the ODE which should be solved. in is not modified in this method 356 * \param dxdt_in The derivative of x at t. dxdt_in is not modified by this method 357 * \param t The value of the time, at which the step should be performed. 358 * \param out The result of the step is written in out. 359 * \param dxdt_out The result of the new derivative at time t+dt. 360 * \param dt The step size. 361 */ 362 363 /** 364 * \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 ) 365 * \brief This method performs one step. The derivative `dxdt_in` of `in` at the time `t` is passed to the 366 * method. The result is updated out-of-place, hence the input is in `in` and the output in `out`. Furthermore, 367 * the derivative is update out-of-place, hence the input is assumed to be in `dxdt_in` and the output in 368 * `dxdt_out`. 369 * Access to this step functionality is provided by explicit_error_stepper_fsal_base and 370 * `do_step_impl` should not be called directly. 371 * An estimation of the error is calculated. 372 * 373 * \param system The system function to solve, hence the r.h.s. of the ODE. It must fulfill the 374 * Simple System concept. 375 * \param in The state of the ODE which should be solved. in is not modified in this method 376 * \param dxdt_in The derivative of x at t. dxdt_in is not modified by this method 377 * \param t The value of the time, at which the step should be performed. 378 * \param out The result of the step is written in out. 379 * \param dxdt_out The result of the new derivative at time t+dt. 380 * \param dt The step size. 381 * \param xerr An estimation of the error. 382 */ 383 384 /** 385 * \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 386 * \brief This method is used for continuous output and it calculates the state `x` at a time `t` from the 387 * knowledge of two states `old_state` and `current_state` at time points `t_old` and `t_new`. It also uses 388 * internal variables to calculate the result. Hence this method must be called after two successful `do_step` 389 * calls. 390 */ 391 392 /** 393 * \fn runge_kutta_dopri5::adjust_size( const StateIn &x ) 394 * \brief Adjust the size of all temporaries in the stepper manually. 395 * \param x A state from which the size of the temporaries to be resized is deduced. 396 */ 397 398 } // odeint 399 } // numeric 400 } // boost 401 402 403 #endif // BOOST_NUMERIC_ODEINT_STEPPER_RUNGE_KUTTA_DOPRI5_HPP_INCLUDED 404