odeint/stepper/bulirsch_stoer_dense_out.hpp

0001 /*
0002  [auto_generated]
0003  boost/numeric/odeint/stepper/bulirsch_stoer_dense_out.hpp
0004
0005  [begin_description]
0006  Implementaiton of the Burlish-Stoer method with dense output
0007  [end_description]
0008
0009  Copyright 2011-2015 Mario Mulansky
0010  Copyright 2011-2013 Karsten Ahnert
0011  Copyright 2012 Christoph Koke
0012
0013  Distributed under the Boost Software License, Version 1.0.
0014  (See accompanying file LICENSE_1_0.txt or
0015  copy at http://www.boost.org/LICENSE_1_0.txt)
0016  */
0017
0018
0019 #ifndef BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_DENSE_OUT_HPP_INCLUDED
0020 #define BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_DENSE_OUT_HPP_INCLUDED
0021
0022
0023 #include <iostream>
0024
0025 #include <algorithm>
0026
0027 #include <boost/config.hpp> // for min/max guidelines
0028
0029 #include <boost/numeric/odeint/util/bind.hpp>
0030
0031 #include <boost/math/special_functions/binomial.hpp>
0032
0033 #include <boost/numeric/odeint/stepper/controlled_runge_kutta.hpp>
0034 #include <boost/numeric/odeint/stepper/modified_midpoint.hpp>
0035 #include <boost/numeric/odeint/stepper/controlled_step_result.hpp>
0036 #include <boost/numeric/odeint/algebra/range_algebra.hpp>
0037 #include <boost/numeric/odeint/algebra/default_operations.hpp>
0038 #include <boost/numeric/odeint/algebra/algebra_dispatcher.hpp>
0039 #include <boost/numeric/odeint/algebra/operations_dispatcher.hpp>
0040
0041 #include <boost/numeric/odeint/util/state_wrapper.hpp>
0042 #include <boost/numeric/odeint/util/is_resizeable.hpp>
0043 #include <boost/numeric/odeint/util/resizer.hpp>
0044 #include <boost/numeric/odeint/util/unit_helper.hpp>
0045
0046 #include <boost/numeric/odeint/integrate/max_step_checker.hpp>
0047
0048 namespace boost {
0049 namespace numeric {
0050 namespace odeint {
0051
0052 template<
0053     class State ,
0054     class Value = double ,
0055     class Deriv = State ,
0056     class Time = Value ,
0057     class Algebra = typename algebra_dispatcher< State >::algebra_type ,
0058     class Operations = typename operations_dispatcher< State >::operations_type ,
0059     class Resizer = initially_resizer
0060     >
0061 class bulirsch_stoer_dense_out {
0062
0063
0064 public:
0065
0066     typedef State state_type;
0067     typedef Value value_type;
0068     typedef Deriv deriv_type;
0069     typedef Time time_type;
0070     typedef Algebra algebra_type;
0071     typedef Operations operations_type;
0072     typedef Resizer resizer_type;
0073     typedef dense_output_stepper_tag stepper_category;
0074 #ifndef DOXYGEN_SKIP
0075     typedef state_wrapper< state_type > wrapped_state_type;
0076     typedef state_wrapper< deriv_type > wrapped_deriv_type;
0077
0078     typedef bulirsch_stoer_dense_out< State , Value , Deriv , Time , Algebra , Operations , Resizer > controlled_error_bs_type;
0079
0080     typedef typename inverse_time< time_type >::type inv_time_type;
0081
0082     typedef std::vector< value_type > value_vector;
0083     typedef std::vector< time_type > time_vector;
0084     typedef std::vector< inv_time_type > inv_time_vector;  //should be 1/time_type for boost.units
0085     typedef std::vector< value_vector > value_matrix;
0086     typedef std::vector< size_t > int_vector;
0087     typedef std::vector< wrapped_state_type > state_vector_type;
0088     typedef std::vector< wrapped_deriv_type > deriv_vector_type;
0089     typedef std::vector< deriv_vector_type > deriv_table_type;
0090 #endif //DOXYGEN_SKIP
0091
0092     const static size_t m_k_max = 8;
0093
0094
0095
0096     bulirsch_stoer_dense_out(
0097         value_type eps_abs = 1E-6 , value_type eps_rel = 1E-6 ,
0098         value_type factor_x = 1.0 , value_type factor_dxdt = 1.0 ,
0099         time_type max_dt = static_cast<time_type>(0) ,
0100         bool control_interpolation = false )
0101         : m_error_checker( eps_abs , eps_rel , factor_x, factor_dxdt ) ,
0102           m_max_dt(max_dt) ,
0103           m_control_interpolation( control_interpolation) ,
0104           m_last_step_rejected( false ) , m_first( true ) ,
0105           m_current_state_x1( true ) ,
0106           m_error( m_k_max ) ,
0107           m_interval_sequence( m_k_max+1 ) ,
0108           m_coeff( m_k_max+1 ) ,
0109           m_cost( m_k_max+1 ) ,
0110           m_facmin_table( m_k_max+1 ) ,
0111           m_table( m_k_max ) ,
0112           m_mp_states( m_k_max+1 ) ,
0113           m_derivs( m_k_max+1 ) ,
0114           m_diffs( 2*m_k_max+2 ) ,
0115           STEPFAC1( 0.65 ) , STEPFAC2( 0.94 ) , STEPFAC3( 0.02 ) , STEPFAC4( 4.0 ) , KFAC1( 0.8 ) , KFAC2( 0.9 )
0116     {
0117         BOOST_USING_STD_MIN();
0118         BOOST_USING_STD_MAX();
0119
0120         for( unsigned short i = 0; i < m_k_max+1; i++ )
0121         {
0122             /* only this specific sequence allows for dense output */
0123             m_interval_sequence[i] = 2 + 4*i;  // 2 6 10 14 ...
0124             m_derivs[i].resize( m_interval_sequence[i] );
0125             if( i == 0 )
0126             {
0127                 m_cost[i] = m_interval_sequence[i];
0128             } else
0129             {
0130                 m_cost[i] = m_cost[i-1] + m_interval_sequence[i];
0131             }
0132             m_facmin_table[i] = pow BOOST_PREVENT_MACRO_SUBSTITUTION( STEPFAC3 , static_cast< value_type >(1) / static_cast< value_type >( 2*i+1 ) );
0133             m_coeff[i].resize(i);
0134             for( size_t k = 0 ; k < i ; ++k  )
0135             {
0136                 const value_type r = static_cast< value_type >( m_interval_sequence[i] ) / static_cast< value_type >( m_interval_sequence[k] );
0137                 m_coeff[i][k] = 1.0 / ( r*r - static_cast< value_type >( 1.0 ) ); // coefficients for extrapolation
0138             }
0139             // crude estimate of optimal order
0140
0141             m_current_k_opt = 4;
0142             /* no calculation because log10 might not exist for value_type!
0143             const value_type logfact( -log10( max BOOST_PREVENT_MACRO_SUBSTITUTION( eps_rel , static_cast< value_type >( 1.0E-12 ) ) ) * 0.6 + 0.5 );
0144             m_current_k_opt = max BOOST_PREVENT_MACRO_SUBSTITUTION( 1 , min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>( m_k_max-1 ) , static_cast<int>( logfact ) ));
0145             */
0146         }
0147         int num = 1;
0148         for( int i = 2*(m_k_max)+1 ; i >=0  ; i-- )
0149         {
0150             m_diffs[i].resize( num );
0151             num += (i+1)%2;
0152         }
0153     }
0154
0155     template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut >
0156     controlled_step_result try_step( System system , const StateIn &in , const DerivIn &dxdt , time_type &t , StateOut &out , DerivOut &dxdt_new , time_type &dt )
0157     {
0158         if( m_max_dt != static_cast<time_type>(0) && detail::less_with_sign(m_max_dt, dt, dt) )
0159         {
0160             // given step size is bigger then max_dt
0161             // set limit and return fail
0162             dt = m_max_dt;
0163             return fail;
0164         }
0165
0166         BOOST_USING_STD_MIN();
0167         BOOST_USING_STD_MAX();
0168         using std::pow;
0169
0170         static const value_type val1( 1.0 );
0171
0172         bool reject( true );
0173
0174         time_vector h_opt( m_k_max+1 );
0175         inv_time_vector work( m_k_max+1 );
0176
0177         m_k_final = 0;
0178         time_type new_h = dt;
0179
0180         //std::cout << "t=" << t <<", dt=" << dt << ", k_opt=" << m_current_k_opt << ", first: " << m_first << std::endl;
0181
0182         for( size_t k = 0 ; k <= m_current_k_opt+1 ; k++ )
0183         {
0184             m_midpoint.set_steps( m_interval_sequence[k] );
0185             if( k == 0 )
0186             {
0187                 m_midpoint.do_step( system , in , dxdt , t , out , dt , m_mp_states[k].m_v , m_derivs[k]);
0188             }
0189             else
0190             {
0191                 m_midpoint.do_step( system , in , dxdt , t , m_table[k-1].m_v , dt , m_mp_states[k].m_v , m_derivs[k] );
0192                 extrapolate( k , m_table , m_coeff , out );
0193                 // get error estimate
0194                 m_algebra.for_each3( m_err.m_v , out , m_table[0].m_v ,
0195                                      typename operations_type::template scale_sum2< value_type , value_type >( val1 , -val1 ) );
0196                 const value_type error = m_error_checker.error( m_algebra , in , dxdt , m_err.m_v , dt );
0197                 h_opt[k] = calc_h_opt( dt , error , k );
0198                 work[k] = static_cast<value_type>( m_cost[k] ) / h_opt[k];
0199
0200                 m_k_final = k;
0201
0202                 if( (k == m_current_k_opt-1) || m_first )
0203                 { // convergence before k_opt ?
0204                     if( error < 1.0 )
0205                     {
0206                         //convergence
0207                         reject = false;
0208                         if( (work[k] < KFAC2*work[k-1]) || (m_current_k_opt <= 2) )
0209                         {
0210                             // leave order as is (except we were in first round)
0211                             m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(k)+1 ) );
0212                             new_h = h_opt[k] * static_cast<value_type>( m_cost[k+1] ) / static_cast<value_type>( m_cost[k] );
0213                         } else {
0214                             m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(k) ) );
0215                             new_h = h_opt[k];
0216                         }
0217                         break;
0218                     }
0219                     else if( should_reject( error , k ) && !m_first )
0220                     {
0221                         reject = true;
0222                         new_h = h_opt[k];
0223                         break;
0224                     }
0225                 }
0226                 if( k == m_current_k_opt )
0227                 { // convergence at k_opt ?
0228                     if( error < 1.0 )
0229                     {
0230                         //convergence
0231                         reject = false;
0232                         if( (work[k-1] < KFAC2*work[k]) )
0233                         {
0234                             m_current_k_opt = max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(m_current_k_opt)-1 );
0235                             new_h = h_opt[m_current_k_opt];
0236                         }
0237                         else if( (work[k] < KFAC2*work[k-1]) && !m_last_step_rejected )
0238                         {
0239                             m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , static_cast<int>(m_current_k_opt)+1 );
0240                             new_h = h_opt[k]*static_cast<value_type>( m_cost[m_current_k_opt] ) / static_cast<value_type>( m_cost[k] );
0241                         } else
0242                             new_h = h_opt[m_current_k_opt];
0243                         break;
0244                     }
0245                     else if( should_reject( error , k ) )
0246                     {
0247                         reject = true;
0248                         new_h = h_opt[m_current_k_opt];
0249                         break;
0250                     }
0251                 }
0252                 if( k == m_current_k_opt+1 )
0253                 { // convergence at k_opt+1 ?
0254                     if( error < 1.0 )
0255                     {   //convergence
0256                         reject = false;
0257                         if( work[k-2] < KFAC2*work[k-1] )
0258                             m_current_k_opt = max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(m_current_k_opt)-1 );
0259                         if( (work[k] < KFAC2*work[m_current_k_opt]) && !m_last_step_rejected )
0260                             m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , static_cast<int>(k) );
0261                         new_h = h_opt[m_current_k_opt];
0262                     } else
0263                     {
0264                         reject = true;
0265                         new_h = h_opt[m_current_k_opt];
0266                     }
0267                     break;
0268                 }
0269             }
0270         }
0271
0272         if( !reject )
0273         {
0274
0275             //calculate dxdt for next step and dense output
0276             typename odeint::unwrap_reference< System >::type &sys = system;
0277             sys( out , dxdt_new , t+dt );
0278
0279             //prepare dense output
0280             value_type error = prepare_dense_output( m_k_final , in , dxdt , out , dxdt_new , dt );
0281
0282             if( error > static_cast<value_type>(10) ) // we are not as accurate for interpolation as for the steps
0283             {
0284                 reject = true;
0285                 new_h = dt * pow BOOST_PREVENT_MACRO_SUBSTITUTION( error , static_cast<value_type>(-1)/(2*m_k_final+2) );
0286             } else {
0287                 t += dt;
0288             }
0289         }
0290         //set next stepsize
0291         if( !m_last_step_rejected || (new_h < dt) )
0292         {
0293             // limit step size
0294             if( m_max_dt != static_cast<time_type>(0) )
0295             {
0296                 new_h = detail::min_abs(m_max_dt, new_h);
0297             }
0298             dt = new_h;
0299         }
0300
0301         m_last_step_rejected = reject;
0302         if( reject )
0303             return fail;
0304         else
0305             return success;
0306     }
0307
0308     template< class StateType >
0309     void initialize( const StateType &x0 , const time_type &t0 , const time_type &dt0 )
0310     {
0311         m_resizer.adjust_size(x0, [this](auto&& arg) { return this->resize_impl<StateType>(std::forward<decltype(arg)>(arg)); });
0312         boost::numeric::odeint::copy( x0 , get_current_state() );
0313         m_t = t0;
0314         m_dt = dt0;
0315         reset();
0316     }
0317
0318
0319     /*  =======================================================
0320      *  the actual step method that should be called from outside (maybe make try_step private?)
0321      */
0322     template< class System >
0323     std::pair< time_type , time_type > do_step( System system )
0324     {
0325         if( m_first )
0326         {
0327             typename odeint::unwrap_reference< System >::type &sys = system;
0328             sys( get_current_state() , get_current_deriv() , m_t );
0329         }
0330
0331         failed_step_checker fail_checker;  // to throw a runtime_error if step size adjustment fails
0332         controlled_step_result res = fail;
0333         m_t_last = m_t;
0334         while( res == fail )
0335         {
0336             res = try_step( system , get_current_state() , get_current_deriv() , m_t , get_old_state() , get_old_deriv() , m_dt );
0337             m_first = false;
0338             fail_checker();  // check for overflow of failed steps
0339         }
0340         toggle_current_state();
0341         return std::make_pair( m_t_last , m_t );
0342     }
0343
0344     /* performs the interpolation from a calculated step */
0345     template< class StateOut >
0346     void calc_state( time_type t , StateOut &x ) const
0347     {
0348         do_interpolation( t , x );
0349     }
0350
0351     const state_type& current_state( void ) const
0352     {
0353         return get_current_state();
0354     }
0355
0356     time_type current_time( void ) const
0357     {
0358         return m_t;
0359     }
0360
0361     const state_type& previous_state( void ) const
0362     {
0363         return get_old_state();
0364     }
0365
0366     time_type previous_time( void ) const
0367     {
0368         return m_t_last;
0369     }
0370
0371     time_type current_time_step( void ) const
0372     {
0373         return m_dt;
0374     }
0375
0376     /** \brief Resets the internal state of the stepper. */
0377     void reset()
0378     {
0379         m_first = true;
0380         m_last_step_rejected = false;
0381     }
0382
0383     template< class StateIn >
0384     void adjust_size( const StateIn &x )
0385     {
0386         resize_impl( x );
0387         m_midpoint.adjust_size( x );
0388     }
0389
0390
0391 protected:
0392
0393     time_type m_max_dt;
0394
0395
0396 private:
0397
0398     template< class StateInOut , class StateVector >
0399     void extrapolate( size_t k , StateVector &table , const value_matrix &coeff , StateInOut &xest , size_t order_start_index = 0 )
0400     //polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf
0401     {
0402         static const value_type val1( 1.0 );
0403         for( int j=k-1 ; j>0 ; --j )
0404         {
0405             m_algebra.for_each3( table[j-1].m_v , table[j].m_v , table[j-1].m_v ,
0406                                  typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][j + order_start_index] ,
0407                                                                                                            -coeff[k + order_start_index][j + order_start_index] ) );
0408         }
0409         m_algebra.for_each3( xest , table[0].m_v , xest ,
0410                              typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][0 + order_start_index] ,
0411                                                                                                        -coeff[k + order_start_index][0 + order_start_index]) );
0412     }
0413
0414
0415     template< class StateVector >
0416     void extrapolate_dense_out( size_t k , StateVector &table , const value_matrix &coeff , size_t order_start_index = 0 )
0417     //polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf
0418     {
0419         // result is written into table[0]
0420         static const value_type val1( 1.0 );
0421         for( int j=k ; j>1 ; --j )
0422         {
0423             m_algebra.for_each3( table[j-1].m_v , table[j].m_v , table[j-1].m_v ,
0424                                  typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][j + order_start_index - 1] ,
0425                                                                                                            -coeff[k + order_start_index][j + order_start_index - 1] ) );
0426         }
0427         m_algebra.for_each3( table[0].m_v , table[1].m_v , table[0].m_v ,
0428                              typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][order_start_index] ,
0429                                                                                                        -coeff[k + order_start_index][order_start_index]) );
0430     }
0431
0432     time_type calc_h_opt( time_type h , value_type error , size_t k ) const
0433     {
0434         BOOST_USING_STD_MIN();
0435         BOOST_USING_STD_MAX();
0436         using std::pow;
0437
0438         value_type expo = static_cast<value_type>(1)/(m_interval_sequence[k-1]);
0439         value_type facmin = m_facmin_table[k];
0440         value_type fac;
0441         if (error == 0.0)
0442             fac = static_cast<value_type>(1)/facmin;
0443         else
0444         {
0445             fac = STEPFAC2 / pow BOOST_PREVENT_MACRO_SUBSTITUTION( error / STEPFAC1 , expo );
0446             fac = max BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<value_type>( facmin/STEPFAC4 ) , min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<value_type>(static_cast<value_type>(1)/facmin) , fac ) );
0447         }
0448         return h*fac;
0449     }
0450
0451     bool in_convergence_window( size_t k ) const
0452     {
0453         if( (k == m_current_k_opt-1) && !m_last_step_rejected )
0454             return true; // decrease order only if last step was not rejected
0455         return ( (k == m_current_k_opt) || (k == m_current_k_opt+1) );
0456     }
0457
0458     bool should_reject( value_type error , size_t k ) const
0459     {
0460         if( k == m_current_k_opt-1 )
0461         {
0462             const value_type d = m_interval_sequence[m_current_k_opt] * m_interval_sequence[m_current_k_opt+1] /
0463                 (m_interval_sequence[0]*m_interval_sequence[0]);
0464             //step will fail, criterion 17.3.17 in NR
0465             return ( error > d*d );
0466         }
0467         else if( k == m_current_k_opt )
0468         {
0469             const value_type d = m_interval_sequence[m_current_k_opt+1] / m_interval_sequence[0];
0470             return ( error > d*d );
0471         } else
0472             return error > 1.0;
0473     }
0474
0475     template< class StateIn1 , class DerivIn1 , class StateIn2 , class DerivIn2 >
0476     value_type prepare_dense_output( int k , const StateIn1 &x_start , const DerivIn1 &dxdt_start ,
0477                                      const StateIn2 & /* x_end */ , const DerivIn2 & /*dxdt_end */ , time_type dt )
0478     /* k is the order to which the result was approximated */
0479     {
0480
0481         /* compute the coefficients of the interpolation polynomial
0482          * we parametrize the interval t .. t+dt by theta = -1 .. 1
0483          * we use 2k+3 values at the interval center theta=0 to obtain the interpolation coefficients
0484          * the values are x(t+dt/2) and the derivatives dx/dt , ... d^(2k+2) x / dt^(2k+2) at the midpoints
0485          * the derivatives are approximated via finite differences
0486          * all values are obtained from interpolation of the results from the increasing orders of the midpoint calls
0487          */
0488
0489         // calculate finite difference approximations to derivatives at the midpoint
0490         for( int j = 0 ; j<=k ; j++ )
0491         {
0492             /* not working with boost units... */
0493             const value_type d = m_interval_sequence[j] / ( static_cast<value_type>(2) * dt );
0494             value_type f = 1.0; //factor 1/2 here because our interpolation interval has length 2 !!!
0495             for( int kappa = 0 ; kappa <= 2*j+1 ; ++kappa )
0496             {
0497                 calculate_finite_difference( j , kappa , f , dxdt_start );
0498                 f *= d;
0499             }
0500
0501             if( j > 0 )
0502                 extrapolate_dense_out( j , m_mp_states , m_coeff );
0503         }
0504
0505         time_type d = dt/2;
0506
0507         // extrapolate finite differences
0508         for( int kappa = 0 ; kappa<=2*k+1 ; kappa++ )
0509         {
0510             for( int j=1 ; j<=(k-kappa/2) ; ++j )
0511                 extrapolate_dense_out( j , m_diffs[kappa] , m_coeff , kappa/2 );
0512
0513             // extrapolation results are now stored in m_diffs[kappa][0]
0514
0515             // divide kappa-th derivative by kappa because we need these terms for dense output interpolation
0516             m_algebra.for_each1( m_diffs[kappa][0].m_v , typename operations_type::template scale< time_type >( static_cast<time_type>(d) ) );
0517
0518             d *= dt/(2*(kappa+2));
0519         }
0520
0521         // dense output coefficients a_0 is stored in m_mp_states[0], a_i for i = 1...2k are stored in m_diffs[i-1][0]
0522
0523         // the error is just the highest order coefficient of the interpolation polynomial
0524         // this is because we use only the midpoint theta=0 as support for the interpolation (remember that theta = -1 .. 1)
0525
0526         value_type error = 0.0;
0527         if( m_control_interpolation )
0528         {
0529             boost::numeric::odeint::copy( m_diffs[2*k+1][0].m_v , m_err.m_v );
0530             error = m_error_checker.error( m_algebra , x_start , dxdt_start , m_err.m_v , dt );
0531         }
0532
0533         return error;
0534     }
0535
0536     template< class DerivIn >
0537     void calculate_finite_difference( size_t j , size_t kappa , value_type fac , const DerivIn &dxdt )
0538     {
0539         const int m = m_interval_sequence[j]/2-1;
0540         if( kappa == 0) // no calculation required for 0th derivative of f
0541         {
0542             m_algebra.for_each2( m_diffs[0][j].m_v , m_derivs[j][m].m_v ,
0543                                  typename operations_type::template scale_sum1< value_type >( fac ) );
0544         }
0545         else
0546         {
0547             // calculate the index of m_diffs for this kappa-j-combination
0548             const int j_diffs = j - kappa/2;
0549
0550             m_algebra.for_each2( m_diffs[kappa][j_diffs].m_v , m_derivs[j][m+kappa].m_v ,
0551                                  typename operations_type::template scale_sum1< value_type >( fac ) );
0552             value_type sign = -1.0;
0553             int c = 1;
0554             //computes the j-th order finite difference for the kappa-th derivative of f at t+dt/2 using function evaluations stored in m_derivs
0555             for( int i = m+static_cast<int>(kappa)-2 ; i >= m-static_cast<int>(kappa) ; i -= 2 )
0556             {
0557                 if( i >= 0 )
0558                 {
0559                     m_algebra.for_each3( m_diffs[kappa][j_diffs].m_v , m_diffs[kappa][j_diffs].m_v , m_derivs[j][i].m_v ,
0560                                          typename operations_type::template scale_sum2< value_type , value_type >( 1.0 ,
0561                                                                                                                    sign * fac * boost::math::binomial_coefficient< value_type >( kappa , c ) ) );
0562                 }
0563                 else
0564                 {
0565                     m_algebra.for_each3( m_diffs[kappa][j_diffs].m_v , m_diffs[kappa][j_diffs].m_v , dxdt ,
0566                                          typename operations_type::template scale_sum2< value_type , value_type >( 1.0 , sign * fac ) );
0567                 }
0568                 sign *= -1;
0569                 ++c;
0570             }
0571         }
0572     }
0573
0574     template< class StateOut >
0575     void do_interpolation( time_type t , StateOut &out ) const
0576     {
0577         // interpolation polynomial is defined for theta = -1 ... 1
0578         // m_k_final is the number of order-iterations done for the last step - it governs the order of the interpolation polynomial
0579         const value_type theta = 2 * get_unit_value( (t - m_t_last) / (m_t - m_t_last) ) - 1;
0580         // we use only values at interval center, that is theta=0, for interpolation
0581         // our interpolation polynomial is thus of order 2k+2, hence we have 2k+3 terms
0582
0583         boost::numeric::odeint::copy( m_mp_states[0].m_v , out );
0584         // add remaining terms: x += a_1 theta + a2 theta^2 + ... + a_{2k} theta^{2k}
0585         value_type theta_pow( theta );
0586         for( size_t i=0 ; i<=2*m_k_final+1 ; ++i )
0587         {
0588             m_algebra.for_each3( out , out , m_diffs[i][0].m_v ,
0589                                  typename operations_type::template scale_sum2< value_type >( static_cast<value_type>(1) , theta_pow ) );
0590             theta_pow *= theta;
0591         }
0592     }
0593
0594     /* Resizer methods */
0595     template< class StateIn >
0596     bool resize_impl( const StateIn &x )
0597     {
0598         bool resized( false );
0599
0600         resized |= adjust_size_by_resizeability( m_x1 , x , typename is_resizeable<state_type>::type() );
0601         resized |= adjust_size_by_resizeability( m_x2 , x , typename is_resizeable<state_type>::type() );
0602         resized |= adjust_size_by_resizeability( m_dxdt1 , x , typename is_resizeable<state_type>::type() );
0603         resized |= adjust_size_by_resizeability( m_dxdt2 , x , typename is_resizeable<state_type>::type() );
0604         resized |= adjust_size_by_resizeability( m_err , x , typename is_resizeable<state_type>::type() );
0605
0606         for( size_t i = 0 ; i < m_k_max ; ++i )
0607             resized |= adjust_size_by_resizeability( m_table[i] , x , typename is_resizeable<state_type>::type() );
0608         for( size_t i = 0 ; i < m_k_max+1 ; ++i )
0609             resized |= adjust_size_by_resizeability( m_mp_states[i] , x , typename is_resizeable<state_type>::type() );
0610         for( size_t i = 0 ; i < m_k_max+1 ; ++i )
0611             for( size_t j = 0 ; j < m_derivs[i].size() ; ++j )
0612                 resized |= adjust_size_by_resizeability( m_derivs[i][j] , x , typename is_resizeable<deriv_type>::type() );
0613         for( size_t i = 0 ; i < 2*m_k_max+2 ; ++i )
0614             for( size_t j = 0 ; j < m_diffs[i].size() ; ++j )
0615                 resized |= adjust_size_by_resizeability( m_diffs[i][j] , x , typename is_resizeable<deriv_type>::type() );
0616
0617         return resized;
0618     }
0619
0620
0621     state_type& get_current_state( void )
0622     {
0623         return m_current_state_x1 ? m_x1.m_v : m_x2.m_v ;
0624     }
0625
0626     const state_type& get_current_state( void ) const
0627     {
0628         return m_current_state_x1 ? m_x1.m_v : m_x2.m_v ;
0629     }
0630
0631     state_type& get_old_state( void )
0632     {
0633         return m_current_state_x1 ? m_x2.m_v : m_x1.m_v ;
0634     }
0635
0636     const state_type& get_old_state( void ) const
0637     {
0638         return m_current_state_x1 ? m_x2.m_v : m_x1.m_v ;
0639     }
0640
0641     deriv_type& get_current_deriv( void )
0642     {
0643         return m_current_state_x1 ? m_dxdt1.m_v : m_dxdt2.m_v ;
0644     }
0645
0646     const deriv_type& get_current_deriv( void ) const
0647     {
0648         return m_current_state_x1 ? m_dxdt1.m_v : m_dxdt2.m_v ;
0649     }
0650
0651     deriv_type& get_old_deriv( void )
0652     {
0653         return m_current_state_x1 ? m_dxdt2.m_v : m_dxdt1.m_v ;
0654     }
0655
0656     const deriv_type& get_old_deriv( void ) const
0657     {
0658         return m_current_state_x1 ? m_dxdt2.m_v : m_dxdt1.m_v ;
0659     }
0660
0661
0662     void toggle_current_state( void )
0663     {
0664         m_current_state_x1 = ! m_current_state_x1;
0665     }
0666
0667
0668
0669     default_error_checker< value_type, algebra_type , operations_type > m_error_checker;
0670     modified_midpoint_dense_out< state_type , value_type , deriv_type , time_type , algebra_type , operations_type , resizer_type > m_midpoint;
0671
0672     bool m_control_interpolation;
0673
0674     bool m_last_step_rejected;
0675     bool m_first;
0676
0677     time_type m_t;
0678     time_type m_dt;
0679     time_type m_dt_last;
0680     time_type m_t_last;
0681
0682     size_t m_current_k_opt;
0683     size_t m_k_final;
0684
0685     algebra_type m_algebra;
0686
0687     resizer_type m_resizer;
0688
0689     wrapped_state_type m_x1 , m_x2;
0690     wrapped_deriv_type m_dxdt1 , m_dxdt2;
0691     wrapped_state_type m_err;
0692     bool m_current_state_x1;
0693
0694
0695
0696     value_vector m_error; // errors of repeated midpoint steps and extrapolations
0697     int_vector m_interval_sequence; // stores the successive interval counts
0698     value_matrix m_coeff;
0699     int_vector m_cost; // costs for interval count
0700     value_vector m_facmin_table; // for precomputed facmin to save pow calls
0701
0702     state_vector_type m_table; // sequence of states for extrapolation
0703
0704     //for dense output:
0705     state_vector_type m_mp_states; // sequence of approximations of x at distance center
0706     deriv_table_type m_derivs; // table of function values
0707     deriv_table_type m_diffs; // table of function values
0708
0709     //wrapped_state_type m_a1 , m_a2 , m_a3 , m_a4;
0710
0711     value_type STEPFAC1 , STEPFAC2 , STEPFAC3 , STEPFAC4 , KFAC1 , KFAC2;
0712 };
0713
0714
0715
0716 /********** DOXYGEN **********/
0717
0718 /**
0719  * \class bulirsch_stoer_dense_out
0720  * \brief The Bulirsch-Stoer algorithm.
0721  *
0722  * The Bulirsch-Stoer is a controlled stepper that adjusts both step size
0723  * and order of the method. The algorithm uses the modified midpoint and
0724  * a polynomial extrapolation compute the solution. This class also provides
0725  * dense output facility.
0726  *
0727  * \tparam State The state type.
0728  * \tparam Value The value type.
0729  * \tparam Deriv The type representing the time derivative of the state.
0730  * \tparam Time The time representing the independent variable - the time.
0731  * \tparam Algebra The algebra type.
0732  * \tparam Operations The operations type.
0733  * \tparam Resizer The resizer policy type.
0734  */
0735
0736     /**
0737      * \fn bulirsch_stoer_dense_out::bulirsch_stoer_dense_out( value_type eps_abs , value_type eps_rel , value_type factor_x , value_type factor_dxdt , bool control_interpolation )
0738      * \brief Constructs the bulirsch_stoer class, including initialization of
0739      * the error bounds.
0740      *
0741      * \param eps_abs Absolute tolerance level.
0742      * \param eps_rel Relative tolerance level.
0743      * \param factor_x Factor for the weight of the state.
0744      * \param factor_dxdt Factor for the weight of the derivative.
0745      * \param control_interpolation Set true to additionally control the error of
0746      * the interpolation.
0747      */
0748
0749     /**
0750      * \fn bulirsch_stoer_dense_out::try_step( System system , const StateIn &in , const DerivIn &dxdt , time_type &t , StateOut &out , DerivOut &dxdt_new , time_type &dt )
0751      * \brief Tries to perform one step.
0752      *
0753      * This method tries to do one step with step size dt. If the error estimate
0754      * is to large, the step is rejected and the method returns fail and the
0755      * step size dt is reduced. If the error estimate is acceptably small, the
0756      * step is performed, success is returned and dt might be increased to make
0757      * the steps as large as possible. This method also updates t if a step is
0758      * performed. Also, the internal order of the stepper is adjusted if required.
0759      *
0760      * \param system The system function to solve, hence the r.h.s. of the ODE.
0761      * It must fulfill the Simple System concept.
0762      * \param in The state of the ODE which should be solved.
0763      * \param dxdt The derivative of state.
0764      * \param t The value of the time. Updated if the step is successful.
0765      * \param out Used to store the result of the step.
0766      * \param dt The step size. Updated.
0767      * \return success if the step was accepted, fail otherwise.
0768      */
0769
0770     /**
0771      * \fn bulirsch_stoer_dense_out::initialize( const StateType &x0 , const time_type &t0 , const time_type &dt0 )
0772      * \brief Initializes the dense output stepper.
0773      *
0774      * \param x0 The initial state.
0775      * \param t0 The initial time.
0776      * \param dt0 The initial time step.
0777      */
0778
0779     /**
0780      * \fn bulirsch_stoer_dense_out::do_step( System system )
0781      * \brief Does one time step. This is the main method that should be used to
0782      * integrate an ODE with this stepper.
0783      * \note initialize has to be called before using this method to set the
0784      * initial conditions x,t and the stepsize.
0785      * \param system The system function to solve, hence the r.h.s. of the
0786      * ordinary differential equation. It must fulfill the Simple System concept.
0787      * \return Pair with start and end time of the integration step.
0788      */
0789
0790     /**
0791      * \fn bulirsch_stoer_dense_out::calc_state( time_type t , StateOut &x ) const
0792      * \brief Calculates the solution at an intermediate point within the last step
0793      * \param t The time at which the solution should be calculated, has to be
0794      * in the current time interval.
0795      * \param x The output variable where the result is written into.
0796      */
0797
0798     /**
0799      * \fn bulirsch_stoer_dense_out::current_state( void ) const
0800      * \brief Returns the current state of the solution.
0801      * \return The current state of the solution x(t).
0802      */
0803
0804     /**
0805      * \fn bulirsch_stoer_dense_out::current_time( void ) const
0806      * \brief Returns the current time of the solution.
0807      * \return The current time of the solution t.
0808      */
0809
0810     /**
0811      * \fn bulirsch_stoer_dense_out::previous_state( void ) const
0812      * \brief Returns the last state of the solution.
0813      * \return The last state of the solution x(t-dt).
0814      */
0815
0816     /**
0817      * \fn bulirsch_stoer_dense_out::previous_time( void ) const
0818      * \brief Returns the last time of the solution.
0819      * \return The last time of the solution t-dt.
0820      */
0821
0822     /**
0823      * \fn bulirsch_stoer_dense_out::current_time_step( void ) const
0824      * \brief Returns the current step size.
0825      * \return The current step size.
0826      */
0827
0828     /**
0829      * \fn bulirsch_stoer_dense_out::adjust_size( const StateIn &x )
0830      * \brief Adjust the size of all temporaries in the stepper manually.
0831      * \param x A state from which the size of the temporaries to be resized is deduced.
0832      */
0833
0834 }
0835 }
0836 }
0837
0838 #endif // BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_HPP_INCLUDED