boost/numeric/odeint/stepper/bulirsch_stoer_dense_out.hpp
/* [auto_generated] boost/numeric/odeint/stepper/bulirsch_stoer_dense_out.hpp [begin_description] Implementaiton of the Burlish-Stoer method with dense output [end_description] Copyright 2011-2013 Mario Mulansky Copyright 2011-2013 Karsten Ahnert 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_BULIRSCH_STOER_DENSE_OUT_HPP_INCLUDED #define BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_DENSE_OUT_HPP_INCLUDED #include <iostream> #include <algorithm> #include <boost/config.hpp> // for min/max guidelines #include <boost/numeric/odeint/util/bind.hpp> #include <boost/math/special_functions/binomial.hpp> #include <boost/numeric/odeint/stepper/controlled_runge_kutta.hpp> #include <boost/numeric/odeint/stepper/modified_midpoint.hpp> #include <boost/numeric/odeint/stepper/controlled_step_result.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/util/state_wrapper.hpp> #include <boost/numeric/odeint/util/is_resizeable.hpp> #include <boost/numeric/odeint/util/resizer.hpp> #include <boost/numeric/odeint/util/unit_helper.hpp> #include <boost/type_traits.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 bulirsch_stoer_dense_out { public: typedef State state_type; typedef Value value_type; typedef Deriv deriv_type; typedef Time time_type; typedef Algebra algebra_type; typedef Operations operations_type; typedef Resizer resizer_type; typedef dense_output_stepper_tag stepper_category; #ifndef DOXYGEN_SKIP typedef state_wrapper< state_type > wrapped_state_type; typedef state_wrapper< deriv_type > wrapped_deriv_type; typedef bulirsch_stoer_dense_out< State , Value , Deriv , Time , Algebra , Operations , Resizer > controlled_error_bs_type; typedef typename inverse_time< time_type >::type inv_time_type; typedef std::vector< value_type > value_vector; typedef std::vector< time_type > time_vector; typedef std::vector< inv_time_type > inv_time_vector; //should be 1/time_type for boost.units typedef std::vector< value_vector > value_matrix; typedef std::vector< size_t > int_vector; typedef std::vector< wrapped_state_type > state_vector_type; typedef std::vector< wrapped_deriv_type > deriv_vector_type; typedef std::vector< deriv_vector_type > deriv_table_type; #endif //DOXYGEN_SKIP const static size_t m_k_max = 8; bulirsch_stoer_dense_out( value_type eps_abs = 1E-6 , value_type eps_rel = 1E-6 , value_type factor_x = 1.0 , value_type factor_dxdt = 1.0 , bool control_interpolation = false ) : m_error_checker( eps_abs , eps_rel , factor_x, factor_dxdt ) , m_control_interpolation( control_interpolation) , m_last_step_rejected( false ) , m_first( true ) , m_current_state_x1( true ) , m_error( m_k_max ) , m_interval_sequence( m_k_max+1 ) , m_coeff( m_k_max+1 ) , m_cost( m_k_max+1 ) , m_table( m_k_max ) , m_mp_states( m_k_max+1 ) , m_derivs( m_k_max+1 ) , m_diffs( 2*m_k_max+1 ) , STEPFAC1( 0.65 ) , STEPFAC2( 0.94 ) , STEPFAC3( 0.02 ) , STEPFAC4( 4.0 ) , KFAC1( 0.8 ) , KFAC2( 0.9 ) { BOOST_USING_STD_MIN(); BOOST_USING_STD_MAX(); for( unsigned short i = 0; i < m_k_max+1; i++ ) { /* only this specific sequence allows for dense output */ m_interval_sequence[i] = 2 + 4*i; // 2 6 10 14 ... m_derivs[i].resize( m_interval_sequence[i] ); if( i == 0 ) m_cost[i] = m_interval_sequence[i]; else m_cost[i] = m_cost[i-1] + m_interval_sequence[i]; m_coeff[i].resize(i); for( size_t k = 0 ; k < i ; ++k ) { const value_type r = static_cast< value_type >( m_interval_sequence[i] ) / static_cast< value_type >( m_interval_sequence[k] ); m_coeff[i][k] = 1.0 / ( r*r - static_cast< value_type >( 1.0 ) ); // coefficients for extrapolation } // crude estimate of optimal order m_current_k_opt = 4; /* no calculation because log10 might not exist for value_type! const value_type logfact( -log10( max BOOST_PREVENT_MACRO_SUBSTITUTION( eps_rel , static_cast< value_type >( 1.0E-12 ) ) ) * 0.6 + 0.5 ); 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 ) )); */ } int num = 1; for( int i = 2*(m_k_max) ; i >=0 ; i-- ) { m_diffs[i].resize( num ); num += (i+1)%2; } } template< class System , class StateIn , class DerivIn , class StateOut , class DerivOut > controlled_step_result try_step( System system , const StateIn &in , const DerivIn &dxdt , time_type &t , StateOut &out , DerivOut &dxdt_new , time_type &dt ) { BOOST_USING_STD_MIN(); BOOST_USING_STD_MAX(); using std::pow; static const value_type val1( 1.0 ); typename odeint::unwrap_reference< System >::type &sys = system; bool reject( true ); time_vector h_opt( m_k_max+1 ); inv_time_vector work( m_k_max+1 ); m_k_final = 0; time_type new_h = dt; //std::cout << "t=" << t <<", dt=" << dt << ", k_opt=" << m_current_k_opt << ", first: " << m_first << std::endl; for( size_t k = 0 ; k <= m_current_k_opt+1 ; k++ ) { m_midpoint.set_steps( m_interval_sequence[k] ); if( k == 0 ) { m_midpoint.do_step( system , in , dxdt , t , out , dt , m_mp_states[k].m_v , m_derivs[k]); } else { m_midpoint.do_step( system , in , dxdt , t , m_table[k-1].m_v , dt , m_mp_states[k].m_v , m_derivs[k] ); extrapolate( k , m_table , m_coeff , out ); // get error estimate m_algebra.for_each3( m_err.m_v , out , m_table[0].m_v , typename operations_type::template scale_sum2< value_type , value_type >( val1 , -val1 ) ); const value_type error = m_error_checker.error( m_algebra , in , dxdt , m_err.m_v , dt ); h_opt[k] = calc_h_opt( dt , error , k ); work[k] = static_cast<value_type>( m_cost[k] ) / h_opt[k]; m_k_final = k; if( (k == m_current_k_opt-1) || m_first ) { // convergence before k_opt ? if( error < 1.0 ) { //convergence reject = false; if( (work[k] < KFAC2*work[k-1]) || (m_current_k_opt <= 2) ) { // leave order as is (except we were in first round) 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 ) ); new_h = h_opt[k] * static_cast<value_type>( m_cost[k+1] ) / static_cast<value_type>( m_cost[k] ); } else { 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) ) ); new_h = h_opt[k]; } break; } else if( should_reject( error , k ) && !m_first ) { reject = true; new_h = h_opt[k]; break; } } if( k == m_current_k_opt ) { // convergence at k_opt ? if( error < 1.0 ) { //convergence reject = false; if( (work[k-1] < KFAC2*work[k]) ) { m_current_k_opt = max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(m_current_k_opt)-1 ); new_h = h_opt[m_current_k_opt]; } else if( (work[k] < KFAC2*work[k-1]) && !m_last_step_rejected ) { m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , static_cast<int>(m_current_k_opt)+1 ); new_h = h_opt[k]*static_cast<value_type>( m_cost[m_current_k_opt] ) / static_cast<value_type>( m_cost[k] ); } else new_h = h_opt[m_current_k_opt]; break; } else if( should_reject( error , k ) ) { reject = true; new_h = h_opt[m_current_k_opt]; break; } } if( k == m_current_k_opt+1 ) { // convergence at k_opt+1 ? if( error < 1.0 ) { //convergence reject = false; if( work[k-2] < KFAC2*work[k-1] ) m_current_k_opt = max BOOST_PREVENT_MACRO_SUBSTITUTION( 2 , static_cast<int>(m_current_k_opt)-1 ); if( (work[k] < KFAC2*work[m_current_k_opt]) && !m_last_step_rejected ) m_current_k_opt = min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<int>(m_k_max)-1 , static_cast<int>(k) ); new_h = h_opt[m_current_k_opt]; } else { reject = true; new_h = h_opt[m_current_k_opt]; } break; } } } if( !reject ) { //calculate dxdt for next step and dense output sys( out , dxdt_new , t+dt ); //prepare dense output value_type error = prepare_dense_output( m_k_final , in , dxdt , out , dxdt_new , dt ); if( error > static_cast<value_type>(10) ) // we are not as accurate for interpolation as for the steps { reject = true; new_h = dt * pow BOOST_PREVENT_MACRO_SUBSTITUTION( error , static_cast<value_type>(-1)/(2*m_k_final+2) ); } else { t += dt; } } //set next stepsize if( !m_last_step_rejected || (new_h < dt) ) dt = new_h; m_last_step_rejected = reject; if( reject ) return fail; else return success; } template< class StateType > void initialize( const StateType &x0 , const time_type &t0 , const time_type &dt0 ) { m_resizer.adjust_size( x0 , detail::bind( &controlled_error_bs_type::template resize_impl< StateType > , detail::ref( *this ) , detail::_1 ) ); boost::numeric::odeint::copy( x0 , get_current_state() ); m_t = t0; m_dt = dt0; reset(); } /* ======================================================= * the actual step method that should be called from outside (maybe make try_step private?) */ template< class System > std::pair< time_type , time_type > do_step( System system ) { const size_t max_count = 1000; if( m_first ) { typename odeint::unwrap_reference< System >::type &sys = system; sys( get_current_state() , get_current_deriv() , m_t ); } controlled_step_result res = fail; m_t_last = m_t; size_t count = 0; while( res == fail ) { res = try_step( system , get_current_state() , get_current_deriv() , m_t , get_old_state() , get_old_deriv() , m_dt ); m_first = false; if( count++ == max_count ) throw std::overflow_error( "bulirsch_stoer : too much iterations!"); } toggle_current_state(); return std::make_pair( m_t_last , m_t ); } /* performs the interpolation from a calculated step */ template< class StateOut > void calc_state( time_type t , StateOut &x ) const { do_interpolation( t , x ); } const state_type& current_state( void ) const { return get_current_state(); } time_type current_time( void ) const { return m_t; } const state_type& previous_state( void ) const { return get_old_state(); } time_type previous_time( void ) const { return m_t_last; } time_type current_time_step( void ) const { return m_dt; } /** \brief Resets the internal state of the stepper. */ void reset() { m_first = true; m_last_step_rejected = false; } template< class StateIn > void adjust_size( const StateIn &x ) { resize_impl( x ); m_midpoint.adjust_size(); } private: template< class StateInOut , class StateVector > void extrapolate( size_t k , StateVector &table , const value_matrix &coeff , StateInOut &xest , size_t order_start_index = 0 ) //polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf { static const value_type val1( 1.0 ); for( int j=k-1 ; j>0 ; --j ) { m_algebra.for_each3( table[j-1].m_v , table[j].m_v , table[j-1].m_v , typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][j + order_start_index] , -coeff[k + order_start_index][j + order_start_index] ) ); } m_algebra.for_each3( xest , table[0].m_v , xest , typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][0 + order_start_index] , -coeff[k + order_start_index][0 + order_start_index]) ); } template< class StateVector > void extrapolate_dense_out( size_t k , StateVector &table , const value_matrix &coeff , size_t order_start_index = 0 ) //polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf { // result is written into table[0] static const value_type val1( 1.0 ); for( int j=k ; j>1 ; --j ) { m_algebra.for_each3( table[j-1].m_v , table[j].m_v , table[j-1].m_v , typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][j + order_start_index - 1] , -coeff[k + order_start_index][j + order_start_index - 1] ) ); } m_algebra.for_each3( table[0].m_v , table[1].m_v , table[0].m_v , typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k + order_start_index][order_start_index] , -coeff[k + order_start_index][order_start_index]) ); } time_type calc_h_opt( time_type h , value_type error , size_t k ) const { BOOST_USING_STD_MIN(); BOOST_USING_STD_MAX(); using std::pow; value_type expo=1.0/(m_interval_sequence[k-1]); value_type facmin = pow BOOST_PREVENT_MACRO_SUBSTITUTION( STEPFAC3 , expo ); value_type fac; if (error == 0.0) fac=1.0/facmin; else { fac = STEPFAC2 / pow BOOST_PREVENT_MACRO_SUBSTITUTION( error / STEPFAC1 , expo ); fac = max BOOST_PREVENT_MACRO_SUBSTITUTION( facmin/STEPFAC4 , min BOOST_PREVENT_MACRO_SUBSTITUTION( 1.0/facmin , fac ) ); } return h*fac; } bool in_convergence_window( size_t k ) const { if( (k == m_current_k_opt-1) && !m_last_step_rejected ) return true; // decrease order only if last step was not rejected return ( (k == m_current_k_opt) || (k == m_current_k_opt+1) ); } bool should_reject( value_type error , size_t k ) const { if( k == m_current_k_opt-1 ) { const value_type d = m_interval_sequence[m_current_k_opt] * m_interval_sequence[m_current_k_opt+1] / (m_interval_sequence[0]*m_interval_sequence[0]); //step will fail, criterion 17.3.17 in NR return ( error > d*d ); } else if( k == m_current_k_opt ) { const value_type d = m_interval_sequence[m_current_k_opt+1] / m_interval_sequence[0]; return ( error > d*d ); } else return error > 1.0; } template< class StateIn1 , class DerivIn1 , class StateIn2 , class DerivIn2 > value_type prepare_dense_output( int k , const StateIn1 &x_start , const DerivIn1 &dxdt_start , const StateIn2 & /* x_end */ , const DerivIn2 & /*dxdt_end */ , time_type dt ) /* k is the order to which the result was approximated */ { /* compute the coefficients of the interpolation polynomial * we parametrize the interval t .. t+dt by theta = -1 .. 1 * we use 2k+3 values at the interval center theta=0 to obtain the interpolation coefficients * the values are x(t+dt/2) and the derivatives dx/dt , ... d^(2k+2) x / dt^(2k+2) at the midpoints * the derivatives are approximated via finite differences * all values are obtained from interpolation of the results from the increasing orders of the midpoint calls */ // calculate finite difference approximations to derivatives at the midpoint for( int j = 0 ; j<=k ; j++ ) { /* not working with boost units... */ const value_type d = m_interval_sequence[j] / ( static_cast<value_type>(2) * dt ); value_type f = 1.0; //factor 1/2 here because our interpolation interval has length 2 !!! for( int kappa = 0 ; kappa <= 2*j+1 ; ++kappa ) { calculate_finite_difference( j , kappa , f , dxdt_start ); f *= d; } if( j > 0 ) extrapolate_dense_out( j , m_mp_states , m_coeff ); } time_type d = dt/2; // extrapolate finite differences for( int kappa = 0 ; kappa<=2*k+1 ; kappa++ ) { for( int j=1 ; j<=(k-kappa/2) ; ++j ) extrapolate_dense_out( j , m_diffs[kappa] , m_coeff , kappa/2 ); // extrapolation results are now stored in m_diffs[kappa][0] // divide kappa-th derivative by kappa because we need these terms for dense output interpolation m_algebra.for_each1( m_diffs[kappa][0].m_v , typename operations_type::template scale< time_type >( static_cast<time_type>(d) ) ); d *= dt/(2*(kappa+2)); } // 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] // the error is just the highest order coefficient of the interpolation polynomial // this is because we use only the midpoint theta=0 as support for the interpolation (remember that theta = -1 .. 1) value_type error = 0.0; if( m_control_interpolation ) { boost::numeric::odeint::copy( m_diffs[2*k+1][0].m_v , m_err.m_v ); error = m_error_checker.error( m_algebra , x_start , dxdt_start , m_err.m_v , dt ); } return error; } template< class DerivIn > void calculate_finite_difference( size_t j , size_t kappa , value_type fac , const DerivIn &dxdt ) { const int m = m_interval_sequence[j]/2-1; if( kappa == 0) // no calculation required for 0th derivative of f { m_algebra.for_each2( m_diffs[0][j].m_v , m_derivs[j][m].m_v , typename operations_type::template scale_sum1< value_type >( fac ) ); } else { // calculate the index of m_diffs for this kappa-j-combination const int j_diffs = j - kappa/2; m_algebra.for_each2( m_diffs[kappa][j_diffs].m_v , m_derivs[j][m+kappa].m_v , typename operations_type::template scale_sum1< value_type >( fac ) ); value_type sign = -1.0; int c = 1; //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 for( int i = m+static_cast<int>(kappa)-2 ; i >= m-static_cast<int>(kappa) ; i -= 2 ) { if( i >= 0 ) { m_algebra.for_each3( m_diffs[kappa][j_diffs].m_v , m_diffs[kappa][j_diffs].m_v , m_derivs[j][i].m_v , typename operations_type::template scale_sum2< value_type , value_type >( 1.0 , sign * fac * boost::math::binomial_coefficient< value_type >( kappa , c ) ) ); } else { m_algebra.for_each3( m_diffs[kappa][j_diffs].m_v , m_diffs[kappa][j_diffs].m_v , dxdt , typename operations_type::template scale_sum2< value_type , value_type >( 1.0 , sign * fac ) ); } sign *= -1; ++c; } } } template< class StateOut > void do_interpolation( time_type t , StateOut &out ) const { // interpolation polynomial is defined for theta = -1 ... 1 // m_k_final is the number of order-iterations done for the last step - it governs the order of the interpolation polynomial const value_type theta = 2 * get_unit_value( (t - m_t_last) / (m_t - m_t_last) ) - 1; // we use only values at interval center, that is theta=0, for interpolation // our interpolation polynomial is thus of order 2k+2, hence we have 2k+3 terms boost::numeric::odeint::copy( m_mp_states[0].m_v , out ); // add remaining terms: x += a_1 theta + a2 theta^2 + ... + a_{2k} theta^{2k} value_type theta_pow( theta ); for( size_t i=0 ; i<=2*m_k_final+1 ; ++i ) { m_algebra.for_each3( out , out , m_diffs[i][0].m_v , typename operations_type::template scale_sum2< value_type >( static_cast<value_type>(1) , theta_pow ) ); theta_pow *= theta; } } /* Resizer methods */ template< class StateIn > bool resize_impl( const StateIn &x ) { bool resized( false ); resized |= adjust_size_by_resizeability( m_x1 , x , typename is_resizeable<state_type>::type() ); resized |= adjust_size_by_resizeability( m_x2 , x , typename is_resizeable<state_type>::type() ); resized |= adjust_size_by_resizeability( m_dxdt1 , x , typename is_resizeable<state_type>::type() ); resized |= adjust_size_by_resizeability( m_dxdt2 , x , typename is_resizeable<state_type>::type() ); resized |= adjust_size_by_resizeability( m_err , x , typename is_resizeable<state_type>::type() ); for( size_t i = 0 ; i < m_k_max ; ++i ) resized |= adjust_size_by_resizeability( m_table[i] , x , typename is_resizeable<state_type>::type() ); for( size_t i = 0 ; i < m_k_max+1 ; ++i ) resized |= adjust_size_by_resizeability( m_mp_states[i] , x , typename is_resizeable<state_type>::type() ); for( size_t i = 0 ; i < m_k_max+1 ; ++i ) for( size_t j = 0 ; j < m_derivs[i].size() ; ++j ) resized |= adjust_size_by_resizeability( m_derivs[i][j] , x , typename is_resizeable<deriv_type>::type() ); for( size_t i = 0 ; i < 2*m_k_max+1 ; ++i ) for( size_t j = 0 ; j < m_diffs[i].size() ; ++j ) resized |= adjust_size_by_resizeability( m_diffs[i][j] , x , typename is_resizeable<deriv_type>::type() ); return resized; } state_type& get_current_state( void ) { return m_current_state_x1 ? m_x1.m_v : m_x2.m_v ; } const state_type& get_current_state( void ) const { return m_current_state_x1 ? m_x1.m_v : m_x2.m_v ; } state_type& get_old_state( void ) { return m_current_state_x1 ? m_x2.m_v : m_x1.m_v ; } const state_type& get_old_state( void ) const { return m_current_state_x1 ? m_x2.m_v : m_x1.m_v ; } deriv_type& get_current_deriv( void ) { return m_current_state_x1 ? m_dxdt1.m_v : m_dxdt2.m_v ; } const deriv_type& get_current_deriv( void ) const { return m_current_state_x1 ? m_dxdt1.m_v : m_dxdt2.m_v ; } deriv_type& get_old_deriv( void ) { return m_current_state_x1 ? m_dxdt2.m_v : m_dxdt1.m_v ; } const deriv_type& get_old_deriv( void ) const { return m_current_state_x1 ? m_dxdt2.m_v : m_dxdt1.m_v ; } void toggle_current_state( void ) { m_current_state_x1 = ! m_current_state_x1; } default_error_checker< value_type, algebra_type , operations_type > m_error_checker; modified_midpoint_dense_out< state_type , value_type , deriv_type , time_type , algebra_type , operations_type , resizer_type > m_midpoint; bool m_control_interpolation; bool m_last_step_rejected; bool m_first; time_type m_t; time_type m_dt; time_type m_dt_last; time_type m_t_last; size_t m_current_k_opt; size_t m_k_final; algebra_type m_algebra; resizer_type m_resizer; wrapped_state_type m_x1 , m_x2; wrapped_deriv_type m_dxdt1 , m_dxdt2; wrapped_state_type m_err; bool m_current_state_x1; value_vector m_error; // errors of repeated midpoint steps and extrapolations int_vector m_interval_sequence; // stores the successive interval counts value_matrix m_coeff; int_vector m_cost; // costs for interval count state_vector_type m_table; // sequence of states for extrapolation //for dense output: state_vector_type m_mp_states; // sequence of approximations of x at distance center deriv_table_type m_derivs; // table of function values deriv_table_type m_diffs; // table of function values //wrapped_state_type m_a1 , m_a2 , m_a3 , m_a4; const value_type STEPFAC1 , STEPFAC2 , STEPFAC3 , STEPFAC4 , KFAC1 , KFAC2; }; /********** DOXYGEN **********/ /** * \class bulirsch_stoer_dense_out * \brief The Bulirsch-Stoer algorithm. * * The Bulirsch-Stoer is a controlled stepper that adjusts both step size * and order of the method. The algorithm uses the modified midpoint and * a polynomial extrapolation compute the solution. This class also provides * dense output facility. * * \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 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 ) * \brief Constructs the bulirsch_stoer class, including initialization of * the error bounds. * * \param eps_abs Absolute tolerance level. * \param eps_rel Relative tolerance level. * \param factor_x Factor for the weight of the state. * \param factor_dxdt Factor for the weight of the derivative. * \param control_interpolation Set true to additionally control the error of * the interpolation. */ /** * \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 ) * \brief Tries to perform one step. * * This method tries to do one step with step size dt. If the error estimate * is to large, the step is rejected and the method returns fail and the * step size dt is reduced. If the error estimate is acceptably small, the * step is performed, success is returned and dt might be increased to make * the steps as large as possible. This method also updates t if a step is * performed. Also, the internal order of the stepper is adjusted if required. * * \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. * \param dxdt The derivative of state. * \param t The value of the time. Updated if the step is successful. * \param out Used to store the result of the step. * \param dt The step size. Updated. * \return success if the step was accepted, fail otherwise. */ /** * \fn bulirsch_stoer_dense_out::initialize( const StateType &x0 , const time_type &t0 , const time_type &dt0 ) * \brief Initializes the dense output stepper. * * \param x0 The initial state. * \param t0 The initial time. * \param dt0 The initial time step. */ /** * \fn bulirsch_stoer_dense_out::do_step( System system ) * \brief Does one time step. This is the main method that should be used to * integrate an ODE with this stepper. * \note initialize has to be called before using this method to set the * initial conditions x,t and the stepsize. * \param system The system function to solve, hence the r.h.s. of the * ordinary differential equation. It must fulfill the Simple System concept. * \return Pair with start and end time of the integration step. */ /** * \fn bulirsch_stoer_dense_out::calc_state( time_type t , StateOut &x ) const * \brief Calculates the solution at an intermediate point within the last step * \param t The time at which the solution should be calculated, has to be * in the current time interval. * \param x The output variable where the result is written into. */ /** * \fn bulirsch_stoer_dense_out::current_state( void ) const * \brief Returns the current state of the solution. * \return The current state of the solution x(t). */ /** * \fn bulirsch_stoer_dense_out::current_time( void ) const * \brief Returns the current time of the solution. * \return The current time of the solution t. */ /** * \fn bulirsch_stoer_dense_out::previous_state( void ) const * \brief Returns the last state of the solution. * \return The last state of the solution x(t-dt). */ /** * \fn bulirsch_stoer_dense_out::previous_time( void ) const * \brief Returns the last time of the solution. * \return The last time of the solution t-dt. */ /** * \fn bulirsch_stoer_dense_out::current_time_step( void ) const * \brief Returns the current step size. * \return The current step size. */ /** * \fn bulirsch_stoer_dense_out::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. */ } } } #endif // BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_HPP_INCLUDED