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/*
[auto_generated]
boost/numeric/odeint/stepper/bulirsch_stoer.hpp
[begin_description]
Implementation of the Burlish-Stoer method. As described in
Ernst Hairer, Syvert Paul Norsett, Gerhard Wanner
Solving Ordinary Differential Equations I. Nonstiff Problems.
Springer Series in Comput. Mathematics, Vol. 8, Springer-Verlag 1987, Second revised edition 1993.
[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_HPP_INCLUDED
#define BOOST_NUMERIC_ODEINT_STEPPER_BULIRSCH_STOER_HPP_INCLUDED
#include <iostream>
#include <algorithm>
#include <boost/config.hpp> // for min/max guidelines
#include <boost/numeric/odeint/util/bind.hpp>
#include <boost/numeric/odeint/util/unwrap_reference.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/numeric/odeint/util/detail/less_with_sign.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 {
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;
#ifndef DOXYGEN_SKIP
typedef state_wrapper< state_type > wrapped_state_type;
typedef state_wrapper< deriv_type > wrapped_deriv_type;
typedef controlled_stepper_tag stepper_category;
typedef bulirsch_stoer< 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_table_type;
#endif //DOXYGEN_SKIP
const static size_t m_k_max = 8;
bulirsch_stoer(
value_type eps_abs = 1E-6 , value_type eps_rel = 1E-6 ,
value_type factor_x = 1.0 , value_type factor_dxdt = 1.0 ,
time_type max_dt = static_cast<time_type>(0))
: m_error_checker( eps_abs , eps_rel , factor_x, factor_dxdt ) , m_midpoint() ,
m_last_step_rejected( false ) , m_first( true ) ,
m_max_dt(max_dt) ,
m_interval_sequence( m_k_max+1 ) ,
m_coeff( m_k_max+1 ) ,
m_cost( m_k_max+1 ) ,
m_table( m_k_max ) ,
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();
/* initialize sequence of stage numbers and work */
for( unsigned short i = 0; i < m_k_max+1; i++ )
{
m_interval_sequence[i] = 2 * (i+1);
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( static_cast<value_type>( 1 ) , min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<value_type>( m_k_max-1 ) , logfact ));
*/
}
}
/*
* Version 1 : try_step( sys , x , t , dt )
*
* The overloads are needed to solve the forwarding problem
*/
template< class System , class StateInOut >
controlled_step_result try_step( System system , StateInOut &x , time_type &t , time_type &dt )
{
return try_step_v1( system , x , t, dt );
}
/**
* \brief Second version to solve the forwarding problem, can be used with Boost.Range as StateInOut.
*/
template< class System , class StateInOut >
controlled_step_result try_step( System system , const StateInOut &x , time_type &t , time_type &dt )
{
return try_step_v1( system , x , t, dt );
}
/*
* Version 2 : try_step( sys , x , dxdt , t , dt )
*
* this version does not solve the forwarding problem, boost.range can not be used
*/
template< class System , class StateInOut , class DerivIn >
controlled_step_result try_step( System system , StateInOut &x , const DerivIn &dxdt , time_type &t , time_type &dt )
{
m_xnew_resizer.adjust_size( x , detail::bind( &controlled_error_bs_type::template resize_m_xnew< StateInOut > , detail::ref( *this ) , detail::_1 ) );
controlled_step_result res = try_step( system , x , dxdt , t , m_xnew.m_v , dt );
if( res == success )
{
boost::numeric::odeint::copy( m_xnew.m_v , x );
}
return res;
}
/*
* Version 3 : try_step( sys , in , t , out , dt )
*
* this version does not solve the forwarding problem, boost.range can not be used
*/
template< class System , class StateIn , class StateOut >
typename boost::disable_if< boost::is_same< StateIn , time_type > , controlled_step_result >::type
try_step( System system , const StateIn &in , time_type &t , StateOut &out , time_type &dt )
{
typename odeint::unwrap_reference< System >::type &sys = system;
m_dxdt_resizer.adjust_size( in , detail::bind( &controlled_error_bs_type::template resize_m_dxdt< StateIn > , detail::ref( *this ) , detail::_1 ) );
sys( in , m_dxdt.m_v , t );
return try_step( system , in , m_dxdt.m_v , t , out , dt );
}
/*
* Full version : try_step( sys , in , dxdt_in , t , out , dt )
*
* contains the actual implementation
*/
template< class System , class StateIn , class DerivIn , class StateOut >
controlled_step_result try_step( System system , const StateIn &in , const DerivIn &dxdt , time_type &t , StateOut &out , time_type &dt )
{
if( m_max_dt != static_cast<time_type>(0) && detail::less_with_sign(m_max_dt, dt, dt) )
{
// given step size is bigger then max_dt
// set limit and return fail
dt = m_max_dt;
return fail;
}
BOOST_USING_STD_MIN();
BOOST_USING_STD_MAX();
static const value_type val1( 1.0 );
if( m_resizer.adjust_size( in , detail::bind( &controlled_error_bs_type::template resize_impl< StateIn > , detail::ref( *this ) , detail::_1 ) ) )
{
reset(); // system resized -> reset
}
if( dt != m_dt_last )
{
reset(); // step size changed from outside -> reset
}
bool reject( true );
time_vector h_opt( m_k_max+1 );
inv_time_vector work( m_k_max+1 );
time_type new_h = dt;
/* m_current_k_opt is the estimated current optimal stage number */
for( size_t k = 0 ; k <= m_current_k_opt+1 ; k++ )
{
/* the stage counts are stored in m_interval_sequence */
m_midpoint.set_steps( m_interval_sequence[k] );
if( k == 0 )
{
m_midpoint.do_step( system , in , dxdt , t , out , dt );
/* the first step, nothing more to do */
}
else
{
m_midpoint.do_step( system , in , dxdt , t , m_table[k-1].m_v , dt );
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];
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];
new_h *= 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];
new_h *= m_cost[m_current_k_opt]/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 )
{
t += dt;
}
if( !m_last_step_rejected || boost::numeric::odeint::detail::less_with_sign(new_h, dt, dt) )
{
// limit step size
if( m_max_dt != static_cast<time_type>(0) )
{
new_h = detail::min_abs(m_max_dt, new_h);
}
m_dt_last = new_h;
dt = new_h;
}
m_last_step_rejected = reject;
m_first = false;
if( reject )
return fail;
else
return success;
}
/** \brief Resets the internal state of the stepper */
void reset()
{
m_first = true;
m_last_step_rejected = false;
}
/* Resizer methods */
template< class StateIn >
void adjust_size( const StateIn &x )
{
resize_m_dxdt( x );
resize_m_xnew( x );
resize_impl( x );
m_midpoint.adjust_size( x );
}
private:
template< class StateIn >
bool resize_m_dxdt( const StateIn &x )
{
return adjust_size_by_resizeability( m_dxdt , x , typename is_resizeable<deriv_type>::type() );
}
template< class StateIn >
bool resize_m_xnew( const StateIn &x )
{
return adjust_size_by_resizeability( m_xnew , x , typename is_resizeable<state_type>::type() );
}
template< class StateIn >
bool resize_impl( const StateIn &x )
{
bool resized( false );
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() );
resized |= adjust_size_by_resizeability( m_err , x , typename is_resizeable<state_type>::type() );
return resized;
}
template< class System , class StateInOut >
controlled_step_result try_step_v1( System system , StateInOut &x , time_type &t , time_type &dt )
{
typename odeint::unwrap_reference< System >::type &sys = system;
m_dxdt_resizer.adjust_size( x , detail::bind( &controlled_error_bs_type::template resize_m_dxdt< StateInOut > , detail::ref( *this ) , detail::_1 ) );
sys( x , m_dxdt.m_v ,t );
return try_step( system , x , m_dxdt.m_v , t , dt );
}
template< class StateInOut >
void extrapolate( size_t k , state_table_type &table , const value_matrix &coeff , StateInOut &xest )
/* polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf
uses the obtained intermediate results to extrapolate to dt->0
*/
{
static const value_type val1 = static_cast< value_type >( 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][j] , -coeff[k][j] ) );
}
m_algebra.for_each3( xest , table[0].m_v , xest ,
typename operations_type::template scale_sum2< value_type , value_type >( val1 + coeff[k][0] , -coeff[k][0]) );
}
time_type calc_h_opt( time_type h , value_type error , size_t k ) const
/* calculates the optimal step size for a given error and stage number */
{
BOOST_USING_STD_MIN();
BOOST_USING_STD_MAX();
using std::pow;
value_type expo( 1.0/(2*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( static_cast<value_type>(facmin/STEPFAC4) , min BOOST_PREVENT_MACRO_SUBSTITUTION( static_cast<value_type>(1.0/facmin) , fac ) );
}
return h*fac;
}
controlled_step_result set_k_opt( size_t k , const inv_time_vector &work , const time_vector &h_opt , time_type &dt )
/* calculates the optimal stage number */
{
if( k == 1 )
{
m_current_k_opt = 2;
return success;
}
if( (work[k-1] < KFAC1*work[k]) || (k == m_k_max) )
{ // order decrease
m_current_k_opt = k-1;
dt = h_opt[ m_current_k_opt ];
return success;
}
else if( (work[k] < KFAC2*work[k-1]) || m_last_step_rejected || (k == m_k_max-1) )
{ // same order - also do this if last step got rejected
m_current_k_opt = k;
dt = h_opt[ m_current_k_opt ];
return success;
}
else
{ // order increase - only if last step was not rejected
m_current_k_opt = k+1;
dt = h_opt[ m_current_k_opt-1 ] * m_cost[ m_current_k_opt ] / m_cost[ m_current_k_opt-1 ] ;
return success;
}
}
bool in_convergence_window( size_t k ) const
{
if( (k == m_current_k_opt-1) && !m_last_step_rejected )
return true; // decrease stepsize 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] / m_interval_sequence[0];
return ( error > d*d );
} else
return error > 1.0;
}
default_error_checker< value_type, algebra_type , operations_type > m_error_checker;
modified_midpoint< state_type , value_type , deriv_type , time_type , algebra_type , operations_type , resizer_type > m_midpoint;
bool m_last_step_rejected;
bool m_first;
time_type m_dt_last;
time_type m_t_last;
time_type m_max_dt;
size_t m_current_k_opt;
algebra_type m_algebra;
resizer_type m_dxdt_resizer;
resizer_type m_xnew_resizer;
resizer_type m_resizer;
wrapped_state_type m_xnew;
wrapped_state_type m_err;
wrapped_deriv_type m_dxdt;
int_vector m_interval_sequence; // stores the successive interval counts
value_matrix m_coeff;
int_vector m_cost; // costs for interval count
state_table_type m_table; // sequence of states for extrapolation
value_type STEPFAC1 , STEPFAC2 , STEPFAC3 , STEPFAC4 , KFAC1 , KFAC2;
};
/******** DOXYGEN ********/
/**
* \class bulirsch_stoer
* \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.
*
* \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::bulirsch_stoer( value_type eps_abs , value_type eps_rel , value_type factor_x , value_type factor_dxdt )
* \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.
*/
/**
* \fn bulirsch_stoer::try_step( System system , StateInOut &x , time_type &t , 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 x The state of the ODE which should be solved. Overwritten if
* the step is successful.
* \param t The value of the time. Updated if the step is successful.
* \param dt The step size. Updated.
* \return success if the step was accepted, fail otherwise.
*/
/**
* \fn bulirsch_stoer::try_step( System system , StateInOut &x , const DerivIn &dxdt , time_type &t , 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 x The state of the ODE which should be solved. Overwritten if
* the step is successful.
* \param dxdt The derivative of state.
* \param t The value of the time. Updated if the step is successful.
* \param dt The step size. Updated.
* \return success if the step was accepted, fail otherwise.
*/
/**
* \fn bulirsch_stoer::try_step( System system , const StateIn &in , time_type &t , StateOut &out , time_type &dt )
* \brief Tries to perform one step.
*
* \note This method is disabled if state_type=time_type to avoid ambiguity.
*
* 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 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::try_step( System system , const StateIn &in , const DerivIn &dxdt , time_type &t , StateOut &out , 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::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
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