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+/*
+ * abm_precision.cpp
+ *
+ * example to check the order of the multi-step methods
+ *
+ * Copyright 2009-2013 Karsten Ahnert
+ * Copyright 2009-2013 Mario Mulansky
+ *
+ * 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)
+ */
+
+#include <iostream>
+#include <cmath>
+
+#include <boost/array.hpp>
+#include <boost/numeric/odeint.hpp>
+
+using namespace boost::numeric::odeint;
+
+const int Steps = 4;
+
+typedef double value_type;
+
+typedef boost::array< double , 2 > state_type;
+
+typedef runge_kutta_fehlberg78<state_type> initializing_stepper_type;
+typedef adams_bashforth_moulton< Steps , state_type > stepper_type;
+//typedef adams_bashforth< Steps , state_type > stepper_type;
+
+// harmonic oscillator, analytic solution x[0] = sin( t )
+struct osc
+{
+ void operator()( const state_type &x , state_type &dxdt , const double t ) const
+ {
+ dxdt[0] = x[1];
+ dxdt[1] = -x[0];
+ }
+};
+
+int main()
+{
+ stepper_type stepper;
+ initializing_stepper_type init_stepper;
+ const int o = stepper.order()+1; //order of the error is order of approximation + 1
+
+ const state_type x0 = {{ 0.0 , 1.0 }};
+ state_type x1 = x0;
+ double t = 0.0;
+ double dt = 0.25;
+ // initialization, does a number of steps already to fill internal buffer, t is increased
+ // we use the rk78 as initializing stepper
+ stepper.initialize( boost::ref(init_stepper) , osc() , x1 , t , dt );
+ // do a number of steps to fill the buffer with results from adams bashforth
+ for( size_t n=0 ; n < stepper.steps ; ++n )
+ {
+ stepper.do_step( osc() , x1 , t , dt );
+ t += dt;
+ }
+ double A = std::sqrt( x1[0]*x1[0] + x1[1]*x1[1] );
+ double phi = std::asin(x1[0]/A) - t;
+ // now we do the actual step
+ stepper.do_step( osc() , x1 , t , dt );
+ // only examine the error of the adams-bashforth-moulton step, not the initialization
+ const double f = 2.0 * std::abs( A*sin(t+dt+phi) - x1[0] ) / std::pow( dt , o ); // upper bound
+
+ std::cout << "# " << o << " , " << f << std::endl;
+
+ /* as long as we have errors above machine precision */
+ while( f*std::pow( dt , o ) > 1E-16 )
+ {
+ x1 = x0;
+ t = 0.0;
+ stepper.initialize( boost::ref(init_stepper) , osc() , x1 , t , dt );
+ A = std::sqrt( x1[0]*x1[0] + x1[1]*x1[1] );
+ phi = std::asin(x1[0]/A) - t;
+ // now we do the actual step
+ stepper.do_step( osc() , x1 , t , dt );
+ // only examine the error of the adams-bashforth-moulton step, not the initialization
+ std::cout << dt << '\t' << std::abs( A*sin(t+dt+phi) - x1[0] ) << std::endl;
+ dt *= 0.5;
+ }
+}