What is the recommended way to calculate a multidimensional integral using boost odeint with high accuracy? The following code integrates f=x*y from -1 to 2 but the error relative to an analytic solution is over 1 % (gcc 4.8.2, -std=c++0x):
#include "array"
#include "boost/numeric/odeint.hpp"
#include "iostream"
using integral_type = std::array<double, 1>;
int main() {
integral_type outer_integral{0};
double current_x = 0;
boost::numeric::odeint::integrate(
[&](
const integral_type&,
integral_type& dfdx,
const double x
) {
integral_type inner_integral{0};
boost::numeric::odeint::integrate(
[¤t_x](
const integral_type&,
integral_type& dfdy,
const double y
) {
dfdy[0] = current_x * y;
},
inner_integral,
-1.0,
2.0,
1e-3
);
dfdx[0] = inner_integral[0];
},
outer_integral,
-1.0,
2.0,
1e-3,
[¤t_x](const integral_type&, const double x) {
current_x = x; // update x in inner integrator
}
);
std::cout
<< "Exact: 2.25, numerical: "
<< outer_integral[0]
<< std::endl;
return 0;
}
prints:
Exact: 2.25, numerical: 2.19088
Should I just use more stringent stopping condition in the inner integrals or is there a faster/more accurate way to do this? Thanks!
Firstly, there might be better numerical methods to compute high-dimensional integrals than an ODE scheme (http://en.wikipedia.org/wiki/Numerical_integration), but then I think this is a neat application of odeint, I find.
However, the problem with your code is that it assumes that you use an observer in the outer integrate to update the x-value for the inner integration. However, the integrate function uses a dense-output stepper internally which means that the actual time steps and the observer calls are not in synchrony. So the x for the inner integration is not updated at the right moments. A quick fix would be to use an integrate_const with a runge_kutta4 stepper, which uses constant step size and ensure that the observer calls, and thus x-updates, are called after each step of the outer loop. However, this is a bit of a hack relying on some internal details of the integrate routines. A better way would be to design the program in such a way that the state is indeed 2-dimensional, but where each integration works only on one of the two variables.
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