Boost C++ Libraries

Synopsis

    #include <boost/math/quadrature/tanh_sinh.hpp>
    #include <boost/math/quadrature/exp_sinh.hpp>
    #include <boost/math/quadrature/sinh_sinh.hpp>

    namespace boost{ namespace math{ namespace quadrature {

    template<class Real>
    class tanh_sinh
    {
    public:
        tanh_sinh(size_t max_refinements = 15, const Real& min_complement = tools::min_value<Real>() * 4)

        template<class F>
        auto integrate(const F f, Real a, Real b,
                       Real tolerance = tools::root_epsilon<Real>(),
                       Real* error = nullptr,
                       Real* L1 = nullptr,
                       std::size_t* levels = nullptr)->decltype(std::declval<F>()(std::declval<Real>())) const;

        template<class F>
        auto integrate(const F f, Real
                       tolerance = tools::root_epsilon<Real>(),
                       Real* error = nullptr,
                       Real* L1 = nullptr,
                       std::size_t* levels = nullptr)->decltype(std::declval<F>()(std::declval<Real>())) const;

    };

    template<class Real>
    class exp_sinh
    {
    public:
        exp_sinh(size_t max_refinements = 9);

        template<class F>
        auto integrate(const F f, Real a, Real b,
                       Real tol = sqrt(std::numeric_limits<Real>::epsilon()),
                       Real* error = nullptr,
                       Real* L1 = nullptr,
                       size_t* levels = nullptr)->decltype(std::declval<F>()(std::declval<Real>())) const;
        template<class F>
        auto integrate(const F f,
                       Real tol = sqrt(std::numeric_limits<Real>::epsilon()),
                       Real* error = nullptr,
                       Real* L1 = nullptr,
                       size_t* levels = nullptr)->decltype(std::declval<F>()(std::declval<Real>())) const;
    };

    template<class Real>
    class sinh_sinh
    {
    public:
        sinh_sinh(size_t max_refinements = 9);

        template<class F>
        auto integrate(const F f,
                       Real tol = sqrt(std::numeric_limits<Real>::epsilon()),
                       Real* error = nullptr,
                       Real* L1 = nullptr,
                       size_t* levels = nullptr)->decltype(std::declval<F>()(std::declval<Real>())) const;
    };

}}}

These three integration routines provide robust general purpose quadrature, each having a "native" range over which quadrature is performed. For example, the sinh_sinh quadrature integrates over the entire real line, the tanh_sinh over (-1, 1), and the exp_sinh over (0, ∞). The latter integrators also have auxiliary ranges which are handled via a change of variables on the function being integrated, so that the tanh_sinh can handle integration over (a, b), and exp_sinh over /(a, ∞) and(-∞, b)/.

Like the other quadrature routines in Boost, these routines support both real and complex-valued integrands.

The integrate methods which do not specify a range always integrate over the native range of the method, and generally are the most efficient and produce the smallest code, on the other hand the methods which do specify the bounds of integration are the most general, and use argument transformations which are generally very robust. The following table summarizes the ranges supported by each method: