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#include <boost/math/interpolators/quintic_hermite.hpp> namespace boost::math::interpolators { template<class RandomAccessContainer> class quintic_hermite { public: using Real = typename RandomAccessContainer::value_type; quintic_hermite(RandomAccessContainer && x, RandomAccessContainer && y, RandomAccessContainer && dydx, RandomAccessContainer && d2ydx2) inline Real operator()(Real x) const; inline Real prime(Real x) const; inline Real double_prime(Real x) const; std::pair<Real, Real> domain() const; friend std::ostream& operator<<(std::ostream & os, const quintic_hermite & m); void push_back(Real x, Real y, Real dydx, Real d2ydx2); }; template<class RandomAccessContainer> class cardinal_quintic_hermite { public: using Real = typename RandomAccessContainer::value_type; cardinal_quintic_hermite(RandomAccessContainer && y, RandomAccessContainer && dydx, RandomAccessContainer && d2ydx2, Real x0, Real dx); inline Real operator()(Real x) const; inline Real prime(Real x) const; inline Real double_prime(Real x) const; std::pair<Real, Real> domain() const; }; template<class RandomAccessContainer> class cardinal_quintic_hermite_aos { public: using Point = typename RandomAccessContainer::value_type; using Real = typename Point::value_type; cardinal_quintic_hermite_aos(RandomAccessContainer && data, Real x0, Real dx) inline Real operator()(Real x) const; inline Real prime(Real x) const; inline Real double_prime(Real x) const; std::pair<Real, Real> domain() const; }
The quintic Hermite interpolator takes a list of possibly non-uniformly spaced abscissas, ordinates, and their velocities and accelerations which are used to construct a quintic interpolating polynomial between segments. This is useful for taking solution skeletons from ODE steppers and turning them into a continuous function, provided that the right-hand side f(x, y) is differentiable along the solution path. The interpolant is C2 and its evaluation has 𝑶(log(N)) complexity. An example usage is as follows:
std::vector<double> x{1,2,3, 4, 5, 6}; std::vector<double> y{7,8,9,10,11,12}; std::vector<double> dydx{1,1,1,1,1,1}; std::vector<double> d2ydx2{0,0,0,0,0,0}; using boost::math::interpolators::quintic_hermite; auto spline = quintic_hermite(std::move(x), std::move(y), std::move(dydx), std::move(d2ydx2)); // evaluate at a point: double z = spline(3.4); // evaluate derivative at a point: double zprime = spline.prime(3.4);
Periodically, it is helpful to see what data the interpolator has. This can be achieved via
std::cout << spline << "\n";
Note that the interpolator is pimpl'd, so that copying the class is cheap,
and hence it can be shared between threads. (The call operator and .prime()
are threadsafe.)
The interpolator can be updated in constant time. Hence we can use boost::circular_buffer
to do real-time interpolation.
#include <boost/circular_buffer.hpp> ... boost::circular_buffer<double> initial_x{1,2,3,4}; boost::circular_buffer<double> initial_y{4,5,6,7}; boost::circular_buffer<double> initial_dydx{1,1,1,1}; boost::circular_buffer<double> initial_d2ydx2{0,0,0,0}; auto circular_akima = quintic_hermite(std::move(initial_x), std::move(initial_y), std::move(initial_dydx), std::move(initial_d2ydx2)); // interpolate via call operation: double y = circular_akima(3.5); // add new data: circular_akima.push_back(5, 8, 1, 0); // interpolate at 4.5: y = circular_akima(4.5);
For equispaced data, we can use cardinal_quintic_hermite
or cardinal_quintic_hermite_aos
to get constant-time evaluation. This is useful in memory-constrained or performance
critical applications where data is equispaced.
The following google benchmark demonstrates the cost of the call operator for this interpolator:
Run on (16 X 4300 MHz CPU s) CPU Caches: L1 Data 32K (x8) L1 Instruction 32K (x8) L2 Unified 1024K (x8) L3 Unified 11264K (x1) Load Average: 0.92, 0.64, 0.35 -------------------------------------------------- Benchmark Time -------------------------------------------------- QuinticHermite<double>/8 8.14 ns QuinticHermite<double>/16 8.69 ns QuinticHermite<double>/32 9.42 ns QuinticHermite<double>/64 9.90 ns QuinticHermite<double>/128 10.4 ns QuinticHermite<double>/256 10.9 ns QuinticHermite<double>/512 11.6 ns QuinticHermite<double>/1024 12.3 ns QuinticHermite<double>/2048 12.8 ns QuinticHermite<double>/4096 13.6 ns QuinticHermite<double>/8192 14.6 ns QuinticHermite<double>/16384 15.5 ns QuinticHermite<double>/32768 17.4 ns QuinticHermite<double>/65536 18.5 ns QuinticHermite<double>/131072 18.8 ns QuinticHermite<double>/262144 19.8 ns QuinticHermite<double>/524288 20.5 ns QuinticHermite<double>/1048576 21.6 ns QuinticHermite<double>/2097152 22.5 ns QuinticHermite<double>/4194304 24.2 ns QuinticHermite<double>/8388608 26.6 ns QuinticHermite<double>/16777216 26.7 ns QuinticHermite<double>_BigO 1.14 lgN CardinalQuinticHermite<double>/256 5.22 ns CardinalQuinticHermite<double>/512 5.21 ns CardinalQuinticHermite<double>/1024 5.21 ns CardinalQuinticHermite<double>/2048 5.32 ns CardinalQuinticHermite<double>/4096 5.33 ns CardinalQuinticHermite<double>/8192 5.50 ns CardinalQuinticHermite<double>/16384 5.74 ns CardinalQuinticHermite<double>/32768 7.74 ns CardinalQuinticHermite<double>/65536 10.6 ns CardinalQuinticHermite<double>/131072 10.7 ns CardinalQuinticHermite<double>/262144 10.6 ns CardinalQuinticHermite<double>/524288 10.5 ns CardinalQuinticHermite<double>/1048576 10.6 ns CardinalQuinticHermite<double>_BigO 7.57 (1) CardinalQuinticHermiteAOS<double>/256 5.27 ns CardinalQuinticHermiteAOS<double>/512 5.26 ns CardinalQuinticHermiteAOS<double>/1024 5.26 ns CardinalQuinticHermiteAOS<double>/2048 5.28 ns CardinalQuinticHermiteAOS<double>/4096 5.30 ns CardinalQuinticHermiteAOS<double>/8192 5.41 ns CardinalQuinticHermiteAOS<double>/16384 5.89 ns CardinalQuinticHermiteAOS<double>/32768 5.97 ns CardinalQuinticHermiteAOS<double>/65536 5.96 ns CardinalQuinticHermiteAOS<double>/131072 5.92 ns CardinalQuinticHermiteAOS<double>/262144 5.94 ns CardinalQuinticHermiteAOS<double>/524288 5.96 ns CardinalQuinticHermiteAOS<double>/1048576 5.93 ns CardinalQuinticHermiteAOS<double>_BigO 5.64 (1)