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Quintic Hermite interpolation


#include <boost/math/interpolators/quintic_hermite.hpp>

namespace boost::math::interpolators {

template<class RandomAccessContainer>
class quintic_hermite {
    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 {
    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 {
    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;


Quintic Hermite Interpolation

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 =;

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.

Complexity and Performance

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)