[/
Copyright (c) 2020 Nick Thompson
Use, modification and distribution are subject to 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)
]
[section:quintic_hermite Quintic Hermite interpolation]
[heading Synopsis]
``
#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;
}
``
[heading 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 /C/[super 2] and its evaluation has [bigo](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);
[$../graphs/quintic_sine_approximation.svg]
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.
[heading 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)
```
[endsect]
[/section:quintic_hermite]