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## PCHIP interpolation

#### Synopsis

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

    namespace boost::math::interpolators {

    template <class RandomAccessContainer>
    class pchip
    {
    public:

        using Real = RandomAccessContainer::value_type;

        pchip(RandomAccessContainer&& abscissas, RandomAccessContainer&& ordinates,
              Real left_endpoint_derivative = std::numeric_limits<Real>::quiet_NaN(),
              Real right_endpoint_derivative = std::numeric_limits<Real>::quiet_NaN());

        Real operator()(Real x) const;

        Real prime(Real x) const;

        void push_back(Real x, Real y);

        friend std::ostream& operator<<(std::ostream & os, const pchip & m);
    };

    } // namespaces

#### PCHIP Interpolation

The PCHIP interpolant takes non-equispaced data and interpolates between them via cubic Hermite polynomials whose slopes are chosen so that the resulting interpolant is monotonic; see Fritsch and Carlson for details. The interpolant is C1 and evaluation has 𝑶(log(N)) complexity. An example usage is as follows:

    std::vector<double> x{1, 5, 9 , 12};
    std::vector<double> y{8,17, 4, -3};
    using boost::math::interpolators::pchip;
    auto spline = pchip(std::move(x), std::move(y));
    // 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, and the slopes it has chosen. 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; `push_back` is not.)

This interpolant 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};
    auto circular_pchip = pchip(std::move(initial_x), std::move(initial_y));
    // interpolate via call operation:
    double y = circular_pchip(3.5);
    // add new data:
    circular_pchip.push_back(5, 8);
    // interpolate at 4.5:
    y = circular_pchip(4.5);

#### Complexity and Performance

This interpolator chooses the slopes and forwards data to the cubic Hermite interpolator, so the performance is stated in the documentation for `cubic_hermite.hpp`.