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- [/
- Copyright (c) 2019 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:cardinal_quintic_b Cardinal Quintic B-spline interpolation]
- [heading Synopsis]
- ``
- #include <boost/math/interpolators/cardinal_quintic_b_spline.hpp>
- ``
- namespace boost{ namespace math{ namespace interpolators {
- template <class Real>
- class cardinal_quintic_b_spline
- {
- public:
- // If you don't know the value of the derivative at the endpoints, leave them as NaNs and the routine will estimate them.
- // y[0] = y(a), y[n - 1] = y(b), step_size = (b - a)/(n -1).
- cardinal_quintic_b_spline(const Real* const y,
- size_t n,
- Real t0 /* initial time, left endpoint */,
- Real h /*spacing, stepsize*/,
- std::pair<Real, Real> left_endpoint_derivatives = {std::numeric_limits<Real>::quiet_NaN(), std::numeric_limit<Real>::quiet_NaN()},
- std::pair<Real, Real> right_endpoint_derivatives = {std::numeric_limits<Real>::quiet_NaN(), std::numeric_limit<Real>::quiet_NaN()})
- cardinal_quintic_b_spline(std::vector<Real> const & y,
- Real t0 /* initial time, left endpoint */,
- Real h /*spacing, stepsize*/,
- std::pair<Real, Real> left_endpoint_derivatives = {std::numeric_limits<Real>::quiet_NaN(), std::numeric_limit<Real>::quiet_NaN()},
- std::pair<Real, Real> right_endpoint_derivatives = {std::numeric_limits<Real>::quiet_NaN(), std::numeric_limit<Real>::quiet_NaN()})
- Real operator()(Real t) const;
- Real prime(Real t) const;
- Real double_prime(Real t) const;
- };
- }}}
- [heading Cardinal Quintic B-Spline Interpolation]
- The cardinal quintic B-spline interpolator is very nearly the same as the cubic B-spline interpolator,
- with the modification that the basis functions are constructed by convolving a box function with itself five times,
- rather than three times as is done with the cubic B-spline.
- The basis functions of the quintic B-spline interpolator are more smooth than the cubic /B/-spline interpolator, and hence this is very useful for computing second derivatives.
- For example, the second derivative of the cubic spline interpolator is a piecewise linear function, whereas the second derivative of the quintic /B/-spline is a cubic spline.
- The graph of the second derivative of the quintic /B/-spline is therefore more visually appealing, though whether it is in fact more accurate depends on the smoothness of your data.
- And example usage is as follows:
- #include <boost/math/interpolators/cardinal_quintic_b_spline.hpp>
- using boost::math::interpolators::cardinal_quintic_b_spline;
- std::vector<double> v(512);
- // fill v with data . . .
- double t0 = 0; // initial time
- double h = 0.125; // spacing
- std::pair<double, double> left_endpoint_derivatives{first_derivative_at_t0, second_derivative_at_t0};
- std::pair<double, double> right_endpoint_derivatives{first_derivative_at_tf, second_derivative_at_tf};
- auto qs = cardinal_quintic_b_spline<double>(v, t0, h, left_endpoint_derivatives, right_endpoint_derivatives);
- // Evaluate the interpolant at a point:
- double y = qs(0.1);
- // Evaluate the derivative of the interpolant:
- double yp = qs.prime(0.1);
- // Evaluate the second derivative of the interpolant:
- double ypp = qs.double_prime(0.1);
- This routine will estimate the endpoint derivatives if they are not provided.
- /Try to avoid this if possible./
- The endpoint derivatives must be evaluated by finite differences and this is not robust again perturbations in the data.
- So if you have some way of knowing the endpoint derivatives, make sure to provide them.
- [heading References]
- Cox, Maurice G. ['Numerical methods for the interpolation and approximation of data by spline functions.] Diss. City, University of London, 1975.
- [endsect] [/section:cardinal_quintic_b]
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