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// Copyright Nick Thompson, 2017 // 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) // This implements the compactly supported cubic b spline algorithm described in // Kress, Rainer. "Numerical analysis, volume 181 of Graduate Texts in Mathematics." (1998). // Splines of compact support are faster to evaluate and are better conditioned than classical cubic splines. // Let f be the function we are trying to interpolate, and s be the interpolating spline. // The routine constructs the interpolant in O(N) time, and evaluating s at a point takes constant time. // The order of accuracy depends on the regularity of the f, however, assuming f is // four-times continuously differentiable, the error is of O(h^4). // In addition, we can differentiate the spline and obtain a good interpolant for f'. // The main restriction of this method is that the samples of f must be evenly spaced. // Look for barycentric rational interpolation for non-evenly sampled data. // Properties: // - s(x_j) = f(x_j) // - All cubic polynomials interpolated exactly #ifndef BOOST_MATH_INTERPOLATORS_CUBIC_B_SPLINE_HPP #define BOOST_MATH_INTERPOLATORS_CUBIC_B_SPLINE_HPP #include #include BOOST_MATH_HEADER_DEPRECATED(""); namespace boost{ namespace math{ template class cubic_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. // f[0] = f(a), f[length -1] = b, step_size = (b - a)/(length -1). template cubic_b_spline(const BidiIterator f, BidiIterator end_p, Real left_endpoint, Real step_size, Real left_endpoint_derivative = std::numeric_limits::quiet_NaN(), Real right_endpoint_derivative = std::numeric_limits::quiet_NaN()); cubic_b_spline(const Real* const f, size_t length, Real left_endpoint, Real step_size, Real left_endpoint_derivative = std::numeric_limits::quiet_NaN(), Real right_endpoint_derivative = std::numeric_limits::quiet_NaN()); cubic_b_spline() = default; Real operator()(Real x) const; Real prime(Real x) const; Real double_prime(Real x) const; private: std::shared_ptr> m_imp; }; template cubic_b_spline::cubic_b_spline(const Real* const f, size_t length, Real left_endpoint, Real step_size, Real left_endpoint_derivative, Real right_endpoint_derivative) : m_imp(std::make_shared>(f, f + length, left_endpoint, step_size, left_endpoint_derivative, right_endpoint_derivative)) { } template template cubic_b_spline::cubic_b_spline(BidiIterator f, BidiIterator end_p, Real left_endpoint, Real step_size, Real left_endpoint_derivative, Real right_endpoint_derivative) : m_imp(std::make_shared>(f, end_p, left_endpoint, step_size, left_endpoint_derivative, right_endpoint_derivative)) { } template Real cubic_b_spline::operator()(Real x) const { return m_imp->operator()(x); } template Real cubic_b_spline::prime(Real x) const { return m_imp->prime(x); } template Real cubic_b_spline::double_prime(Real x) const { return m_imp->double_prime(x); } }} #endif