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// Copyright Nick Thompson, 2020 // 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) // See: https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/ // And: https://doi.org/10.1145/321607.321609 #ifndef BOOST_MATH_INTERPOLATORS_MAKIMA_HPP #define BOOST_MATH_INTERPOLATORS_MAKIMA_HPP #include #include #include namespace boost { namespace math { namespace interpolators { template class makima { public: using Real = typename RandomAccessContainer::value_type; makima(RandomAccessContainer && x, RandomAccessContainer && y, Real left_endpoint_derivative = std::numeric_limits::quiet_NaN(), Real right_endpoint_derivative = std::numeric_limits::quiet_NaN()) { using std::isnan; using std::abs; if (x.size() < 4) { throw std::domain_error("Must be at least four data points."); } RandomAccessContainer s(x.size(), std::numeric_limits::quiet_NaN()); Real m2 = (y[3]-y[2])/(x[3]-x[2]); Real m1 = (y[2]-y[1])/(x[2]-x[1]); Real m0 = (y[1]-y[0])/(x[1]-x[0]); // Quadratic extrapolation: m_{-1} = 2m_0 - m_1: Real mm1 = 2*m0 - m1; // Quadratic extrapolation: m_{-2} = 2*m_{-1}-m_0: Real mm2 = 2*mm1 - m0; Real w1 = abs(m1-m0) + abs(m1+m0)/2; Real w2 = abs(mm1-mm2) + abs(mm1+mm2)/2; if (isnan(left_endpoint_derivative)) { s[0] = (w1*mm1 + w2*m0)/(w1+w2); if (isnan(s[0])) { s[0] = 0; } } else { s[0] = left_endpoint_derivative; } w1 = abs(m2-m1) + abs(m2+m1)/2; w2 = abs(m0-mm1) + abs(m0+mm1)/2; s[1] = (w1*m0 + w2*m1)/(w1+w2); if (isnan(s[1])) { s[1] = 0; } for (decltype(s.size()) i = 2; i < s.size()-2; ++i) { Real mim2 = (y[i-1]-y[i-2])/(x[i-1]-x[i-2]); Real mim1 = (y[i ]-y[i-1])/(x[i ]-x[i-1]); Real mi = (y[i+1]-y[i ])/(x[i+1]-x[i ]); Real mip1 = (y[i+2]-y[i+1])/(x[i+2]-x[i+1]); w1 = abs(mip1-mi) + abs(mip1+mi)/2; w2 = abs(mim1-mim2) + abs(mim1+mim2)/2; s[i] = (w1*mim1 + w2*mi)/(w1+w2); if (isnan(s[i])) { s[i] = 0; } } // Quadratic extrapolation at the other end: decltype(s.size()) n = s.size(); Real mnm4 = (y[n-3]-y[n-4])/(x[n-3]-x[n-4]); Real mnm3 = (y[n-2]-y[n-3])/(x[n-2]-x[n-3]); Real mnm2 = (y[n-1]-y[n-2])/(x[n-1]-x[n-2]); Real mnm1 = 2*mnm2 - mnm3; Real mn = 2*mnm1 - mnm2; w1 = abs(mnm1 - mnm2) + abs(mnm1+mnm2)/2; w2 = abs(mnm3 - mnm4) + abs(mnm3+mnm4)/2; s[n-2] = (w1*mnm3 + w2*mnm2)/(w1 + w2); if (isnan(s[n-2])) { s[n-2] = 0; } w1 = abs(mn - mnm1) + abs(mn+mnm1)/2; w2 = abs(mnm2 - mnm3) + abs(mnm2+mnm3)/2; if (isnan(right_endpoint_derivative)) { s[n-1] = (w1*mnm2 + w2*mnm1)/(w1+w2); if (isnan(s[n-1])) { s[n-1] = 0; } } else { s[n-1] = right_endpoint_derivative; } impl_ = std::make_shared>(std::move(x), std::move(y), std::move(s)); } Real operator()(Real x) const { return impl_->operator()(x); } Real prime(Real x) const { return impl_->prime(x); } friend std::ostream& operator<<(std::ostream & os, const makima & m) { os << *m.impl_; return os; } void push_back(Real x, Real y) { using std::abs; using std::isnan; if (x <= impl_->x_.back()) { throw std::domain_error("Calling push_back must preserve the monotonicity of the x's"); } impl_->x_.push_back(x); impl_->y_.push_back(y); impl_->dydx_.push_back(std::numeric_limits::quiet_NaN()); // dydx_[n-2] was computed by extrapolation. Now dydx_[n-2] -> dydx_[n-3], and it can be computed by the same formula. decltype(impl_->size()) n = impl_->size(); auto i = n - 3; Real mim2 = (impl_->y_[i-1]-impl_->y_[i-2])/(impl_->x_[i-1]-impl_->x_[i-2]); Real mim1 = (impl_->y_[i ]-impl_->y_[i-1])/(impl_->x_[i ]-impl_->x_[i-1]); Real mi = (impl_->y_[i+1]-impl_->y_[i ])/(impl_->x_[i+1]-impl_->x_[i ]); Real mip1 = (impl_->y_[i+2]-impl_->y_[i+1])/(impl_->x_[i+2]-impl_->x_[i+1]); Real w1 = abs(mip1-mi) + abs(mip1+mi)/2; Real w2 = abs(mim1-mim2) + abs(mim1+mim2)/2; impl_->dydx_[i] = (w1*mim1 + w2*mi)/(w1+w2); if (isnan(impl_->dydx_[i])) { impl_->dydx_[i] = 0; } Real mnm4 = (impl_->y_[n-3]-impl_->y_[n-4])/(impl_->x_[n-3]-impl_->x_[n-4]); Real mnm3 = (impl_->y_[n-2]-impl_->y_[n-3])/(impl_->x_[n-2]-impl_->x_[n-3]); Real mnm2 = (impl_->y_[n-1]-impl_->y_[n-2])/(impl_->x_[n-1]-impl_->x_[n-2]); Real mnm1 = 2*mnm2 - mnm3; Real mn = 2*mnm1 - mnm2; w1 = abs(mnm1 - mnm2) + abs(mnm1+mnm2)/2; w2 = abs(mnm3 - mnm4) + abs(mnm3+mnm4)/2; impl_->dydx_[n-2] = (w1*mnm3 + w2*mnm2)/(w1 + w2); if (isnan(impl_->dydx_[n-2])) { impl_->dydx_[n-2] = 0; } w1 = abs(mn - mnm1) + abs(mn+mnm1)/2; w2 = abs(mnm2 - mnm3) + abs(mnm2+mnm3)/2; impl_->dydx_[n-1] = (w1*mnm2 + w2*mnm1)/(w1+w2); if (isnan(impl_->dydx_[n-1])) { impl_->dydx_[n-1] = 0; } } private: std::shared_ptr> impl_; }; } } } #endif