MathExtended.h

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/*
 * Some code in this header is taken from SciPy. Their license can be found
 * in the THIRDPARTY_LEGAL_NOTICES.txt file at the root of this repository.
 */
 
#pragma once
 
#include "Math.h"
 
#include <array>
#include <cstddef>
 
namespace sourcepp::math {
 
// Evaluate Chebyshev series
template<std::size_t L>
[[nodiscard]] constexpr double chebyshev(double x, const std::array<double, L>& array) {
    const double* p = array.data();
    double b0 = *p++;
    double b1 = 0.0;
    int i = L - 1;
    double b2;
    do {
        b2 = b1;
        b1 = b0;
        b0 = x * b1 - b2 + *p++;
    } while (--i);
    return 0.5 * (b0 - b2);
}
 
namespace detail {
 
/* Chebyshev coefficients for exp(-x) I0(x)
 * in the interval [0,8].
 *
 * lim(x->0){ exp(-x) I0(x) } = 1.
 */
constexpr std::array<double, 30> besselI0A{
    -4.41534164647933937950E-18, 3.33079451882223809783E-17,  -2.43127984654795469359E-16,
    1.71539128555513303061E-15,  -1.16853328779934516808E-14, 7.67618549860493561688E-14,
    -4.85644678311192946090E-13, 2.95505266312963983461E-12,  -1.72682629144155570723E-11,
    9.67580903537323691224E-11,  -5.18979560163526290666E-10, 2.65982372468238665035E-9,
    -1.30002500998624804212E-8,  6.04699502254191894932E-8,   -2.67079385394061173391E-7,
    1.11738753912010371815E-6,   -4.41673835845875056359E-6,  1.64484480707288970893E-5,
    -5.75419501008210370398E-5,  1.88502885095841655729E-4,   -5.76375574538582365885E-4,
    1.63947561694133579842E-3,   -4.32430999505057594430E-3,  1.05464603945949983183E-2,
    -2.37374148058994688156E-2,  4.93052842396707084878E-2,   -9.49010970480476444210E-2,
    1.71620901522208775349E-1,   -3.04682672343198398683E-1,  6.76795274409476084995E-1,
};
 
/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
 * in the inverted interval [8,infinity].
 *
 * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
 */
constexpr std::array<double, 25> besselI0B{
    -7.23318048787475395456E-18, -4.83050448594418207126E-18, 4.46562142029675999901E-17,
    3.46122286769746109310E-17,  -2.82762398051658348494E-16, -3.42548561967721913462E-16,
    1.77256013305652638360E-15,  3.81168066935262242075E-15,  -9.55484669882830764870E-15,
    -4.15056934728722208663E-14, 1.54008621752140982691E-14,  3.85277838274214270114E-13,
    7.18012445138366623367E-13,  -1.79417853150680611778E-12, -1.32158118404477131188E-11,
    -3.14991652796324136454E-11, 1.18891471078464383424E-11,  4.94060238822496958910E-10,
    3.39623202570838634515E-9,   2.26666899049817806459E-8,   2.04891858946906374183E-7,
    2.89137052083475648297E-6,   6.88975834691682398426E-5,   3.36911647825569408990E-3,
    8.04490411014108831608E-1,
};
 
} // namespace detail
 
constexpr double besselI0(double x) {
    if (x < 0) {
        x = -x;
    }
    if (x <= 8.0) {
        double y = (x / 2.0) - 2.0;
        return (std::exp(x) * chebyshev(y, detail::besselI0A));
    }
    return std::exp(x) * chebyshev(32.0 / x - 2.0, detail::besselI0B) / std::sqrt(x);
}
 
constexpr double kaiserWindow(double x, double b) {
    const auto d = besselI0(b);
    if (d == 0.0) {
        return 0.0;
    }
    return besselI0(b * std::sqrt(1 - x * x)) / d;
}
 
} // namespace sourcepp::math