venv/lib/python3.11/site-packages/scipy/special/xsf/cephes/kn.h

/* Translated into C++ by SciPy developers in 2024.
 * Original header with Copyright information appears below.
 */

/*                                                     kn.c
 *
 *     Modified Bessel function, third kind, integer order
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, kn();
 * int n;
 *
 * y = kn( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of the third kind
 * of order n of the argument.
 *
 * The range is partitioned into the two intervals [0,9.55] and
 * (9.55, infinity).  An ascending power series is used in the
 * low range, and an asymptotic expansion in the high range.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        90000       1.8e-8      3.0e-10
 *
 *  Error is high only near the crossover point x = 9.55
 * between the two expansions used.
 */

/*
 * Cephes Math Library Release 2.8:  June, 2000
 * Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
 */

/*
 * Algorithm for Kn.
 *                        n-1
 *                    -n   -  (n-k-1)!    2   k
 * K (x)  =  0.5 (x/2)     >  -------- (-x /4)
 *  n                      -     k!
 *                        k=0
 *
 *                     inf.                                   2   k
 *        n         n   -                                   (x /4)
 *  + (-1)  0.5(x/2)    >  {p(k+1) + p(n+k+1) - 2log(x/2)} ---------
 *                      -                                  k! (n+k)!
 *                     k=0
 *
 * where  p(m) is the psi function: p(1) = -EUL and
 *
 *                       m-1
 *                        -
 *       p(m)  =  -EUL +  >  1/k
 *                        -
 *                       k=1
 *
 * For large x,
 *                                          2        2     2
 *                                       u-1     (u-1 )(u-3 )
 * K (z)  =  sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...}
 *  v                                        1            2
 *                                     1! (8z)     2! (8z)
 * asymptotically, where
 *
 *            2
 *     u = 4 v .
 *
 */
#pragma once

#include "../config.h"
#include "../error.h"

#include "const.h"

namespace xsf {
namespace cephes {

    namespace detail {

        constexpr int kn_MAXFAC = 31;

    }

    XSF_HOST_DEVICE inline double kn(int nn, double x) {
        double k, kf, nk1f, nkf, zn, t, s, z0, z;
        double ans, fn, pn, pk, zmn, tlg, tox;
        int i, n;

        if (nn < 0)
            n = -nn;
        else
            n = nn;

        if (n > detail::kn_MAXFAC) {
        overf:
            set_error("kn", SF_ERROR_OVERFLOW, NULL);
            return (std::numeric_limits<double>::infinity());
        }

        if (x <= 0.0) {
            if (x < 0.0) {
                set_error("kn", SF_ERROR_DOMAIN, NULL);
                return std::numeric_limits<double>::quiet_NaN();
            } else {
                set_error("kn", SF_ERROR_SINGULAR, NULL);
                return std::numeric_limits<double>::infinity();
            }
        }

        if (x > 9.55)
            goto asymp;

        ans = 0.0;
        z0 = 0.25 * x * x;
        fn = 1.0;
        pn = 0.0;
        zmn = 1.0;
        tox = 2.0 / x;

        if (n > 0) {
            /* compute factorial of n and psi(n) */
            pn = -detail::SCIPY_EULER;
            k = 1.0;
            for (i = 1; i < n; i++) {
                pn += 1.0 / k;
                k += 1.0;
                fn *= k;
            }

            zmn = tox;

            if (n == 1) {
                ans = 1.0 / x;
            } else {
                nk1f = fn / n;
                kf = 1.0;
                s = nk1f;
                z = -z0;
                zn = 1.0;
                for (i = 1; i < n; i++) {
                    nk1f = nk1f / (n - i);
                    kf = kf * i;
                    zn *= z;
                    t = nk1f * zn / kf;
                    s += t;
                    if ((std::numeric_limits<double>::max() - std::abs(t)) < std::abs(s)) {
                        goto overf;
                    }
                    if ((tox > 1.0) && ((std::numeric_limits<double>::max() / tox) < zmn)) {
                        goto overf;
                    }
                    zmn *= tox;
                }
                s *= 0.5;
                t = std::abs(s);
                if ((zmn > 1.0) && ((std::numeric_limits<double>::max() / zmn) < t)) {
                    goto overf;
                }
                if ((t > 1.0) && ((std::numeric_limits<double>::max() / t) < zmn)) {
                    goto overf;
                }
                ans = s * zmn;
            }
        }

        tlg = 2.0 * log(0.5 * x);
        pk = -detail::SCIPY_EULER;
        if (n == 0) {
            pn = pk;
            t = 1.0;
        } else {
            pn = pn + 1.0 / n;
            t = 1.0 / fn;
        }
        s = (pk + pn - tlg) * t;
        k = 1.0;
        do {
            t *= z0 / (k * (k + n));
            pk += 1.0 / k;
            pn += 1.0 / (k + n);
            s += (pk + pn - tlg) * t;
            k += 1.0;
        } while (fabs(t / s) > detail::MACHEP);

        s = 0.5 * s / zmn;
        if (n & 1) {
            s = -s;
        }
        ans += s;

        return (ans);

        /* Asymptotic expansion for Kn(x) */
        /* Converges to 1.4e-17 for x > 18.4 */

    asymp:

        if (x > detail::MAXLOG) {
            set_error("kn", SF_ERROR_UNDERFLOW, NULL);
            return (0.0);
        }
        k = n;
        pn = 4.0 * k * k;
        pk = 1.0;
        z0 = 8.0 * x;
        fn = 1.0;
        t = 1.0;
        s = t;
        nkf = std::numeric_limits<double>::infinity();
        i = 0;
        do {
            z = pn - pk * pk;
            t = t * z / (fn * z0);
            nk1f = std::abs(t);
            if ((i >= n) && (nk1f > nkf)) {
                goto adone;
            }
            nkf = nk1f;
            s += t;
            fn += 1.0;
            pk += 2.0;
            i += 1;
        } while (std::abs(t / s) > detail::MACHEP);

    adone:
        ans = std::exp(-x) * std::sqrt(M_PI / (2.0 * x)) * s;
        return (ans);
    }

} // namespace cephes
} // namespace xsf