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

/*                                                     iv.c
 *
 *     Modified Bessel function of noninteger order
 *
 *
 *
 * SYNOPSIS:
 *
 * double v, x, y, iv();
 *
 * y = iv( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order v of the
 * argument.  If x is negative, v must be integer valued.
 *
 */
/*                                                     iv.c    */
/*     Modified Bessel function of noninteger order            */
/* If x < 0, then v must be an integer. */

/*
 * Parts of the code are copyright:
 *
 *     Cephes Math Library Release 2.8:  June, 2000
 *     Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
 *
 * And other parts:
 *
 *     Copyright (c) 2006 Xiaogang Zhang
 *     Use, modification and distribution are subject to the
 *     Boost Software License, Version 1.0.
 *
 *     Boost Software License - Version 1.0 - August 17th, 2003
 *
 *     Permission is hereby granted, free of charge, to any person or
 *     organization obtaining a copy of the software and accompanying
 *     documentation covered by this license (the "Software") to use, reproduce,
 *     display, distribute, execute, and transmit the Software, and to prepare
 *     derivative works of the Software, and to permit third-parties to whom the
 *     Software is furnished to do so, all subject to the following:
 *
 *     The copyright notices in the Software and this entire statement,
 *     including the above license grant, this restriction and the following
 *     disclaimer, must be included in all copies of the Software, in whole or
 *     in part, and all derivative works of the Software, unless such copies or
 *     derivative works are solely in the form of machine-executable object code
 *     generated by a source language processor.
 *
 *     THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
 *     OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
 *     MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE AND
 *     NON-INFRINGEMENT. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR ANYONE
 *     DISTRIBUTING THE SOFTWARE BE LIABLE FOR ANY DAMAGES OR OTHER LIABILITY,
 *     WHETHER IN CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
 *     CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 *     SOFTWARE.
 *
 * And the rest are:
 *
 *     Copyright (C) 2009 Pauli Virtanen
 *     Distributed under the same license as Scipy.
 *
 */
#pragma once

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

#include "const.h"
#include "gamma.h"
#include "trig.h"

namespace special {
namespace cephes {

    namespace detail {

        /*
         * Compute Iv from (AMS5 9.7.1), asymptotic expansion for large |z|
         * Iv ~ exp(x)/sqrt(2 pi x) ( 1 + (4*v*v-1)/8x + (4*v*v-1)(4*v*v-9)/8x/2! + ...)
         */
        SPECFUN_HOST_DEVICE inline double iv_asymptotic(double v, double x) {
            double mu;
            double sum, term, prefactor, factor;
            int k;

            prefactor = std::exp(x) / std::sqrt(2 * M_PI * x);

            if (prefactor == std::numeric_limits<double>::infinity()) {
                return prefactor;
            }

            mu = 4 * v * v;
            sum = 1.0;
            term = 1.0;
            k = 1;

            do {
                factor = (mu - (2 * k - 1) * (2 * k - 1)) / (8 * x) / k;
                if (k > 100) {
                    /* didn't converge */
                    set_error("iv(iv_asymptotic)", SF_ERROR_NO_RESULT, NULL);
                    break;
                }
                term *= -factor;
                sum += term;
                ++k;
            } while (std::abs(term) > MACHEP * std::abs(sum));
            return sum * prefactor;
        }

        /*
         * Uniform asymptotic expansion factors, (AMS5 9.3.9; AMS5 9.3.10)
         *
         * Computed with:
         * --------------------
         import numpy as np
         t = np.poly1d([1,0])
         def up1(p):
         return .5*t*t*(1-t*t)*p.deriv() + 1/8. * ((1-5*t*t)*p).integ()
         us = [np.poly1d([1])]
         for k in range(10):
         us.append(up1(us[-1]))
         n = us[-1].order
         for p in us:
         print "{" + ", ".join(["0"]*(n-p.order) + map(repr, p)) + "},"
         print "N_UFACTORS", len(us)
         print "N_UFACTOR_TERMS", us[-1].order + 1
         * --------------------
         */
        constexpr int iv_N_UFACTORS = 11;
        constexpr int iv_N_UFACTOR_TERMS = 31;

        constexpr double iv_asymptotic_ufactors[iv_N_UFACTORS][iv_N_UFACTOR_TERMS] = {
            {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
            {0,   0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
             0,   0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.20833333333333334,
             0.0, 0.125, 0.0},
            {0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0.3342013888888889,
             0.0,
             -0.40104166666666669,
             0.0,
             0.0703125,
             0.0,
             0.0},
            {0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   -1.0258125964506173,
             0.0, 1.8464626736111112,
             0.0, -0.89121093750000002,
             0.0, 0.0732421875,
             0.0, 0.0,
             0.0},
            {0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             4.6695844234262474,
             0.0,
             -11.207002616222995,
             0.0,
             8.78912353515625,
             0.0,
             -2.3640869140624998,
             0.0,
             0.112152099609375,
             0.0,
             0.0,
             0.0,
             0.0},
            {0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   -28.212072558200244,
             0.0, 84.636217674600744,
             0.0, -91.818241543240035,
             0.0, 42.534998745388457,
             0.0, -7.3687943594796312,
             0.0, 0.22710800170898438,
             0.0, 0.0,
             0.0, 0.0,
             0.0},
            {0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             0,
             212.5701300392171,
             0.0,
             -765.25246814118157,
             0.0,
             1059.9904525279999,
             0.0,
             -699.57962737613275,
             0.0,
             218.19051174421159,
             0.0,
             -26.491430486951554,
             0.0,
             0.57250142097473145,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0},
            {0,   0,
             0,   0,
             0,   0,
             0,   0,
             0,   -1919.4576623184068,
             0.0, 8061.7221817373083,
             0.0, -13586.550006434136,
             0.0, 11655.393336864536,
             0.0, -5305.6469786134048,
             0.0, 1200.9029132163525,
             0.0, -108.09091978839464,
             0.0, 1.7277275025844574,
             0.0, 0.0,
             0.0, 0.0,
             0.0, 0.0,
             0.0},
            {0,
             0,
             0,
             0,
             0,
             0,
             20204.291330966149,
             0.0,
             -96980.598388637503,
             0.0,
             192547.0012325315,
             0.0,
             -203400.17728041555,
             0.0,
             122200.46498301747,
             0.0,
             -41192.654968897557,
             0.0,
             7109.5143024893641,
             0.0,
             -493.915304773088,
             0.0,
             6.074042001273483,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0},
            {0,   0,
             0,   -242919.18790055133,
             0.0, 1311763.6146629769,
             0.0, -2998015.9185381061,
             0.0, 3763271.2976564039,
             0.0, -2813563.2265865342,
             0.0, 1268365.2733216248,
             0.0, -331645.17248456361,
             0.0, 45218.768981362737,
             0.0, -2499.8304818112092,
             0.0, 24.380529699556064,
             0.0, 0.0,
             0.0, 0.0,
             0.0, 0.0,
             0.0, 0.0,
             0.0},
            {3284469.8530720375,
             0.0,
             -19706819.11843222,
             0.0,
             50952602.492664628,
             0.0,
             -74105148.211532637,
             0.0,
             66344512.274729028,
             0.0,
             -37567176.660763353,
             0.0,
             13288767.166421819,
             0.0,
             -2785618.1280864552,
             0.0,
             308186.40461266245,
             0.0,
             -13886.089753717039,
             0.0,
             110.01714026924674,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0,
             0.0}};

        /*
         * Compute Iv, Kv from (AMS5 9.7.7 + 9.7.8), asymptotic expansion for large v
         */
        SPECFUN_HOST_DEVICE inline void ikv_asymptotic_uniform(double v, double x, double *i_value, double *k_value) {
            double i_prefactor, k_prefactor;
            double t, t2, eta, z;
            double i_sum, k_sum, term, divisor;
            int k, n;
            int sign = 1;

            if (v < 0) {
                /* Negative v; compute I_{-v} and K_{-v} and use (AMS 9.6.2) */
                sign = -1;
                v = -v;
            }

            z = x / v;
            t = 1 / std::sqrt(1 + z * z);
            t2 = t * t;
            eta = std::sqrt(1 + z * z) + std::log(z / (1 + 1 / t));

            i_prefactor = std::sqrt(t / (2 * M_PI * v)) * std::exp(v * eta);
            i_sum = 1.0;

            k_prefactor = std::sqrt(M_PI * t / (2 * v)) * std::exp(-v * eta);
            k_sum = 1.0;

            divisor = v;
            for (n = 1; n < iv_N_UFACTORS; ++n) {
                /*
                 * Evaluate u_k(t) with Horner's scheme;
                 * (using the knowledge about which coefficients are zero)
                 */
                term = 0;
                for (k = iv_N_UFACTOR_TERMS - 1 - 3 * n; k < iv_N_UFACTOR_TERMS - n; k += 2) {
                    term *= t2;
                    term += iv_asymptotic_ufactors[n][k];
                }
                for (k = 1; k < n; k += 2) {
                    term *= t2;
                }
                if (n % 2 == 1) {
                    term *= t;
                }

                /* Sum terms */
                term /= divisor;
                i_sum += term;
                k_sum += (n % 2 == 0) ? term : -term;

                /* Check convergence */
                if (std::abs(term) < MACHEP) {
                    break;
                }

                divisor *= v;
            }

            if (std::abs(term) > 1e-3 * std::abs(i_sum)) {
                /* Didn't converge */
                set_error("ikv_asymptotic_uniform", SF_ERROR_NO_RESULT, NULL);
            }
            if (std::abs(term) > MACHEP * std::abs(i_sum)) {
                /* Some precision lost */
                set_error("ikv_asymptotic_uniform", SF_ERROR_LOSS, NULL);
            }

            if (k_value != NULL) {
                /* symmetric in v */
                *k_value = k_prefactor * k_sum;
            }

            if (i_value != NULL) {
                if (sign == 1) {
                    *i_value = i_prefactor * i_sum;
                } else {
                    /* (AMS 9.6.2) */
                    *i_value = (i_prefactor * i_sum + (2 / M_PI) * special::cephes::sinpi(v) * k_prefactor * k_sum);
                }
            }
        }

        /*
         * The following code originates from the Boost C++ library,
         * from file `boost/math/special_functions/detail/bessel_ik.hpp`,
         * converted from C++ to C.
         */

        /*
         * Modified Bessel functions of the first and second kind of fractional order
         *
         * Calculate K(v, x) and K(v+1, x) by method analogous to
         * Temme, Journal of Computational Physics, vol 21, 343 (1976)
         */
        SPECFUN_HOST_DEVICE inline int temme_ik_series(double v, double x, double *K, double *K1) {
            double f, h, p, q, coef, sum, sum1, tolerance;
            double a, b, c, d, sigma, gamma1, gamma2;
            std::uint64_t k;
            double gp;
            double gm;

            /*
             * |x| <= 2, Temme series converge rapidly
             * |x| > 2, the larger the |x|, the slower the convergence
             */
            SPECFUN_ASSERT(std::abs(x) <= 2);
            SPECFUN_ASSERT(std::abs(v) <= 0.5f);

            gp = special::cephes::Gamma(v + 1) - 1;
            gm = special::cephes::Gamma(-v + 1) - 1;

            a = std::log(x / 2);
            b = std::exp(v * a);
            sigma = -a * v;
            c = std::abs(v) < MACHEP ? 1 : special::cephes::sinpi(v) / (v * M_PI);
            d = std::abs(sigma) < MACHEP ? 1 : std::sinh(sigma) / sigma;
            gamma1 = std::abs(v) < MACHEP ? -SCIPY_EULER : (0.5 / v) * (gp - gm) * c;
            gamma2 = (2 + gp + gm) * c / 2;

            /* initial values */
            p = (gp + 1) / (2 * b);
            q = (1 + gm) * b / 2;
            f = (std::cosh(sigma) * gamma1 + d * (-a) * gamma2) / c;
            h = p;
            coef = 1;
            sum = coef * f;
            sum1 = coef * h;

            /* series summation */
            tolerance = MACHEP;
            for (k = 1; k < MAXITER; k++) {
                f = (k * f + p + q) / (k * k - v * v);
                p /= k - v;
                q /= k + v;
                h = p - k * f;
                coef *= x * x / (4 * k);
                sum += coef * f;
                sum1 += coef * h;
                if (std::abs(coef * f) < std::abs(sum) * tolerance) {
                    break;
                }
            }
            if (k == MAXITER) {
                set_error("ikv_temme(temme_ik_series)", SF_ERROR_NO_RESULT, NULL);
            }

            *K = sum;
            *K1 = 2 * sum1 / x;

            return 0;
        }

        /* Evaluate continued fraction fv = I_(v+1) / I_v, derived from
         * Abramowitz and Stegun, Handbook of Mathematical Functions, 1972, 9.1.73 */
        SPECFUN_HOST_DEVICE inline int CF1_ik(double v, double x, double *fv) {
            double C, D, f, a, b, delta, tiny, tolerance;
            std::uint64_t k;

            /*
             * |x| <= |v|, CF1_ik converges rapidly
             * |x| > |v|, CF1_ik needs O(|x|) iterations to converge
             */

            /*
             * modified Lentz's method, see
             * Lentz, Applied Optics, vol 15, 668 (1976)
             */
            tolerance = 2 * MACHEP;
            tiny = 1 / std::sqrt(std::numeric_limits<double>::max());
            C = f = tiny; /* b0 = 0, replace with tiny */
            D = 0;
            for (k = 1; k < MAXITER; k++) {
                a = 1;
                b = 2 * (v + k) / x;
                C = b + a / C;
                D = b + a * D;
                if (C == 0) {
                    C = tiny;
                }
                if (D == 0) {
                    D = tiny;
                }
                D = 1 / D;
                delta = C * D;
                f *= delta;
                if (std::abs(delta - 1) <= tolerance) {
                    break;
                }
            }
            if (k == MAXITER) {
                set_error("ikv_temme(CF1_ik)", SF_ERROR_NO_RESULT, NULL);
            }

            *fv = f;

            return 0;
        }

        /*
         * Calculate K(v, x) and K(v+1, x) by evaluating continued fraction
         * z1 / z0 = U(v+1.5, 2v+1, 2x) / U(v+0.5, 2v+1, 2x), see
         * Thompson and Barnett, Computer Physics Communications, vol 47, 245 (1987)
         */
        SPECFUN_HOST_DEVICE inline int CF2_ik(double v, double x, double *Kv, double *Kv1) {

            double S, C, Q, D, f, a, b, q, delta, tolerance, current, prev;
            std::uint64_t k;

            /*
             * |x| >= |v|, CF2_ik converges rapidly
             * |x| -> 0, CF2_ik fails to converge
             */

            SPECFUN_ASSERT(std::abs(x) > 1);

            /*
             * Steed's algorithm, see Thompson and Barnett,
             * Journal of Computational Physics, vol 64, 490 (1986)
             */
            tolerance = MACHEP;
            a = v * v - 0.25;
            b = 2 * (x + 1);                /* b1 */
            D = 1 / b;                      /* D1 = 1 / b1 */
            f = delta = D;                  /* f1 = delta1 = D1, coincidence */
            prev = 0;                       /* q0 */
            current = 1;                    /* q1 */
            Q = C = -a;                     /* Q1 = C1 because q1 = 1 */
            S = 1 + Q * delta;              /* S1 */
            for (k = 2; k < MAXITER; k++) { /* starting from 2 */
                /* continued fraction f = z1 / z0 */
                a -= 2 * (k - 1);
                b += 2;
                D = 1 / (b + a * D);
                delta *= b * D - 1;
                f += delta;

                /* series summation S = 1 + \sum_{n=1}^{\infty} C_n * z_n / z_0 */
                q = (prev - (b - 2) * current) / a;
                prev = current;
                current = q; /* forward recurrence for q */
                C *= -a / k;
                Q += C * q;
                S += Q * delta;

                /* S converges slower than f */
                if (std::abs(Q * delta) < std::abs(S) * tolerance) {
                    break;
                }
            }
            if (k == MAXITER) {
                set_error("ikv_temme(CF2_ik)", SF_ERROR_NO_RESULT, NULL);
            }

            *Kv = std::sqrt(M_PI / (2 * x)) * std::exp(-x) / S;
            *Kv1 = *Kv * (0.5 + v + x + (v * v - 0.25) * f) / x;

            return 0;
        }

        /* Flags for what to compute */
        enum { ikv_temme_need_i = 0x1, ikv_temme_need_k = 0x2 };

        /*
         * Compute I(v, x) and K(v, x) simultaneously by Temme's method, see
         * Temme, Journal of Computational Physics, vol 19, 324 (1975)
         */
        SPECFUN_HOST_DEVICE inline void ikv_temme(double v, double x, double *Iv_p, double *Kv_p) {
            /* Kv1 = K_(v+1), fv = I_(v+1) / I_v */
            /* Ku1 = K_(u+1), fu = I_(u+1) / I_u */
            double u, Iv, Kv, Kv1, Ku, Ku1, fv;
            double W, current, prev, next;
            int reflect = 0;
            unsigned n, k;
            int kind;

            kind = 0;
            if (Iv_p != NULL) {
                kind |= ikv_temme_need_i;
            }
            if (Kv_p != NULL) {
                kind |= ikv_temme_need_k;
            }

            if (v < 0) {
                reflect = 1;
                v = -v; /* v is non-negative from here */
                kind |= ikv_temme_need_k;
            }
            n = std::round(v);
            u = v - n; /* -1/2 <= u < 1/2 */

            if (x < 0) {
                if (Iv_p != NULL)
                    *Iv_p = std::numeric_limits<double>::quiet_NaN();
                if (Kv_p != NULL)
                    *Kv_p = std::numeric_limits<double>::quiet_NaN();
                set_error("ikv_temme", SF_ERROR_DOMAIN, NULL);
                return;
            }
            if (x == 0) {
                Iv = (v == 0) ? 1 : 0;
                if (kind & ikv_temme_need_k) {
                    set_error("ikv_temme", SF_ERROR_OVERFLOW, NULL);
                    Kv = std::numeric_limits<double>::infinity();
                } else {
                    Kv = std::numeric_limits<double>::quiet_NaN(); /* any value will do */
                }

                if (reflect && (kind & ikv_temme_need_i)) {
                    double z = (u + n % 2);

                    Iv = special::cephes::sinpi(z) == 0 ? Iv : std::numeric_limits<double>::infinity();
                    if (std::isinf(Iv)) {
                        set_error("ikv_temme", SF_ERROR_OVERFLOW, NULL);
                    }
                }

                if (Iv_p != NULL) {
                    *Iv_p = Iv;
                }
                if (Kv_p != NULL) {
                    *Kv_p = Kv;
                }
                return;
            }
            /* x is positive until reflection */
            W = 1 / x;                            /* Wronskian */
            if (x <= 2) {                         /* x in (0, 2] */
                temme_ik_series(u, x, &Ku, &Ku1); /* Temme series */
            } else {                              /* x in (2, \infty) */
                CF2_ik(u, x, &Ku, &Ku1);          /* continued fraction CF2_ik */
            }
            prev = Ku;
            current = Ku1;
            for (k = 1; k <= n; k++) { /* forward recurrence for K */
                next = 2 * (u + k) * current / x + prev;
                prev = current;
                current = next;
            }
            Kv = prev;
            Kv1 = current;
            if (kind & ikv_temme_need_i) {
                double lim = (4 * v * v + 10) / (8 * x);

                lim *= lim;
                lim *= lim;
                lim /= 24;
                if ((lim < MACHEP * 10) && (x > 100)) {
                    /*
                     * x is huge compared to v, CF1 may be very slow
                     * to converge so use asymptotic expansion for large
                     * x case instead.  Note that the asymptotic expansion
                     * isn't very accurate - so it's deliberately very hard
                     * to get here - probably we're going to overflow:
                     */
                    Iv = iv_asymptotic(v, x);
                } else {
                    CF1_ik(v, x, &fv);        /* continued fraction CF1_ik */
                    Iv = W / (Kv * fv + Kv1); /* Wronskian relation */
                }
            } else {
                Iv = std::numeric_limits<double>::quiet_NaN(); /* any value will do */
            }

            if (reflect) {
                double z = (u + n % 2);

                if (Iv_p != NULL) {
                    *Iv_p = Iv + (2 / M_PI) * special::cephes::sinpi(z) * Kv; /* reflection formula */
                }
                if (Kv_p != NULL) {
                    *Kv_p = Kv;
                }
            } else {
                if (Iv_p != NULL) {
                    *Iv_p = Iv;
                }
                if (Kv_p != NULL) {
                    *Kv_p = Kv;
                }
            }
            return;
        }

    } // namespace detail

    SPECFUN_HOST_DEVICE inline double iv(double v, double x) {
        int sign;
        double t, ax, res;

        if (std::isnan(v) || std::isnan(x)) {
            return std::numeric_limits<double>::quiet_NaN();
        }

        /* If v is a negative integer, invoke symmetry */
        t = std::floor(v);
        if (v < 0.0) {
            if (t == v) {
                v = -v; /* symmetry */
                t = -t;
            }
        }
        /* If x is negative, require v to be an integer */
        sign = 1;
        if (x < 0.0) {
            if (t != v) {
                set_error("iv", SF_ERROR_DOMAIN, NULL);
                return (std::numeric_limits<double>::quiet_NaN());
            }
            if (v != 2.0 * std::floor(v / 2.0)) {
                sign = -1;
            }
        }

        /* Avoid logarithm singularity */
        if (x == 0.0) {
            if (v == 0.0) {
                return 1.0;
            }
            if (v < 0.0) {
                set_error("iv", SF_ERROR_OVERFLOW, NULL);
                return std::numeric_limits<double>::infinity();
            } else
                return 0.0;
        }

        ax = std::abs(x);
        if (std::abs(v) > 50) {
            /*
             * Uniform asymptotic expansion for large orders.
             *
             * This appears to overflow slightly later than the Boost
             * implementation of Temme's method.
             */
            detail::ikv_asymptotic_uniform(v, ax, &res, NULL);
        } else {
            /* Otherwise: Temme's method */
            detail::ikv_temme(v, ax, &res, NULL);
        }
        res *= sign;
        return res;
    }

} // namespace cephes
} // namespace special
