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

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

/*                                                     jv.c
 *
 *     Bessel function of noninteger order
 *
 *
 *
 * SYNOPSIS:
 *
 * double v, x, y, jv();
 *
 * y = jv( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order v of the argument,
 * where v is real.  Negative x is allowed if v is an integer.
 *
 * Several expansions are included: the ascending power
 * series, the Hankel expansion, and two transitional
 * expansions for large v.  If v is not too large, it
 * is reduced by recurrence to a region of best accuracy.
 * The transitional expansions give 12D accuracy for v > 500.
 *
 *
 *
 * ACCURACY:
 * Results for integer v are indicated by *, where x and v
 * both vary from -125 to +125.  Otherwise,
 * x ranges from 0 to 125, v ranges as indicated by "domain."
 * Error criterion is absolute, except relative when |jv()| > 1.
 *
 * arithmetic  v domain  x domain    # trials      peak       rms
 *    IEEE      0,125     0,125      100000      4.6e-15    2.2e-16
 *    IEEE   -125,0       0,125       40000      5.4e-11    3.7e-13
 *    IEEE      0,500     0,500       20000      4.4e-15    4.0e-16
 * Integer v:
 *    IEEE   -125,125   -125,125      50000      3.5e-15*   1.9e-16*
 *
 */

/*
 * Cephes Math Library Release 2.8:  June, 2000
 * Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
 */
#pragma once

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

#include "airy.h"
#include "cbrt.h"
#include "rgamma.h"
#include "j0.h"
#include "j1.h"
#include "polevl.h"

namespace xsf {
namespace cephes {

    namespace detail {

        constexpr double jv_BIG = 1.44115188075855872E+17;

        /* Reduce the order by backward recurrence.
         * AMS55 #9.1.27 and 9.1.73.
         */

        XSF_HOST_DEVICE inline double jv_recur(double *n, double x, double *newn, int cancel) {
            double pkm2, pkm1, pk, qkm2, qkm1;

            /* double pkp1; */
            double k, ans, qk, xk, yk, r, t, kf;
            constexpr double big = jv_BIG;
            int nflag, ctr;
            int miniter, maxiter;

            /* Continued fraction for Jn(x)/Jn-1(x)
             * AMS 9.1.73
             *
             *    x       -x^2      -x^2
             * ------  ---------  ---------   ...
             * 2 n +   2(n+1) +   2(n+2) +
             *
             * Compute it with the simplest possible algorithm.
             *
             * This continued fraction starts to converge when (|n| + m) > |x|.
             * Hence, at least |x|-|n| iterations are necessary before convergence is
             * achieved. There is a hard limit set below, m <= 30000, which is chosen
             * so that no branch in `jv` requires more iterations to converge.
             * The exact maximum number is (500/3.6)^2 - 500 ~ 19000
             */

            maxiter = 22000;
            miniter = std::abs(x) - std::abs(*n);
            if (miniter < 1) {
                miniter = 1;
            }

            if (*n < 0.0) {
                nflag = 1;
            } else {
                nflag = 0;
            }

        fstart:
            pkm2 = 0.0;
            qkm2 = 1.0;
            pkm1 = x;
            qkm1 = *n + *n;
            xk = -x * x;
            yk = qkm1;
            ans = 0.0; /* ans=0.0 ensures that t=1.0 in the first iteration */
            ctr = 0;
            do {
                yk += 2.0;
                pk = pkm1 * yk + pkm2 * xk;
                qk = qkm1 * yk + qkm2 * xk;
                pkm2 = pkm1;
                pkm1 = pk;
                qkm2 = qkm1;
                qkm1 = qk;

                /* check convergence */
                if (qk != 0 && ctr > miniter)
                    r = pk / qk;
                else
                    r = 0.0;

                if (r != 0) {
                    t = std::abs((ans - r) / r);
                    ans = r;
                } else {
                    t = 1.0;
                }

                if (++ctr > maxiter) {
                    set_error("jv", SF_ERROR_UNDERFLOW, NULL);
                    goto done;
                }
                if (t < MACHEP) {
                    goto done;
                }

                /* renormalize coefficients */
                if (std::abs(pk) > big) {
                    pkm2 /= big;
                    pkm1 /= big;
                    qkm2 /= big;
                    qkm1 /= big;
                }
            } while (t > MACHEP);

        done:
            if (ans == 0)
                ans = 1.0;

            /* Change n to n-1 if n < 0 and the continued fraction is small */
            if (nflag > 0) {
                if (std::abs(ans) < 0.125) {
                    nflag = -1;
                    *n = *n - 1.0;
                    goto fstart;
                }
            }

            kf = *newn;

            /* backward recurrence
             *              2k
             *  J   (x)  =  --- J (x)  -  J   (x)
             *   k-1         x   k         k+1
             */

            pk = 1.0;
            pkm1 = 1.0 / ans;
            k = *n - 1.0;
            r = 2 * k;
            do {
                pkm2 = (pkm1 * r - pk * x) / x;
                /*      pkp1 = pk; */
                pk = pkm1;
                pkm1 = pkm2;
                r -= 2.0;
                /*
                 * t = fabs(pkp1) + fabs(pk);
                 * if( (k > (kf + 2.5)) && (fabs(pkm1) < 0.25*t) )
                 * {
                 * k -= 1.0;
                 * t = x*x;
                 * pkm2 = ( (r*(r+2.0)-t)*pk - r*x*pkp1 )/t;
                 * pkp1 = pk;
                 * pk = pkm1;
                 * pkm1 = pkm2;
                 * r -= 2.0;
                 * }
                 */
                k -= 1.0;
            } while (k > (kf + 0.5));

            /* Take the larger of the last two iterates
             * on the theory that it may have less cancellation error.
             */

            if (cancel) {
                if ((kf >= 0.0) && (std::abs(pk) > std::abs(pkm1))) {
                    k += 1.0;
                    pkm2 = pk;
                }
            }
            *newn = k;
            return (pkm2);
        }

        /* Ascending power series for Jv(x).
         * AMS55 #9.1.10.
         */

        XSF_HOST_DEVICE inline double jv_jvs(double n, double x) {
            double t, u, y, z, k;
            int ex, sgngam;

            z = -x * x / 4.0;
            u = 1.0;
            y = u;
            k = 1.0;
            t = 1.0;

            while (t > MACHEP) {
                u *= z / (k * (n + k));
                y += u;
                k += 1.0;
                if (y != 0)
                    t = std::abs(u / y);
            }
            t = std::frexp(0.5 * x, &ex);
            ex = ex * n;
            if ((ex > -1023) && (ex < 1023) && (n > 0.0) && (n < (MAXGAM - 1.0))) {
                t = std::pow(0.5 * x, n) * xsf::cephes::rgamma(n + 1.0);
                y *= t;
            } else {
                t = n * std::log(0.5 * x) - lgam_sgn(n + 1.0, &sgngam);
                if (y < 0) {
                    sgngam = -sgngam;
                    y = -y;
                }
                t += std::log(y);
                if (t < -MAXLOG) {
                    return (0.0);
                }
                if (t > MAXLOG) {
                    set_error("Jv", SF_ERROR_OVERFLOW, NULL);
                    return (std::numeric_limits<double>::infinity());
                }
                y = sgngam * std::exp(t);
            }
            return (y);
        }

        /* Hankel's asymptotic expansion
         * for large x.
         * AMS55 #9.2.5.
         */

        XSF_HOST_DEVICE inline double jv_hankel(double n, double x) {
            double t, u, z, k, sign, conv;
            double p, q, j, m, pp, qq;
            int flag;

            m = 4.0 * n * n;
            j = 1.0;
            z = 8.0 * x;
            k = 1.0;
            p = 1.0;
            u = (m - 1.0) / z;
            q = u;
            sign = 1.0;
            conv = 1.0;
            flag = 0;
            t = 1.0;
            pp = 1.0e38;
            qq = 1.0e38;

            while (t > MACHEP) {
                k += 2.0;
                j += 1.0;
                sign = -sign;
                u *= (m - k * k) / (j * z);
                p += sign * u;
                k += 2.0;
                j += 1.0;
                u *= (m - k * k) / (j * z);
                q += sign * u;
                t = std::abs(u / p);
                if (t < conv) {
                    conv = t;
                    qq = q;
                    pp = p;
                    flag = 1;
                }
                /* stop if the terms start getting larger */
                if ((flag != 0) && (t > conv)) {
                    goto hank1;
                }
            }

        hank1:
            u = x - (0.5 * n + 0.25) * M_PI;
            t = std::sqrt(2.0 / (M_PI * x)) * (pp * std::cos(u) - qq * std::sin(u));
            return (t);
        }

        /* Asymptotic expansion for transition region,
         * n large and x close to n.
         * AMS55 #9.3.23.
         */

        constexpr double jv_PF2[] = {-9.0000000000000000000e-2, 8.5714285714285714286e-2};

        constexpr double jv_PF3[] = {1.3671428571428571429e-1, -5.4920634920634920635e-2, -4.4444444444444444444e-3};

        constexpr double jv_PF4[] = {1.3500000000000000000e-3, -1.6036054421768707483e-1, 4.2590187590187590188e-2,
                                     2.7330447330447330447e-3};

        constexpr double jv_PG1[] = {-2.4285714285714285714e-1, 1.4285714285714285714e-2};

        constexpr double jv_PG2[] = {-9.0000000000000000000e-3, 1.9396825396825396825e-1, -1.1746031746031746032e-2};

        constexpr double jv_PG3[] = {1.9607142857142857143e-2, -1.5983694083694083694e-1, 6.3838383838383838384e-3};

        XSF_HOST_DEVICE inline double jv_jnt(double n, double x) {
            double z, zz, z3;
            double cbn, n23, cbtwo;
            double ai, aip, bi, bip; /* Airy functions */
            double nk, fk, gk, pp, qq;
            double F[5], G[4];
            int k;

            cbn = cbrt(n);
            z = (x - n) / cbn;
            cbtwo = cbrt(2.0);

            /* Airy function */
            zz = -cbtwo * z;
            xsf::cephes::airy(zz, &ai, &aip, &bi, &bip);

            /* polynomials in expansion */
            zz = z * z;
            z3 = zz * z;
            F[0] = 1.0;
            F[1] = -z / 5.0;
            F[2] = xsf::cephes::polevl(z3, jv_PF2, 1) * zz;
            F[3] = xsf::cephes::polevl(z3, jv_PF3, 2);
            F[4] = xsf::cephes::polevl(z3, jv_PF4, 3) * z;
            G[0] = 0.3 * zz;
            G[1] = xsf::cephes::polevl(z3, jv_PG1, 1);
            G[2] = xsf::cephes::polevl(z3, jv_PG2, 2) * z;
            G[3] = xsf::cephes::polevl(z3, jv_PG3, 2) * zz;

            pp = 0.0;
            qq = 0.0;
            nk = 1.0;
            n23 = cbrt(n * n);

            for (k = 0; k <= 4; k++) {
                fk = F[k] * nk;
                pp += fk;
                if (k != 4) {
                    gk = G[k] * nk;
                    qq += gk;
                }
                nk /= n23;
            }

            fk = cbtwo * ai * pp / cbn + cbrt(4.0) * aip * qq / n;
            return (fk);
        }

        /* Asymptotic expansion for large n.
         * AMS55 #9.3.35.
         */

        constexpr double jv_lambda[] = {1.0,
                                        1.041666666666666666666667E-1,
                                        8.355034722222222222222222E-2,
                                        1.282265745563271604938272E-1,
                                        2.918490264641404642489712E-1,
                                        8.816272674437576524187671E-1,
                                        3.321408281862767544702647E+0,
                                        1.499576298686255465867237E+1,
                                        7.892301301158651813848139E+1,
                                        4.744515388682643231611949E+2,
                                        3.207490090890661934704328E+3};

        constexpr double jv_mu[] = {1.0,
                                    -1.458333333333333333333333E-1,
                                    -9.874131944444444444444444E-2,
                                    -1.433120539158950617283951E-1,
                                    -3.172272026784135480967078E-1,
                                    -9.424291479571202491373028E-1,
                                    -3.511203040826354261542798E+0,
                                    -1.572726362036804512982712E+1,
                                    -8.228143909718594444224656E+1,
                                    -4.923553705236705240352022E+2,
                                    -3.316218568547972508762102E+3};

        constexpr double jv_P1[] = {-2.083333333333333333333333E-1, 1.250000000000000000000000E-1};

        constexpr double jv_P2[] = {3.342013888888888888888889E-1, -4.010416666666666666666667E-1,
                                    7.031250000000000000000000E-2};

        constexpr double jv_P3[] = {-1.025812596450617283950617E+0, 1.846462673611111111111111E+0,
                                    -8.912109375000000000000000E-1, 7.324218750000000000000000E-2};

        constexpr double jv_P4[] = {4.669584423426247427983539E+0, -1.120700261622299382716049E+1,
                                    8.789123535156250000000000E+0, -2.364086914062500000000000E+0,
                                    1.121520996093750000000000E-1};

        constexpr double jv_P5[] = {-2.8212072558200244877E1, 8.4636217674600734632E1,  -9.1818241543240017361E1,
                                    4.2534998745388454861E1,  -7.3687943594796316964E0, 2.27108001708984375E-1};

        constexpr double jv_P6[] = {2.1257013003921712286E2,  -7.6525246814118164230E2, 1.0599904525279998779E3,
                                    -6.9957962737613254123E2, 2.1819051174421159048E2,  -2.6491430486951555525E1,
                                    5.7250142097473144531E-1};

        constexpr double jv_P7[] = {-1.9194576623184069963E3, 8.0617221817373093845E3,  -1.3586550006434137439E4,
                                    1.1655393336864533248E4,  -5.3056469786134031084E3, 1.2009029132163524628E3,
                                    -1.0809091978839465550E2, 1.7277275025844573975E0};

        XSF_HOST_DEVICE inline double jv_jnx(double n, double x) {
            double zeta, sqz, zz, zp, np;
            double cbn, n23, t, z, sz;
            double pp, qq, z32i, zzi;
            double ak, bk, akl, bkl;
            int sign, doa, dob, nflg, k, s, tk, tkp1, m;
            double u[8];
            double ai, aip, bi, bip;

            /* Test for x very close to n. Use expansion for transition region if so. */
            cbn = cbrt(n);
            z = (x - n) / cbn;
            if (std::abs(z) <= 0.7) {
                return (jv_jnt(n, x));
            }

            z = x / n;
            zz = 1.0 - z * z;
            if (zz == 0.0) {
                return (0.0);
            }

            if (zz > 0.0) {
                sz = std::sqrt(zz);
                t = 1.5 * (std::log((1.0 + sz) / z) - sz); /* zeta ** 3/2          */
                zeta = cbrt(t * t);
                nflg = 1;
            } else {
                sz = std::sqrt(-zz);
                t = 1.5 * (sz - std::acos(1.0 / z));
                zeta = -cbrt(t * t);
                nflg = -1;
            }
            z32i = std::abs(1.0 / t);
            sqz = cbrt(t);

            /* Airy function */
            n23 = cbrt(n * n);
            t = n23 * zeta;

            xsf::cephes::airy(t, &ai, &aip, &bi, &bip);

            /* polynomials in expansion */
            u[0] = 1.0;
            zzi = 1.0 / zz;
            u[1] = xsf::cephes::polevl(zzi, jv_P1, 1) / sz;
            u[2] = xsf::cephes::polevl(zzi, jv_P2, 2) / zz;
            u[3] = xsf::cephes::polevl(zzi, jv_P3, 3) / (sz * zz);
            pp = zz * zz;
            u[4] = xsf::cephes::polevl(zzi, jv_P4, 4) / pp;
            u[5] = xsf::cephes::polevl(zzi, jv_P5, 5) / (pp * sz);
            pp *= zz;
            u[6] = xsf::cephes::polevl(zzi, jv_P6, 6) / pp;
            u[7] = xsf::cephes::polevl(zzi, jv_P7, 7) / (pp * sz);

            pp = 0.0;
            qq = 0.0;
            np = 1.0;
            /* flags to stop when terms get larger */
            doa = 1;
            dob = 1;
            akl = std::numeric_limits<double>::infinity();
            bkl = std::numeric_limits<double>::infinity();

            for (k = 0; k <= 3; k++) {
                tk = 2 * k;
                tkp1 = tk + 1;
                zp = 1.0;
                ak = 0.0;
                bk = 0.0;
                for (s = 0; s <= tk; s++) {
                    if (doa) {
                        if ((s & 3) > 1)
                            sign = nflg;
                        else
                            sign = 1;
                        ak += sign * jv_mu[s] * zp * u[tk - s];
                    }

                    if (dob) {
                        m = tkp1 - s;
                        if (((m + 1) & 3) > 1)
                            sign = nflg;
                        else
                            sign = 1;
                        bk += sign * jv_lambda[s] * zp * u[m];
                    }
                    zp *= z32i;
                }

                if (doa) {
                    ak *= np;
                    t = std::abs(ak);
                    if (t < akl) {
                        akl = t;
                        pp += ak;
                    } else
                        doa = 0;
                }

                if (dob) {
                    bk += jv_lambda[tkp1] * zp * u[0];
                    bk *= -np / sqz;
                    t = std::abs(bk);
                    if (t < bkl) {
                        bkl = t;
                        qq += bk;
                    } else
                        dob = 0;
                }
                if (np < MACHEP)
                    break;
                np /= n * n;
            }

            /* normalizing factor ( 4*zeta/(1 - z**2) )**1/4    */
            t = 4.0 * zeta / zz;
            t = sqrt(sqrt(t));

            t *= ai * pp / cbrt(n) + aip * qq / (n23 * n);
            return (t);
        }

    } // namespace detail

    XSF_HOST_DEVICE inline double jv(double n, double x) {
        double k, q, t, y, an;
        int i, sign, nint;

        nint = 0; /* Flag for integer n */
        sign = 1; /* Flag for sign inversion */
        an = std::abs(n);
        y = std::floor(an);
        if (y == an) {
            nint = 1;
            i = an - 16384.0 * std::floor(an / 16384.0);
            if (n < 0.0) {
                if (i & 1)
                    sign = -sign;
                n = an;
            }
            if (x < 0.0) {
                if (i & 1)
                    sign = -sign;
                x = -x;
            }
            if (n == 0.0)
                return (j0(x));
            if (n == 1.0)
                return (sign * j1(x));
        }

        if ((x < 0.0) && (y != an)) {
            set_error("Jv", SF_ERROR_DOMAIN, NULL);
            y = std::numeric_limits<double>::quiet_NaN();
            goto done;
        }

        if (x == 0 && n < 0 && !nint) {
            set_error("Jv", SF_ERROR_OVERFLOW, NULL);
            return std::numeric_limits<double>::infinity() * rgamma(n + 1);
        }

        y = std::abs(x);

        if (y * y < std::abs(n + 1) * detail::MACHEP) {
            return std::pow(0.5 * x, n) * rgamma(n + 1);
        }

        k = 3.6 * std::sqrt(y);
        t = 3.6 * std::sqrt(an);
        if ((y < t) && (an > 21.0)) {
            return (sign * detail::jv_jvs(n, x));
        }
        if ((an < k) && (y > 21.0))
            return (sign * detail::jv_hankel(n, x));

        if (an < 500.0) {
            /* Note: if x is too large, the continued fraction will fail; but then the
             * Hankel expansion can be used. */
            if (nint != 0) {
                k = 0.0;
                q = detail::jv_recur(&n, x, &k, 1);
                if (k == 0.0) {
                    y = j0(x) / q;
                    goto done;
                }
                if (k == 1.0) {
                    y = j1(x) / q;
                    goto done;
                }
            }

            if (an > 2.0 * y)
                goto rlarger;

            if ((n >= 0.0) && (n < 20.0) && (y > 6.0) && (y < 20.0)) {
                /* Recur backwards from a larger value of n */
            rlarger:
                k = n;

                y = y + an + 1.0;
                if (y < 30.0)
                    y = 30.0;
                y = n + std::floor(y - n);
                q = detail::jv_recur(&y, x, &k, 0);
                y = detail::jv_jvs(y, x) * q;
                goto done;
            }

            if (k <= 30.0) {
                k = 2.0;
            } else if (k < 90.0) {
                k = (3 * k) / 4;
            }
            if (an > (k + 3.0)) {
                if (n < 0.0) {
                    k = -k;
                }
                q = n - std::floor(n);
                k = std::floor(k) + q;
                if (n > 0.0) {
                    q = detail::jv_recur(&n, x, &k, 1);
                } else {
                    t = k;
                    k = n;
                    q = detail::jv_recur(&t, x, &k, 1);
                    k = t;
                }
                if (q == 0.0) {
                    y = 0.0;
                    goto done;
                }
            } else {
                k = n;
                q = 1.0;
            }

            /* boundary between convergence of
             * power series and Hankel expansion
             */
            y = std::abs(k);
            if (y < 26.0)
                t = (0.0083 * y + 0.09) * y + 12.9;
            else
                t = 0.9 * y;

            if (x > t)
                y = detail::jv_hankel(k, x);
            else
                y = detail::jv_jvs(k, x);
            if (n > 0.0)
                y /= q;
            else
                y *= q;
        }

        else {
            /* For large n, use the uniform expansion or the transitional expansion.
             * But if x is of the order of n**2, these may blow up, whereas the
             * Hankel expansion will then work.
             */
            if (n < 0.0) {
                set_error("jv", SF_ERROR_LOSS, NULL);
                y = std::numeric_limits<double>::quiet_NaN();
                goto done;
            }
            t = x / n;
            t /= n;
            if (t > 0.3)
                y = detail::jv_hankel(n, x);
            else
                y = detail::jv_jnx(n, x);
        }

    done:
        return (sign * y);
    }

} // namespace cephes
} // namespace xsf