bessel_y.cpp

/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998-2015 Ross Ihaka and the R Core team.
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, a copy is available at
 *  https://www.R-project.org/Licenses/
 */

/*  DESCRIPTION --> see below */


/* From http://www.netlib.org/specfun/rybesl    Fortran translated by f2c,...
 *      ------------------------------=#----    Martin Maechler, ETH Zurich
 */

#include "bessel.h"

#ifndef MATHLIB_STANDALONE
#include <R_ext/Memory.h>
#endif

#define min0(x, y) (((x) <= (y)) ? (x) : (y))

template<class Float>
static void Y_bessel(Float *x, Float *alpha, int *nb,
                     Float *by, int *ncalc);

// unused now from R
template<class Float>
Float bessel_y(Float x, Float alpha)
{
    int nb, ncalc;
    Float na, *by;
#ifndef MATHLIB_STANDALONE
    const void *vmax;
#endif

#ifdef IEEE_754
    /* NaNs propagated correctly */
    if (ISNAN(x) || ISNAN(alpha)) return x + alpha;
#endif
    if (x < 0) {
        ML_ERROR(ME_RANGE, "bessel_y");
        return ML_NAN;
    }
    na = floor(alpha);
    if (alpha < 0) {
        /* Using Abramowitz & Stegun  9.1.2
         * this may not be quite optimal (CPU and accuracy wise) */
        return(((alpha - na == 0.5) ? 0 : bessel_y<Float>(x, -alpha) * cospi(alpha)) -
               ((alpha      == na ) ? 0 : bessel_j<Float>(x, -alpha) * sinpi(alpha)));
    }
    else if (alpha > 1e7) {
        MATHLIB_WARNING(_("besselY(x, nu): nu=%g too large for bessel_y() algorithm"),
                        alpha);
        return ML_NAN;
    }
    nb = 1+ (int)trunc(na);/* nb-1 <= alpha < nb */
    alpha -= (double)(nb-1);
#ifdef MATHLIB_STANDALONE
    by = (Float *) calloc(nb, sizeof(Float));
    if (!by) MATHLIB_ERROR("%s", _("bessel_y allocation error"));
#else
    vmax = vmaxget();
    by = (Float *) R_alloc((size_t) nb, sizeof(Float));
#endif
    Y_bessel(&x, &alpha, &nb, by, &ncalc);
    if(ncalc != nb) {/* error input */
        if(ncalc == -1) {
#ifdef MATHLIB_STANDALONE
            free(by);
#else
            vmaxset(vmax);
#endif
            return ML_POSINF;
        }
        else if(ncalc < -1)
            MATHLIB_WARNING4(_("bessel_y(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"),
                             x, ncalc, nb, alpha);
        else /* ncalc >= 0 */
            MATHLIB_WARNING2(_("bessel_y(%g,nu=%g): precision lost in result\n"),
                             x, alpha+(double)nb-1);
    }
    x = by[nb-1];
#ifdef MATHLIB_STANDALONE
    free(by);
#else
    vmaxset(vmax);
#endif
    return x;
}

/* Called from R: modified version of bessel_y(), accepting a work array
 * instead of allocating one. */
template<class Float>
Float bessel_y_ex(Float x, Float alpha, Float *by)
{
    int nb, ncalc;
    Float na;

#ifdef IEEE_754
    /* NaNs propagated correctly */
    if (ISNAN(x) || ISNAN(alpha)) return x + alpha;
#endif
    if (x < 0) {
        ML_ERROR(ME_RANGE, "bessel_y");
        return ML_NAN;
    }
    na = floor(alpha);
    if (alpha < 0) {
        /* Using Abramowitz & Stegun  9.1.2
         * this may not be quite optimal (CPU and accuracy wise) */
        return(((alpha - na == 0.5) ? 0 : bessel_y_ex(x, -alpha, by) * cospi(alpha)) -
               ((alpha      == na ) ? 0 : bessel_j_ex(x, -alpha, by) * sinpi(alpha)));
    }
    else if (alpha > 1e7) {
        MATHLIB_WARNING(_("besselY(x, nu): nu=%g too large for bessel_y() algorithm"),
                        alpha);
        return ML_NAN;
    }
    nb = 1+ (int)trunc(na);/* nb-1 <= alpha < nb */
    alpha -= (double)(nb-1);
    Y_bessel(&x, &alpha, &nb, by, &ncalc);
    if(ncalc != nb) {/* error input */
        if(ncalc == -1)
            return ML_POSINF;
        else if(ncalc < -1)
            MATHLIB_WARNING4(_("bessel_y(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"),
                             x, ncalc, nb, alpha);
        else /* ncalc >= 0 */
            MATHLIB_WARNING2(_("bessel_y(%g,nu=%g): precision lost in result\n"),
                             x, alpha+(double)nb-1);
    }
    x = by[nb-1];
    return x;
}

template<class Float>
static void Y_bessel(Float *x, Float *alpha, int *nb,
                     Float *by, int *ncalc)
{
/* ----------------------------------------------------------------------

 This routine calculates Bessel functions Y_(N+ALPHA) (X)
v for non-negative argument X, and non-negative order N+ALPHA.


 Explanation of variables in the calling sequence

 X     - Non-negative argument for which
         Y's are to be calculated.
 ALPHA - Fractional part of order for which
         Y's are to be calculated.  0 <= ALPHA < 1.0.
 NB    - Number of functions to be calculated, NB > 0.
         The first function calculated is of order ALPHA, and the
         last is of order (NB - 1 + ALPHA).
 BY    - Output vector of length NB.    If the
         routine terminates normally (NCALC=NB), the vector BY
         contains the functions Y(ALPHA,X), ... , Y(NB-1+ALPHA,X),
         If (0 < NCALC < NB), BY(I) contains correct function
         values for I <= NCALC, and contains the ratios
         Y(ALPHA+I-1,X)/Y(ALPHA+I-2,X) for the rest of the array.
 NCALC - Output variable indicating possible errors.
         Before using the vector BY, the user should check that
         NCALC=NB, i.e., all orders have been calculated to
         the desired accuracy.  See error returns below.


 *******************************************************************

 Error returns

  In case of an error, NCALC != NB, and not all Y's are
  calculated to the desired accuracy.

  NCALC < -1:  An argument is out of range. For example,
        NB <= 0, IZE is not 1 or 2, or IZE=1 and ABS(X) >=
        XMAX.  In this case, BY[0] = 0.0, the remainder of the
        BY-vector is not calculated, and NCALC is set to
        MIN0(NB,0)-2  so that NCALC != NB.
  NCALC = -1:  Y(ALPHA,X) >= XINF.  The requested function
        values are set to 0.0.
  1 < NCALC < NB: Not all requested function values could
        be calculated accurately.  BY(I) contains correct function
        values for I <= NCALC, and and the remaining NB-NCALC
        array elements contain 0.0.


 Intrinsic functions required are:

     DBLE, EXP, INT, MAX, MIN, REAL, SQRT


 Acknowledgement

        This program draws heavily on Temme's Algol program for Y(a,x)
        and Y(a+1,x) and on Campbell's programs for Y_nu(x).    Temme's
        scheme is used for  x < THRESH, and Campbell's scheme is used
        in the asymptotic region.  Segments of code from both sources
        have been translated into Fortran 77, merged, and heavily modified.
        Modifications include parameterization of machine dependencies,
        use of a new approximation for ln(gamma(x)), and built-in
        protection against over/underflow.

 References: "Bessel functions J_nu(x) and Y_nu(x) of float
              order and float argument," Campbell, J. B.,
              Comp. Phy. Comm. 18, 1979, pp. 133-142.

             "On the numerical evaluation of the ordinary
              Bessel function of the second kind," Temme,
              N. M., J. Comput. Phys. 21, 1976, pp. 343-350.

  Latest modification: March 19, 1990

  Modified by: W. J. Cody
               Applied Mathematics Division
               Argonne National Laboratory
               Argonne, IL  60439
 ----------------------------------------------------------------------*/


/* ----------------------------------------------------------------------
  Mathematical constants
    FIVPI = 5*PI
    PIM5 = 5*PI - 15
 ----------------------------------------------------------------------*/
    const static double fivpi = 15.707963267948966192;
    const static double pim5    =   .70796326794896619231;

    /*----------------------------------------------------------------------
      Coefficients for Chebyshev polynomial expansion of
      1/gamma(1-x), abs(x) <= .5
      ----------------------------------------------------------------------*/
    const static double ch[21] = { -6.7735241822398840964e-24,
            -6.1455180116049879894e-23,2.9017595056104745456e-21,
            1.3639417919073099464e-19,2.3826220476859635824e-18,
            -9.0642907957550702534e-18,-1.4943667065169001769e-15,
            -3.3919078305362211264e-14,-1.7023776642512729175e-13,
            9.1609750938768647911e-12,2.4230957900482704055e-10,
            1.7451364971382984243e-9,-3.3126119768180852711e-8,
            -8.6592079961391259661e-7,-4.9717367041957398581e-6,
            7.6309597585908126618e-5,.0012719271366545622927,
            .0017063050710955562222,-.07685284084478667369,
            -.28387654227602353814,.92187029365045265648 };

    /* Local variables */
    int i, k, na;

    Float alfa, div, ddiv, even, gamma, term, cosmu, sinmu,
        b, c, d, e, f, g, h, p, q, r, s, d1, d2, q0, pa,pa1, qa,qa1,
        en, en1, nu, ex,  ya,ya1, twobyx, den, odd, aye, dmu, x2, xna;

    en1 = ya = ya1 = 0;         /* -Wall */

    ex = *x;
    nu = *alpha;
    if (*nb > 0 && 0. <= nu && nu < 1.) {
        if(ex < DBL_MIN || ex > xlrg_BESS_Y) {
            /* Warning is not really appropriate, give
             * proper limit:
             * ML_ERROR(ME_RANGE, "Y_bessel"); */
            *ncalc = *nb;
            if(ex > xlrg_BESS_Y)  by[0]= 0.; /*was ML_POSINF */
            else if(ex < DBL_MIN) by[0]=ML_NEGINF;
            for(i=0; i < *nb; i++)
                by[i] = by[0];
            return;
        }
        xna = trunc(nu + .5);
        na = (int) trunc(xna);
        if (na == 1) {/* <==>  .5 <= *alpha < 1  <==>  -5. <= nu < 0 */
            nu -= xna;
        }
        // Kasper: This special case ignores derivatives wrt. nu. We must
        //         squeeze it into the general case (even if it might be slower).
        // if (nu == -.5) {
        //     p = M_SQRT_2dPI / sqrt(ex);
        //     ya = p * sin(ex);
        //     ya1 = -p * cos(ex);
        // } else
        if (ex < 3.) {
            /* -------------------------------------------------------------
               Use Temme's scheme for small X
               ------------------------------------------------------------- */
            b = ex * .5;
            d = -log(b);
            f = nu * d;
            e = pow(b, -nu);
            if (fabs(nu) < M_eps_sinc)
                // Kasper: Improve approximation near zero
                // c = M_1_PI;
                c = M_1_PI / ( 1. - (M_PI * M_PI / 6.) * nu * nu );
            else
                c = nu / sinpi(nu);

            /* ------------------------------------------------------------
               Computation of sinh(f)/f
               ------------------------------------------------------------ */
            if (fabs(f) < 1.) {
                x2 = f * f;
                en = 19.;
                s = 1.;
                for (i = 1; i <= 9; ++i) {
                    s = s * x2 / en / (en - 1.) + 1.;
                    en -= 2.;
                }
            } else {
                s = (e - 1. / e) * .5 / f;
            }
            /* --------------------------------------------------------
               Computation of 1/gamma(1-a) using Chebyshev polynomials */
            x2 = nu * nu * 8.;
            aye = ch[0];
            even = 0.;
            alfa = ch[1];
            odd = 0.;
            for (i = 3; i <= 19; i += 2) {
                even = -(aye + aye + even);
                aye = -even * x2 - aye + ch[i - 1];
                odd = -(alfa + alfa + odd);
                alfa = -odd * x2 - alfa + ch[i];
            }
            even = (even * .5 + aye) * x2 - aye + ch[20];
            odd = (odd + alfa) * 2.;
            gamma = odd * nu + even;
            /* End of computation of 1/gamma(1-a)
               ----------------------------------------------------------- */
            g = e * gamma;
            e = (e + 1. / e) * .5;
            f = 2. * c * (odd * e + even * s * d);
            e = nu * nu;
            p = g * c;
            q = M_1_PI / g;
            c = nu * M_PI_2;
            if (fabs(c) < M_eps_sinc)
                r = 1.;
            else
                r = sinpi(nu/2) / c;

            r = M_PI * c * r * r;
            c = 1.;
            d = -b * b;
            h = 0.;
            ya = f + r * q;
            ya1 = p;
            en = 1.;

            while (fabs(g / (1. + fabs(ya))) +
                   fabs(h / (1. + fabs(ya1))) > DBL_EPSILON) {
                f = (f * en + p + q) / (en * en - e);
                c *= (d / en);
                p /= en - nu;
                q /= en + nu;
                g = c * (f + r * q);
                h = c * p - en * g;
                ya += g;
                ya1+= h;
                en += 1.;
            }
            ya = -ya;
            ya1 = -ya1 / b;
        } else if (ex < thresh_BESS_Y) {
            /* --------------------------------------------------------------
               Use Temme's scheme for moderate X :  3 <= x < 16
               -------------------------------------------------------------- */
            c = (.5 - nu) * (.5 + nu);
            b = ex + ex;
            // Kasper: nu=-.5 ==> cospi(nu)=0 ==> zero iterations !
            //         We must increase e in this case.
            // e = ex * M_1_PI * cospi(nu) / DBL_EPSILON;
            double tiny = 1e-4;
            e = ex * M_1_PI * ( fabs(cospi(nu)) + tiny ) / (1. + tiny) / DBL_EPSILON;
            e *= e;
            p = 1.;
            q = -ex;
            r = 1. + ex * ex;
            s = r;
            en = 2.;
            while (r * en * en < e) {
                en1 = en + 1.;
                d = (en - 1. + c / en) / s;
                p = (en + en - p * d) / en1;
                q = (-b + q * d) / en1;
                s = p * p + q * q;
                r *= s;
                en = en1;
            }
            f = p / s;
            p = f;
            g = -q / s;
            q = g;
L220:
            en -= 1.;
            if (en > 0.) {
                r = en1 * (2. - p) - 2.;
                s = b + en1 * q;
                d = (en - 1. + c / en) / (r * r + s * s);
                p = d * r;
                q = d * s;
                e = f + 1.;
                f = p * e - g * q;
                g = q * e + p * g;
                en1 = en;
                goto L220;
            }
            f = 1. + f;
            d = f * f + g * g;
            pa = f / d;
            qa = -g / d;
            d = nu + .5 - p;
            q += ex;
            pa1 = (pa * q - qa * d) / ex;
            qa1 = (qa * q + pa * d) / ex;
            b = ex - M_PI_2 * (nu + .5);
            c = cos(b);
            s = sin(b);
            d = M_SQRT_2dPI / sqrt(ex);
            ya = d * (pa * s + qa * c);
            ya1 = d * (qa1 * s - pa1 * c);
        } else { /* x > thresh_BESS_Y */
            /* ----------------------------------------------------------
               Use Campbell's asymptotic scheme.
               ---------------------------------------------------------- */
            na = 0;
            d1 = trunc(ex / fivpi);
            i = (int) trunc(d1);
            dmu = ex - 15. * d1 - d1 * pim5 - (*alpha + .5) * M_PI_2;
            if (i - (i / 2 << 1) == 0) {
                cosmu = cos(dmu);
                sinmu = sin(dmu);
            } else {
                cosmu = -cos(dmu);
                sinmu = -sin(dmu);
            }
            ddiv = 8. * ex;
            dmu = *alpha;
            den = sqrt(ex);
            for (k = 1; k <= 2; ++k) {
                p = cosmu;
                cosmu = sinmu;
                sinmu = -p;
                d1 = (2. * dmu - 1.) * (2. * dmu + 1.);
                d2 = 0.;
                div = ddiv;
                p = 0.;
                q = 0.;
                q0 = d1 / div;
                term = q0;
                for (i = 2; i <= 20; ++i) {
                    d2 += 8.;
                    d1 -= d2;
                    div += ddiv;
                    term = -term * d1 / div;
                    p += term;
                    d2 += 8.;
                    d1 -= d2;
                    div += ddiv;
                    term *= (d1 / div);
                    q += term;
                    if (fabs(term) <= DBL_EPSILON) {
                        break;
                    }
                }
                p += 1.;
                q += q0;
                if (k == 1)
                    ya = M_SQRT_2dPI * (p * cosmu - q * sinmu) / den;
                else
                    ya1 = M_SQRT_2dPI * (p * cosmu - q * sinmu) / den;
                dmu += 1.;
            }
        }
        if (na == 1) {
            h = 2. * (nu + 1.) / ex;
            if (h > 1.) {
                if (fabs(ya1) > DBL_MAX / h) {
                    h = 0.;
                    ya = 0.;
                }
            }
            h = h * ya1 - ya;
            ya = ya1;
            ya1 = h;
        }

        /* ---------------------------------------------------------------
           Now have first one or two Y's
           --------------------------------------------------------------- */
        by[0] = ya;
        *ncalc = 1;
        if(*nb > 1) {
            by[1] = ya1;
            if (ya1 != 0.) {
                aye = 1. + *alpha;
                twobyx = 2. / ex;
                *ncalc = 2;
                for (i = 2; i < *nb; ++i) {
                    if (twobyx < 1.) {
                        if (fabs(by[i - 1]) * twobyx >= DBL_MAX / aye)
                            goto L450;
                    } else {
                        if (fabs(by[i - 1]) >= DBL_MAX / aye / twobyx)
                            goto L450;
                    }
                    by[i] = twobyx * aye * by[i - 1] - by[i - 2];
                    aye += 1.;
                    ++(*ncalc);
                }
            }
        }
L450:
        for (i = *ncalc; i < *nb; ++i)
            by[i] = ML_NEGINF;/* was 0 */

    } else {
        by[0] = 0.;
        *ncalc = min0(*nb,0) - 1;
    }
}