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/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998-2005 Ross Ihaka and the R Development 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
 *  http://www.r-project.org/Licenses/
 */

/*  DESCRIPTION --> see below */


/* From http://www.netlib.org/specfun/rjbesl    Fortran translated by f2c,...
 *  ------------------------------=#----    Martin Maechler, ETH Zurich
 * Additional code for nu == alpha < 0  MM
 */
#include "nmath.h"
#include "bessel.h"

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

static void J_bessel(double *x, double *alpha, long *nb,
             double *b, long *ncalc);

double bessel_j(double x, double alpha)
{
    long nb, ncalc;
    double na, *bj;
#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_j");
    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(bessel_j(x, -alpha) * cos(M_PI * alpha) +
           ((alpha == na) ? 0 :
           bessel_y(x, -alpha) * sin(M_PI * alpha)));
    }
    nb = 1 + (long)na; /* nb-1 <= alpha < nb */
    alpha -= (nb-1);
#ifdef MATHLIB_STANDALONE
    bj = (double *) calloc(nb, sizeof(double));
    if (!bj) MATHLIB_ERROR("%s", _("bessel_j allocation error"));
#else
    vmax = vmaxget();
    bj = (double *) R_alloc(nb, sizeof(double));
#endif
    J_bessel(&x, &alpha, &nb, bj, &ncalc);
    if(ncalc != nb) {/* error input */
      if(ncalc < 0)
    MATHLIB_WARNING4(_("bessel_j(%g): ncalc (=%ld) != nb (=%ld); alpha=%g. Arg. out of range?\n"),
             x, ncalc, nb, alpha);
      else
    MATHLIB_WARNING2(_("bessel_j(%g,nu=%g): precision lost in result\n"),
             x, alpha+nb-1);
    }
    x = bj[nb-1];
#ifdef MATHLIB_STANDALONE
    free(bj);
#else
    vmaxset(vmax);
#endif
    return x;
}

/* modified version of bessel_j that accepts a work array instead of
   allocating one. */
double bessel_j_ex(double x, double alpha, double *bj)
{
    long nb, ncalc;
    double na;

#ifdef IEEE_754
    /* NaNs propagated correctly */
    if (ISNAN(x) || ISNAN(alpha)) return x + alpha;
#endif
    if (x < 0) {
    ML_ERROR(ME_RANGE, "bessel_j");
    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(bessel_j_ex(x, -alpha, bj) * cos(M_PI * alpha) +
           ((alpha == na) ? 0 :
        bessel_y_ex(x, -alpha, bj) * sin(M_PI * alpha)));
    }
    nb = 1 + (long)na; /* nb-1 <= alpha < nb */
    alpha -= (nb-1);
    J_bessel(&x, &alpha, &nb, bj, &ncalc);
    if(ncalc != nb) {/* error input */
      if(ncalc < 0)
    MATHLIB_WARNING4(_("bessel_j(%g): ncalc (=%ld) != nb (=%ld); alpha=%g. Arg. out of range?\n"),
             x, ncalc, nb, alpha);
      else
    MATHLIB_WARNING2(_("bessel_j(%g,nu=%g): precision lost in result\n"),
             x, alpha+nb-1);
    }
    x = bj[nb-1];
    return x;
}

static void J_bessel(double *x, double *alpha, long *nb,
             double *b, long *ncalc)
{
/*
 Calculates Bessel functions J_{n+alpha} (x)
 for non-negative argument x, and non-negative order n+alpha, n = 0,1,..,nb-1.

  Explanation of variables in the calling sequence.

 X     - Non-negative argument for which J's are to be calculated.
 ALPHA - Fractional part of order for which
     J's are to be calculated.  0 <= ALPHA < 1.
 NB    - Number of functions to be calculated, NB >= 1.
     The first function calculated is of order ALPHA, and the
     last is of order (NB - 1 + ALPHA).
 B     - Output vector of length NB.  If RJBESL
     terminates normally (NCALC=NB), the vector B contains the
     functions J/ALPHA/(X) through J/NB-1+ALPHA/(X).
 NCALC - Output variable indicating possible errors.
     Before using the vector B, the user should check that
     NCALC=NB, i.e., all orders have been calculated to
     the desired accuracy.  See the following

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

 Error return codes

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

    NCALC < 0:  An argument is out of range. For example,
       NBES <= 0, ALPHA < 0 or > 1, or X is too large.
       In this case, b[1] is set to zero, the remainder of the
       B-vector is not calculated, and NCALC is set to
       MIN(NB,0)-1 so that NCALC != NB.

    NB > NCALC > 0: Not all requested function values could
       be calculated accurately.  This usually occurs because NB is
       much larger than ABS(X).  In this case, b[N] is calculated
       to the desired accuracy for N <= NCALC, but precision
       is lost for NCALC < N <= NB.  If b[N] does not vanish
       for N > NCALC (because it is too small to be represented),
       and b[N]/b[NCALC] = 10^(-K), then only the first NSIG - K
       significant figures of b[N] can be trusted.


  Acknowledgement

    This program is based on a program written by David J. Sookne
    (2) that computes values of the Bessel functions J or I of float
    argument and long order.  Modifications include the restriction
    of the computation to the J Bessel function of non-negative float
    argument, the extension of the computation to arbitrary positive
    order, and the elimination of most underflow.

  References:

    Olver, F.W.J., and Sookne, D.J. (1972)
    "A Note on Backward Recurrence Algorithms";
    Math. Comp. 26, 941-947.

    Sookne, D.J. (1973)
    "Bessel Functions of Real Argument and Integer Order";
    NBS Jour. of Res. B. 77B, 125-132.

  Latest modification: March 19, 1990

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

/* ---------------------------------------------------------------------
  Mathematical constants

   PI2    = 2 / PI
   TWOPI1 = first few significant digits of 2 * PI
   TWOPI2 = (2*PI - TWOPI1) to working precision, i.e.,
        TWOPI1 + TWOPI2 = 2 * PI to extra precision.
 --------------------------------------------------------------------- */
    const static double pi2 = .636619772367581343075535;
    const static double twopi1 = 6.28125;
    const static double twopi2 =  .001935307179586476925286767;

/*---------------------------------------------------------------------
 *  Factorial(N)
 *--------------------------------------------------------------------- */
    const static double fact[25] = { 1.,1.,2.,6.,24.,120.,720.,5040.,40320.,
        362880.,3628800.,39916800.,479001600.,6227020800.,87178291200.,
        1.307674368e12,2.0922789888e13,3.55687428096e14,6.402373705728e15,
        1.21645100408832e17,2.43290200817664e18,5.109094217170944e19,
        1.12400072777760768e21,2.585201673888497664e22,
        6.2044840173323943936e23 };

    /* Local variables */
    long nend, intx, nbmx, i, j, k, l, m, n, nstart;

    double nu, twonu, capp, capq, pold, vcos, test, vsin;
    double p, s, t, z, alpem, halfx, aa, bb, cc, psave, plast;
    double tover, t1, alp2em, em, en, xc, xk, xm, psavel, gnu, xin, sum;


    /* Parameter adjustment */
    --b;

    nu = *alpha;
    twonu = nu + nu;

    /*-------------------------------------------------------------------
      Check for out of range arguments.
      -------------------------------------------------------------------*/
    if (*nb > 0 && *x >= 0. && 0. <= nu && nu < 1.) {

    *ncalc = *nb;
    if(*x > xlrg_BESS_IJ) {
        ML_ERROR(ME_RANGE, "J_bessel");
        /* indeed, the limit is 0,
         * but the cutoff happens too early */
        for(i=1; i <= *nb; i++)
        b[i] = 0.; /*was ML_POSINF (really nonsense) */
        return;
    }
    intx = (long) (*x);
    /* Initialize result array to zero. */
    for (i = 1; i <= *nb; ++i)
        b[i] = 0.;

    /*===================================================================
      Branch into  3 cases :
      1) use 2-term ascending series for small X
      2) use asymptotic form for large X when NB is not too large
      3) use recursion otherwise
      ===================================================================*/

    if (*x < rtnsig_BESS) {
      /* ---------------------------------------------------------------
         Two-term ascending series for small X.
         --------------------------------------------------------------- */
        alpem = 1. + nu;

        halfx = (*x > enmten_BESS) ? .5 * *x :  0.;
        aa    = (nu != 0.)    ? pow(halfx, nu) / (nu * gamma_cody(nu)) : 1.;
        bb    = (*x + 1. > 1.)? -halfx * halfx : 0.;
        b[1] = aa + aa * bb / alpem;
        if (*x != 0. && b[1] == 0.)
        *ncalc = 0;

        if (*nb != 1) {
        if (*x <= 0.) {
            for (n = 2; n <= *nb; ++n)
            b[n] = 0.;
        }
        else {
            /* ----------------------------------------------
               Calculate higher order functions.
               ---------------------------------------------- */
            if (bb == 0.)
            tover = (enmten_BESS + enmten_BESS) / *x;
            else
            tover = enmten_BESS / bb;
            cc = halfx;
            for (n = 2; n <= *nb; ++n) {
            aa /= alpem;
            alpem += 1.;
            aa *= cc;
            if (aa <= tover * alpem)
                aa = 0.;

            b[n] = aa + aa * bb / alpem;
            if (b[n] == 0. && *ncalc > n)
                *ncalc = n - 1;
            }
        }
        }
    } else if (*x > 25. && *nb <= intx + 1) {
        /* ------------------------------------------------------------
           Asymptotic series for X > 25 (and not too large nb)
           ------------------------------------------------------------ */
        xc = sqrt(pi2 / *x);
        xin = 1 / (64 * *x * *x);
        if (*x >= 130.) m = 4;
        else if (*x >= 35.) m = 8;
        else        m = 11;
        xm = 4. * (double) m;
        /* ------------------------------------------------
           Argument reduction for SIN and COS routines.
           ------------------------------------------------ */
        t = ftrunc(*x / (twopi1 + twopi2) + .5);
        z = (*x - t * twopi1) - t * twopi2 - (nu + .5) / pi2;
        vsin = sin(z);
        vcos = cos(z);
        gnu = twonu;
        for (i = 1; i <= 2; ++i) {
        s = (xm - 1. - gnu) * (xm - 1. + gnu) * xin * .5;
        t = (gnu - (xm - 3.)) * (gnu + (xm - 3.));
        t1= (gnu - (xm + 1.)) * (gnu + (xm + 1.));
        k = m + m;
        capp = s * t / fact[k];
        capq = s * t1/ fact[k + 1];
        xk = xm;
        for (; k >= 4; k -= 2) {/* k + 2(j-2) == 2m */
            xk -= 4.;
            s = (xk - 1. - gnu) * (xk - 1. + gnu);
            t1 = t;
            t = (gnu - (xk - 3.)) * (gnu + (xk - 3.));
            capp = (capp + 1. / fact[k - 2]) * s * t  * xin;
            capq = (capq + 1. / fact[k - 1]) * s * t1 * xin;

        }
        capp += 1.;
        capq = (capq + 1.) * (gnu * gnu - 1.) * (.125 / *x);
        b[i] = xc * (capp * vcos - capq * vsin);
        if (*nb == 1)
            return;

        /* vsin <--> vcos */ t = vsin; vsin = -vcos; vcos = t;
        gnu += 2.;
        }
        /* -----------------------------------------------
           If  NB > 2, compute J(X,ORDER+I) for I = 2, NB-1
           ----------------------------------------------- */
        if (*nb > 2)
        for (gnu = twonu + 2., j = 3; j <= *nb; j++, gnu += 2.)
            b[j] = gnu * b[j - 1] / *x - b[j - 2];
    }
    else {
        /* rtnsig_BESS <= x && ( x <= 25 || intx+1 < *nb ) :
           --------------------------------------------------------
           Use recurrence to generate results.
           First initialize the calculation of P*S.
           -------------------------------------------------------- */
        nbmx = *nb - intx;
        n = intx + 1;
        en = (double)(n + n) + twonu;
        plast = 1.;
        p = en / *x;
        /* ---------------------------------------------------
           Calculate general significance test.
           --------------------------------------------------- */
        test = ensig_BESS + ensig_BESS;
        if (nbmx >= 3) {
        /* ------------------------------------------------------------
           Calculate P*S until N = NB-1.  Check for possible overflow.
           ---------------------------------------------------------- */
        tover = enten_BESS / ensig_BESS;
        nstart = intx + 2;
        nend = *nb - 1;
        en = (double) (nstart + nstart) - 2. + twonu;
        for (k = nstart; k <= nend; ++k) {
            n = k;
            en += 2.;
            pold = plast;
            plast = p;
            p = en * plast / *x - pold;
            if (p > tover) {
            /* -------------------------------------------
               To avoid overflow, divide P*S by TOVER.
               Calculate P*S until ABS(P) > 1.
               -------------------------------------------*/
            tover = enten_BESS;
            p /= tover;
            plast /= tover;
            psave = p;
            psavel = plast;
            nstart = n + 1;
            do {
                ++n;
                en += 2.;
                pold = plast;
                plast = p;
                p = en * plast / *x - pold;
            } while (p <= 1.);

            bb = en / *x;
            /* -----------------------------------------------
               Calculate backward test and find NCALC,
               the highest N such that the test is passed.
               ----------------------------------------------- */
            test = pold * plast * (.5 - .5 / (bb * bb));
            test /= ensig_BESS;
            p = plast * tover;
            --n;
            en -= 2.;
            nend = imin2(*nb,n);
            for (l = nstart; l <= nend; ++l) {
                pold = psavel;
                psavel = psave;
                psave = en * psavel / *x - pold;
                if (psave * psavel > test) {
                *ncalc = l - 1;
                goto L190;
                }
            }
            *ncalc = nend;
            goto L190;
            }
        }
        n = nend;
        en = (double) (n + n) + twonu;
        /* -----------------------------------------------------
           Calculate special significance test for NBMX > 2.
           -----------------------------------------------------*/
        test = fmax2(test, sqrt(plast * ensig_BESS) * sqrt(p + p));
        }
        /* ------------------------------------------------
           Calculate P*S until significance test passes. */
        do {
        ++n;
        en += 2.;
        pold = plast;
        plast = p;
        p = en * plast / *x - pold;
        } while (p < test);

L190:
        /*---------------------------------------------------------------
          Initialize the backward recursion and the normalization sum.
          --------------------------------------------------------------- */
        ++n;
        en += 2.;
        bb = 0.;
        aa = 1. / p;
        m = n / 2;
        em = (double)m;
        m = (n << 1) - (m << 2);/* = 2 n - 4 (n/2)
                       = 0 for even, 2 for odd n */
        if (m == 0)
        sum = 0.;
        else {
        alpem = em - 1. + nu;
        alp2em = em + em + nu;
        sum = aa * alpem * alp2em / em;
        }
        nend = n - *nb;
        /* if (nend > 0) */
        /* --------------------------------------------------------
           Recur backward via difference equation, calculating
           (but not storing) b[N], until N = NB.
           -------------------------------------------------------- */
        for (l = 1; l <= nend; ++l) {
        --n;
        en -= 2.;
        cc = bb;
        bb = aa;
        aa = en * bb / *x - cc;
        m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
        if (m != 0) {
            em -= 1.;
            alp2em = em + em + nu;
            if (n == 1)
            break;

            alpem = em - 1. + nu;
            if (alpem == 0.)
            alpem = 1.;
            sum = (sum + aa * alp2em) * alpem / em;
        }
        }
        /*--------------------------------------------------
          Store b[NB].
          --------------------------------------------------*/
        b[n] = aa;
        if (nend >= 0) {
        if (*nb <= 1) {
            if (nu + 1. == 1.)
            alp2em = 1.;
            else
            alp2em = nu;
            sum += b[1] * alp2em;
            goto L250;
        }
        else {/*-- nb >= 2 : ---------------------------
            Calculate and store b[NB-1].
            ----------------------------------------*/
            --n;
            en -= 2.;
            b[n] = en * aa / *x - bb;
            if (n == 1)
            goto L240;

            m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
            if (m != 0) {
            em -= 1.;
            alp2em = em + em + nu;
            alpem = em - 1. + nu;
            if (alpem == 0.)
                alpem = 1.;
            sum = (sum + b[n] * alp2em) * alpem / em;
            }
        }
        }

        /* if (n - 2 != 0) */
        /* --------------------------------------------------------
           Calculate via difference equation and store b[N],
           until N = 2.
           -------------------------------------------------------- */
        for (n = n-1; n >= 2; n--) {
        en -= 2.;
        b[n] = en * b[n + 1] / *x - b[n + 2];
        m = m ? 0 : 2; /* m = 2 - m failed on gcc4-20041019 */
        if (m != 0) {
            em -= 1.;
            alp2em = em + em + nu;
            alpem = em - 1. + nu;
            if (alpem == 0.)
            alpem = 1.;
            sum = (sum + b[n] * alp2em) * alpem / em;
        }
        }
        /* ---------------------------------------
           Calculate b[1].
           -----------------------------------------*/
        b[1] = 2. * (nu + 1.) * b[2] / *x - b[3];

L240:
        em -= 1.;
        alp2em = em + em + nu;
        if (alp2em == 0.)
        alp2em = 1.;
        sum += b[1] * alp2em;

L250:
        /* ---------------------------------------------------
           Normalize.  Divide all b[N] by sum.
           ---------------------------------------------------*/
/*      if (nu + 1. != 1.) poor test */
        if(fabs(nu) > 1e-15)
        sum *= (gamma_cody(nu) * pow(.5* *x, -nu));

        aa = enmten_BESS;
        if (sum > 1.)
        aa *= sum;
        for (n = 1; n <= *nb; ++n) {
        if (fabs(b[n]) < aa)
            b[n] = 0.;
        else
            b[n] /= sum;
        }
    }

    }
    else {
      /* Error return -- X, NB, or ALPHA is out of range : */
    b[1] = 0.;
    *ncalc = imin2(*nb,0) - 1;
    }
}