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j1.c

/*                                        j1.c
 *
 *    Bessel function of order one
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, j1();
 *
 * y = j1( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order one of the argument.
 *
 * The domain is divided into the intervals [0, 8] and
 * (8, infinity). In the first interval a 24 term Chebyshev
 * expansion is used. In the second, the asymptotic
 * trigonometric representation is employed using two
 * rational functions of degree 5/5.
 *
 *
 *
 * ACCURACY:
 *
 *                      Absolute error:
 * arithmetic   domain      # trials      peak         rms
 *    DEC       0, 30       10000       4.0e-17     1.1e-17
 *    IEEE      0, 30       30000       2.6e-16     1.1e-16
 *
 *
 */
/*                                       y1.c
 *
 *    Bessel function of second kind of order one
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, y1();
 *
 * y = y1( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of the second kind of order one
 * of the argument.
 *
 * The domain is divided into the intervals [0, 8] and
 * (8, infinity). In the first interval a 25 term Chebyshev
 * expansion is used, and a call to j1() is required.
 * In the second, the asymptotic trigonometric representation
 * is employed using two rational functions of degree 5/5.
 *
 *
 *
 * ACCURACY:
 *
 *                      Absolute error:
 * arithmetic   domain      # trials      peak         rms
 *    DEC       0, 30       10000       8.6e-17     1.3e-17
 *    IEEE      0, 30       30000       1.0e-15     1.3e-16
 *
 * (error criterion relative when |y1| > 1).
 *
 */


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

/*
#define PIO4 .78539816339744830962
#define THPIO4 2.35619449019234492885
#define SQ2OPI .79788456080286535588
*/

#include "mconf.h"

static double RP[4] = {
-8.99971225705559398224E8,
 4.52228297998194034323E11,
-7.27494245221818276015E13,
 3.68295732863852883286E15,
};

static double RQ[8] = {
/* 1.00000000000000000000E0,*/
 6.20836478118054335476E2,
 2.56987256757748830383E5,
 8.35146791431949253037E7,
 2.21511595479792499675E10,
 4.74914122079991414898E12,
 7.84369607876235854894E14,
 8.95222336184627338078E16,
 5.32278620332680085395E18,
};

static double PP[7] = {
 7.62125616208173112003E-4,
 7.31397056940917570436E-2,
 1.12719608129684925192E0,
 5.11207951146807644818E0,
 8.42404590141772420927E0,
 5.21451598682361504063E0,
 1.00000000000000000254E0,
};

static double PQ[7] = {
 5.71323128072548699714E-4,
 6.88455908754495404082E-2,
 1.10514232634061696926E0,
 5.07386386128601488557E0,
 8.39985554327604159757E0,
 5.20982848682361821619E0,
 9.99999999999999997461E-1,
};

static double QP[8] = {
 5.10862594750176621635E-2,
 4.98213872951233449420E0,
 7.58238284132545283818E1,
 3.66779609360150777800E2,
 7.10856304998926107277E2,
 5.97489612400613639965E2,
 2.11688757100572135698E2,
 2.52070205858023719784E1,
};

static double QQ[7] = {
/* 1.00000000000000000000E0,*/
 7.42373277035675149943E1,
 1.05644886038262816351E3,
 4.98641058337653607651E3,
 9.56231892404756170795E3,
 7.99704160447350683650E3,
 2.82619278517639096600E3,
 3.36093607810698293419E2,
};

static double YP[6] = {
 1.26320474790178026440E9,
-6.47355876379160291031E11,
 1.14509511541823727583E14,
-8.12770255501325109621E15,
 2.02439475713594898196E17,
-7.78877196265950026825E17,
};

static double YQ[8] = {
/* 1.00000000000000000000E0,*/
 5.94301592346128195359E2,
 2.35564092943068577943E5,
 7.34811944459721705660E7,
 1.87601316108706159478E10,
 3.88231277496238566008E12,
 6.20557727146953693363E14,
 6.87141087355300489866E16,
 3.97270608116560655612E18,
};

static double Z1 = 1.46819706421238932572E1;
static double Z2 = 4.92184563216946036703E1;

extern double TWOOPI, THPIO4, SQ2OPI;

double j1 (double x)
{
    double z, p, q, xn, w = x;

    if (x < 0)
      w = -x;

    if (w <= 5.0) {
      z = x * x;  
      w = polevl(z, RP, 3) / p1evl(z, RQ, 8);
      w = w * x * (z - Z1) * (z - Z2);
      return w;
    }

    w = 5.0/x;
    z = w * w;
    p = polevl(z, PP, 6)/polevl(z, PQ, 6);
    q = polevl(z, QP, 7)/p1evl(z, QQ, 7);
    xn = x - THPIO4;
    p = p * cos(xn) - w * q * sin(xn);

    return p * SQ2OPI / sqrt(x);
}

double y1 (double x)
{
    double w, z, p, q, xn;

    if (x <= 5.0) {
      if (x <= 0.0) {
          mtherr("y1", CEPHES_DOMAIN);
          return -MAXNUM;
      }
      z = x * x;
      w = x * (polevl(z, YP, 5) / p1evl(z, YQ, 8));
      w += TWOOPI * (j1(x) * log(x)  -  1.0/x);
      return w;
    }

    w = 5.0/x;
    z = w * w;
    p = polevl(z, PP, 6)/polevl(z, PQ, 6);
    q = polevl(z, QP, 7)/p1evl(z, QQ, 7);
    xn = x - THPIO4;
    p = p * sin(xn) + w * q * cos(xn);

    return p * SQ2OPI / sqrt(x);
}

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