CocoRoboDesktop/pdf1.js/node_modules/cephes/cephes/polylog.c

/*							polylog.c
 *
 *	Polylogarithms
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, polylog();
 * int n;
 *
 * y = polylog( n, x );
 *
 *
 * The polylogarithm of order n is defined by the series
 *
 *
 *              inf   k
 *               -   x
 *  Li (x)  =    >   ---  .
 *    n          -     n
 *              k=1   k
 *
 *
 *  For x = 1,
 *
 *               inf
 *                -    1
 *   Li (1)  =    >   ---   =  Riemann zeta function (n)  .
 *     n          -     n
 *               k=1   k
 *
 *
 *  When n = 2, the function is the dilogarithm, related to Spence's integral:
 *
 *                 x                      1-x
 *                 -                        -
 *                | |  -ln(1-t)            | |  ln t
 *   Li (x)  =    |    -------- dt    =    |    ------ dt    =   spence(1-x) .
 *     2        | |       t              | |    1 - t
 *               -                        -
 *                0                        1
 *
 *
 *  See also the program cpolylog.c for the complex polylogarithm,
 *  whose definition is extended to x > 1.
 *
 *  References:
 *
 *  Lewin, L., _Polylogarithms and Associated Functions_,
 *  North Holland, 1981.
 *
 *  Lewin, L., ed., _Structural Properties of Polylogarithms_,
 *  American Mathematical Society, 1991.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain   n   # trials      peak         rms
 *    IEEE      0, 1     2     50000      6.2e-16     8.0e-17
 *    IEEE      0, 1     3    100000      2.5e-16     6.6e-17
 *    IEEE      0, 1     4     30000      1.7e-16     4.9e-17
 *    IEEE      0, 1     5     30000      5.1e-16     7.8e-17
 *
 */

/*
Cephes Math Library Release 2.8:  July, 1999
Copyright 1999 by Stephen L. Moshier
*/

#include "mconf.h"
extern double PI;

/* polylog(4, 1-x) = zeta(4) - x zeta(3) + x^2 A4(x)/B4(x)
   0 <= x <= 0.125
   Theoretical peak absolute error 4.5e-18  */
#if UNK
static double A4[13] = {
    3.056144922089490701751E-2,  3.243086484162581557457E-1,
    2.877847281461875922565E-1,  7.091267785886180663385E-2,
    6.466460072456621248630E-3,  2.450233019296542883275E-4,
    4.031655364627704957049E-6,  2.884169163909467997099E-8,
    8.680067002466594858347E-11, 1.025983405866370985438E-13,
    4.233468313538272640380E-17, 4.959422035066206902317E-21,
    1.059365867585275714599E-25,
};
static double B4[12] = {
    /* 1.000000000000000000000E0, */
    2.821262403600310974875E0,   1.780221124881327022033E0,
    3.778888211867875721773E-1,  3.193887040074337940323E-2,
    1.161252418498096498304E-3,  1.867362374829870620091E-5,
    1.319022779715294371091E-7,  3.942755256555603046095E-10,
    4.644326968986396928092E-13, 1.913336021014307074861E-16,
    2.240041814626069927477E-20, 4.784036597230791011855E-25,
};
#endif
#if DEC
static short A4[52] = {
    0036772, 0056001, 0016601, 0164507, 0037646, 0005710, 0076603, 0176456,
    0037623, 0054205, 0013532, 0026476, 0037221, 0035252, 0101064, 0065407,
    0036323, 0162231, 0042033, 0107244, 0035200, 0073170, 0106141, 0136543,
    0033607, 0043647, 0163672, 0055340, 0031767, 0137614, 0173376, 0072313,
    0027676, 0160156, 0161276, 0034203, 0025347, 0003752, 0123106, 0064266,
    0022503, 0035770, 0160173, 0177501, 0017273, 0056226, 0033704, 0132530,
    0013403, 0022244, 0175205, 0052161,
};
static short B4[48] = {
    /*0040200,0000000,0000000,0000000, */
    0040464, 0107620, 0027471, 0071672, 0040343, 0157111, 0025601, 0137255,
    0037701, 0075244, 0140412, 0160220, 0037002, 0151125, 0036572, 0057163,
    0035630, 0032452, 0050727, 0161653, 0034234, 0122515, 0034323, 0172615,
    0032415, 0120405, 0123660, 0003160, 0030330, 0140530, 0161045, 0150177,
    0026002, 0134747, 0014542, 0002510, 0023134, 0113666, 0035730, 0035732,
    0017723, 0110343, 0041217, 0007764, 0014024, 0007412, 0175575, 0160230,
};
#endif
#if IBMPC
static short A4[52] = {
    0x3d29, 0x23b0, 0x4b80, 0x3f9f, 0x7fa6, 0x0fb0, 0xc179, 0x3fd4, 0x45a8,
    0xa2eb, 0x6b10, 0x3fd2, 0x8d61, 0x5046, 0x2755, 0x3fb2, 0x71d4, 0x2883,
    0x7c93, 0x3f7a, 0x37ac, 0x118c, 0x0ecf, 0x3f30, 0x4b5c, 0xfcf7, 0xe8f4,
    0x3ed0, 0xce99, 0x9edf, 0xf7f1, 0x3e5e, 0xc710, 0xdc57, 0xdc0d, 0x3dd7,
    0xcd17, 0x54c8, 0xe0fd, 0x3d3c, 0x7fe8, 0x1c0f, 0x677f, 0x3c88, 0x96ab,
    0xc6f8, 0x6b92, 0x3bb7, 0xaa8e, 0x9f50, 0x6494, 0x3ac0,
};
static short B4[48] = {
    /*0x0000,0x0000,0x0000,0x3ff0,*/
    0x2e77, 0x05e7, 0x91f2, 0x4006, 0x37d6, 0x2570, 0x7bc9, 0x3ffc,
    0x5c12, 0x9821, 0x2f54, 0x3fd8, 0x4bce, 0xa7af, 0x5a4a, 0x3fa0,
    0xfc75, 0x4a3a, 0x06a5, 0x3f53, 0x7eb2, 0xa71a, 0x94a9, 0x3ef3,
    0x00ce, 0xb4f6, 0xb420, 0x3e81, 0xba10, 0x1c44, 0x182b, 0x3dfb,
    0x40a9, 0xe32c, 0x573c, 0x3d60, 0x077b, 0xc77b, 0x92f6, 0x3cab,
    0xe1fe, 0x6851, 0x721c, 0x3bda, 0xbc13, 0x5f6f, 0x81e1, 0x3ae2,
};
#endif
#if MIEEE
static short A4[52] = {
    0x3f9f, 0x4b80, 0x23b0, 0x3d29, 0x3fd4, 0xc179, 0x0fb0, 0x7fa6, 0x3fd2,
    0x6b10, 0xa2eb, 0x45a8, 0x3fb2, 0x2755, 0x5046, 0x8d61, 0x3f7a, 0x7c93,
    0x2883, 0x71d4, 0x3f30, 0x0ecf, 0x118c, 0x37ac, 0x3ed0, 0xe8f4, 0xfcf7,
    0x4b5c, 0x3e5e, 0xf7f1, 0x9edf, 0xce99, 0x3dd7, 0xdc0d, 0xdc57, 0xc710,
    0x3d3c, 0xe0fd, 0x54c8, 0xcd17, 0x3c88, 0x677f, 0x1c0f, 0x7fe8, 0x3bb7,
    0x6b92, 0xc6f8, 0x96ab, 0x3ac0, 0x6494, 0x9f50, 0xaa8e,
};
static short B4[48] = {
    /*0x3ff0,0x0000,0x0000,0x0000,*/
    0x4006, 0x91f2, 0x05e7, 0x2e77, 0x3ffc, 0x7bc9, 0x2570, 0x37d6,
    0x3fd8, 0x2f54, 0x9821, 0x5c12, 0x3fa0, 0x5a4a, 0xa7af, 0x4bce,
    0x3f53, 0x06a5, 0x4a3a, 0xfc75, 0x3ef3, 0x94a9, 0xa71a, 0x7eb2,
    0x3e81, 0xb420, 0xb4f6, 0x00ce, 0x3dfb, 0x182b, 0x1c44, 0xba10,
    0x3d60, 0x573c, 0xe32c, 0x40a9, 0x3cab, 0x92f6, 0xc77b, 0x077b,
    0x3bda, 0x721c, 0x6851, 0xe1fe, 0x3ae2, 0x81e1, 0x5f6f, 0xbc13,
};
#endif

#ifdef ANSIPROT
extern double spence(double);
extern double polevl(double, void *, int);
extern double p1evl(double, void *, int);
extern double zetac(double);
extern double pow(double, double);
extern double powi(double, int);
extern double log(double);
extern double fac(int i);
extern double fabs(double);
double polylog(int, double);
#else
extern double spence(), polevl(), p1evl(), zetac();
extern double pow(), powi(), log();
extern double fac(); /* factorial */
extern double fabs();
double polylog();
#endif
extern double MACHEP;

double polylog(n, x) int n;
double x;
{
  double h, k, p, s, t, u, xc, z;
  int i, j;

  /*  This recurrence provides formulas for n < 2.

      d                 1
      --   Li (x)  =   ---  Li   (x)  .
      dx     n          x     n-1

  */

  if (n == -1) {
    p = 1.0 - x;
    u = x / p;
    s = u * u + u;
    return s;
  }

  if (n == 0) {
    s = x / (1.0 - x);
    return s;
  }

  /* Not implemented for n < -1.
     Not defined for x > 1.  Use cpolylog if you need that.  */
  if (x > 1.0 || n < -1) {
    mtherr("polylog", DOMAIN);
    return 0.0;
  }

  if (n == 1) {
    s = -log(1.0 - x);
    return s;
  }

  /* Argument +1 */
  if (x == 1.0 && n > 1) {
    s = zetac((double)n) + 1.0;
    return s;
  }

  /* Argument -1.
                        1-n
     Li (-z)  = - (1 - 2   ) Li (z)
       n                       n
   */
  if (x == -1.0 && n > 1) {
    /* Li_n(1) = zeta(n) */
    s = zetac((double)n) + 1.0;
    s = s * (powi(2.0, 1 - n) - 1.0);
    return s;
  }

  /*  Inversion formula:
   *                                                   [n/2]   n-2r
   *                n                  1     n           -  log    (z)
   *  Li (-z) + (-1)  Li (-1/z)  =  - --- log (z)  +  2  >  ----------- Li  (-1)
   *    n               n              n!                -   (n - 2r)!    2r
   *                                                    r=1
   */
  if (x < -1.0 && n > 1) {
    double q, w;
    int r;

    w = log(-x);
    s = 0.0;
    for (r = 1; r <= n / 2; r++) {
      j = 2 * r;
      p = polylog(j, -1.0);
      j = n - j;
      if (j == 0) {
        s = s + p;
        break;
      }
      q = (double)j;
      q = pow(w, q) * p / fac(j);
      s = s + q;
    }
    s = 2.0 * s;
    q = polylog(n, 1.0 / x);
    if (n & 1)
      q = -q;
    s = s - q;
    s = s - pow(w, (double)n) / fac(n);
    return s;
  }

  if (n == 2) {
    if (x < 0.0 || x > 1.0)
      return (spence(1.0 - x));
  }

  /*  The power series converges slowly when x is near 1.  For n = 3, this
      identity helps:

      Li (-x/(1-x)) + Li (1-x) + Li (x)
        3               3          3
                     2                               2                 3
       = Li (1) + (pi /6) log(1-x) - (1/2) log(x) log (1-x) + (1/6) log (1-x)
           3
  */

  if (n == 3) {
    p = x * x * x;
    if (x > 0.8) {
      /* Thanks to Oscar van Vlijmen for detecting an error here.  */
      u = log(x);
      s = u * u * u / 6.0;
      xc = 1.0 - x;
      s = s - 0.5 * u * u * log(xc);
      s = s + PI * PI * u / 6.0;
      s = s - polylog(3, -xc / x);
      s = s - polylog(3, xc);
      s = s + zetac(3.0);
      s = s + 1.0;
      return s;
    }
    /* Power series  */
    t = p / 27.0;
    t = t + .125 * x * x;
    t = t + x;

    s = 0.0;
    k = 4.0;
    do {
      p = p * x;
      h = p / (k * k * k);
      s = s + h;
      k += 1.0;
    } while (fabs(h / s) > 1.1e-16);
    return (s + t);
  }

  if (n == 4) {
    if (x >= 0.875) {
      u = 1.0 - x;
      s = polevl(u, A4, 12) / p1evl(u, B4, 12);
      s = s * u * u - 1.202056903159594285400 * u;
      s += 1.0823232337111381915160;
      return s;
    }
    goto pseries;
  }

  if (x < 0.75)
    goto pseries;

  /*  This expansion in powers of log(x) is especially useful when
      x is near 1.

      See also the pari gp calculator.

                        inf                  j
                         -    z(n-j) (log(x))
      polylog(n,x)  =    >   -----------------
                         -           j!
                        j=0

        where

        z(j) = Riemann zeta function (j), j != 1

                                n-1
                                 -
        z(1) =  -log(-log(x)) +  >  1/k
                                 -
                                k=1
    */

  z = log(x);
  h = -log(-z);
  for (i = 1; i < n; i++)
    h = h + 1.0 / i;
  p = 1.0;
  s = zetac((double)n) + 1.0;
  for (j = 1; j <= n + 1; j++) {
    p = p * z / j;
    if (j == n - 1)
      s = s + h * p;
    else
      s = s + (zetac((double)(n - j)) + 1.0) * p;
  }
  j = n + 3;
  z = z * z;
  for (;;) {
    p = p * z / ((j - 1) * j);
    h = (zetac((double)(n - j)) + 1.0);
    h = h * p;
    s = s + h;
    if (fabs(h / s) < MACHEP)
      break;
    j += 2;
  }
  return s;

pseries:

  p = x * x * x;
  k = 3.0;
  s = 0.0;
  do {
    p = p * x;
    k += 1.0;
    h = p / powi(k, n);
    s = s + h;
  } while (fabs(h / s) > MACHEP);
  s += x * x * x / powi(3.0, n);
  s += x * x / powi(2.0, n);
  s += x;
  return s;
}