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

/*							hyperg.c
 *
 *	Confluent hypergeometric function
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, b, x, y, hyperg();
 *
 * y = hyperg( a, b, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the confluent hypergeometric function
 *
 *                          1           2
 *                       a x    a(a+1) x
 *   F ( a,b;x )  =  1 + ---- + --------- + ...
 *  1 1                  b 1!   b(b+1) 2!
 *
 * Many higher transcendental functions are special cases of
 * this power series.
 *
 * As is evident from the formula, b must not be a negative
 * integer or zero unless a is an integer with 0 >= a > b.
 *
 * The routine attempts both a direct summation of the series
 * and an asymptotic expansion.  In each case error due to
 * roundoff, cancellation, and nonconvergence is estimated.
 * The result with smaller estimated error is returned.
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random points (a, b, x), all three variables
 * ranging from 0 to 30.
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    DEC       0,30         2000       1.2e-15     1.3e-16
 qtst1:
 21800   max =  1.4200E-14   rms =  1.0841E-15  ave = -5.3640E-17
 ltstd:
 25500   max = 1.2759e-14   rms = 3.7155e-16  ave = 1.5384e-18
 *    IEEE      0,30        30000       1.8e-14     1.1e-15
 *
 * Larger errors can be observed when b is near a negative
 * integer or zero.  Certain combinations of arguments yield
 * serious cancellation error in the power series summation
 * and also are not in the region of near convergence of the
 * asymptotic series.  An error message is printed if the
 * self-estimated relative error is greater than 1.0e-12.
 *
 */

/*							hyperg.c */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

#ifdef ANSIPROT
extern double exp(double);
extern double log(double);
extern double gamma(double);
extern double lgam(double);
extern double fabs(double);
double hyp2f0(double, double, double, int, double *);
static double hy1f1p(double, double, double, double *);
static double hy1f1a(double, double, double, double *);
double hyperg(double, double, double);
#else
double exp(), log(), gamma(), lgam(), fabs(), hyp2f0();
static double hy1f1p();
static double hy1f1a();
double hyperg();
#endif
extern double MAXNUM, MACHEP;

double hyperg(a, b, x) double a, b, x;
{
  double asum, psum, acanc, pcanc, temp;

  /* See if a Kummer transformation will help */
  temp = b - a;
  if (fabs(temp) < 0.001 * fabs(a))
    return (exp(x) * hyperg(temp, b, -x));

  psum = hy1f1p(a, b, x, &pcanc);
  if (pcanc < 1.0e-15)
    goto done;

  /* try asymptotic series */

  asum = hy1f1a(a, b, x, &acanc);

  /* Pick the result with less estimated error */

  if (acanc < pcanc) {
    pcanc = acanc;
    psum = asum;
  }

done:
  if (pcanc > 1.0e-12)
    mtherr("hyperg", PLOSS);

  return (psum);
}

/* Power series summation for confluent hypergeometric function		*/

static double hy1f1p(a, b, x, err) double a, b, x;
double *err;
{
  double n, a0, sum, t, u, temp;
  double an, bn, maxt, pcanc;

  /* set up for power series summation */
  an = a;
  bn = b;
  a0 = 1.0;
  sum = 1.0;
  n = 1.0;
  t = 1.0;
  maxt = 0.0;

  while (t > MACHEP) {
    if (bn == 0) /* check bn first since if both	*/
    {
      mtherr("hyperg", SING);
      return (MAXNUM); /* an and bn are zero it is	*/
    }
    if (an == 0) /* a singularity		*/
      return (sum);
    if (n > 200)
      goto pdone;
    u = x * (an / (bn * n));

    /* check for blowup */
    temp = fabs(u);
    if ((temp > 1.0) && (maxt > (MAXNUM / temp))) {
      pcanc = 1.0; /* estimate 100% error */
      goto blowup;
    }

    a0 *= u;
    sum += a0;
    t = fabs(a0);
    if (t > maxt)
      maxt = t;
    /*
            if( (maxt/fabs(sum)) > 1.0e17 )
                    {
                    pcanc = 1.0;
                    goto blowup;
                    }
    */
    an += 1.0;
    bn += 1.0;
    n += 1.0;
  }

pdone:

  /* estimate error due to roundoff and cancellation */
  if (sum != 0.0)
    maxt /= fabs(sum);
  maxt *= MACHEP; /* this way avoids multiply overflow */
  pcanc = fabs(MACHEP * n + maxt);

blowup:

  *err = pcanc;

  return (sum);
}

/*							hy1f1a()	*/
/* asymptotic formula for hypergeometric function:
 *
 *        (    -a
 *  --    ( |z|
 * |  (b) ( -------- 2f0( a, 1+a-b, -1/x )
 *        (  --
 *        ( |  (b-a)
 *
 *
 *                                x    a-b                     )
 *                               e  |x|                        )
 *                             + -------- 2f0( b-a, 1-a, 1/x ) )
 *                                --                           )
 *                               |  (a)                        )
 */

static double hy1f1a(a, b, x, err) double a, b, x;
double *err;
{
  double h1, h2, t, u, temp, acanc, asum, err1, err2;

  if (x == 0) {
    acanc = 1.0;
    asum = MAXNUM;
    goto adone;
  }
  temp = log(fabs(x));
  t = x + temp * (a - b);
  u = -temp * a;

  if (b > 0) {
    temp = lgam(b);
    t += temp;
    u += temp;
  }

  h1 = hyp2f0(a, a - b + 1, -1.0 / x, 1, &err1);

  temp = exp(u) / gamma(b - a);
  h1 *= temp;
  err1 *= temp;

  h2 = hyp2f0(b - a, 1.0 - a, 1.0 / x, 2, &err2);

  if (a < 0)
    temp = exp(t) / gamma(a);
  else
    temp = exp(t - lgam(a));

  h2 *= temp;
  err2 *= temp;

  if (x < 0.0)
    asum = h1;
  else
    asum = h2;

  acanc = fabs(err1) + fabs(err2);

  if (b < 0) {
    temp = gamma(b);
    asum *= temp;
    acanc *= fabs(temp);
  }

  if (asum != 0.0)
    acanc /= fabs(asum);

  acanc *= 30.0; /* fudge factor, since error of asymptotic formula
                  * often seems this much larger than advertised */

adone:

  *err = acanc;
  return (asum);
}

/*							hyp2f0()	*/

double hyp2f0(a, b, x, type, err) double a, b, x;
int type; /* determines what converging factor to use */
double *err;
{
  double a0, alast, t, tlast, maxt;
  double n, an, bn, u, sum, temp;

  an = a;
  bn = b;
  a0 = 1.0e0;
  alast = 1.0e0;
  sum = 0.0;
  n = 1.0e0;
  t = 1.0e0;
  tlast = 1.0e9;
  maxt = 0.0;

  do {
    if (an == 0)
      goto pdone;
    if (bn == 0)
      goto pdone;

    u = an * (bn * x / n);

    /* check for blowup */
    temp = fabs(u);
    if ((temp > 1.0) && (maxt > (MAXNUM / temp)))
      goto error;

    a0 *= u;
    t = fabs(a0);

    /* terminating condition for asymptotic series */
    if (t > tlast)
      goto ndone;

    tlast = t;
    sum += alast; /* the sum is one term behind */
    alast = a0;

    if (n > 200)
      goto ndone;

    an += 1.0e0;
    bn += 1.0e0;
    n += 1.0e0;
    if (t > maxt)
      maxt = t;
  } while (t > MACHEP);

pdone: /* series converged! */

  /* estimate error due to roundoff and cancellation */
  *err = fabs(MACHEP * (n + maxt));

  alast = a0;
  goto done;

ndone: /* series did not converge */

  /* The following "Converging factors" are supposed to improve accuracy,
   * but do not actually seem to accomplish very much. */

  n -= 1.0;
  x = 1.0 / x;

  switch (type) /* "type" given as subroutine argument */
  {
  case 1:
    alast *= (0.5 + (0.125 + 0.25 * b - 0.5 * a + 0.25 * x - 0.25 * n) / x);
    break;

  case 2:
    alast *= 2.0 / 3.0 - b + 2.0 * a + x - n;
    break;

  default:;
  }

  /* estimate error due to roundoff, cancellation, and nonconvergence */
  *err = MACHEP * (n + maxt) + fabs(a0);

done:
  sum += alast;
  return (sum);

/* series blew up: */
error:
  *err = MAXNUM;
  mtherr("hyperg", TLOSS);
  return (sum);
}