/*                                                     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"

extern double MACHEP, MAXNUM;

static double hy1f1p(double a, double b, double x, double *acanc);
static double hy1f1a(double a, double b, double x, double *acanc);

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));


    /* Try power & asymptotic series, starting from the one that is likely OK */
    if (fabs(x) < 10 + fabs(a) + fabs(b)) {
	psum = hy1f1p(a, b, x, &pcanc);
	if (pcanc < 1.0e-15)
	    goto done;
	asum = hy1f1a(a, b, x, &acanc);
    }
    else {
	psum = hy1f1a(a, b, x, &pcanc);
	if (pcanc < 1.0e-15)
	    goto done;
	asum = hy1f1p(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, maxn;
    double an, bn, maxt;
    double y, c, sumc;


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

    maxn = 200.0 + 2 * fabs(a) + 2 * fabs(b);

    while (t > MACHEP) {
	if (bn == 0) {		/* check bn first since if both   */
	    mtherr("hyperg", SING);
	    return (NPY_INFINITY);	/* an and bn are zero it is     */
	}
	if (an == 0)		/* a singularity            */
	    return (sum);
	if (n > maxn) {
	    /* too many terms; take the last one as error estimate */
	    c = fabs(c) + fabs(t) * 50.0;
	    goto pdone;
	}
	u = x * (an / (bn * n));

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

	a0 *= u;

	y = a0 - c;
	sumc = sum + y;
	c = (sumc - sum) - y;
	sum = sumc;

	t = fabs(a0);

	an += 1.0;
	bn += 1.0;
	n += 1.0;
    }

  pdone:

    /* estimate error due to roundoff and cancellation */
    if (sum != 0.0) {
	*err = fabs(c / sum);
    }
    else {
	*err = fabs(c);
    }

    if (*err != *err) {
	/* nan */
	*err = 1.0;
    }

    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 = NPY_INFINITY;
	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);

    if (acanc != acanc)
	/* nan */
	acanc = 1.0;

    if (asum == NPY_INFINITY || asum == -NPY_INFINITY)
	/* infinity */
	acanc = 0;

    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:
	 * the series is divergent (if a or b is not a negative integer),
	 * but its leading part can be used as an asymptotic expansion
	 */
	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 = NPY_INFINITY;
    mtherr("hyperg", TLOSS);
    return (sum);
}
