/*                                                      hyp2f1.c
 *
 *      Gauss hypergeometric function   F
 *                                     2 1
 *
 *
 * SYNOPSIS:
 *
 * double a, b, c, x, y, hyp2f1();
 *
 * y = hyp2f1( a, b, c, x );
 *
 *
 * DESCRIPTION:
 *
 *
 *  hyp2f1( a, b, c, x )  =   F ( a, b; c; x )
 *                           2 1
 *
 *           inf.
 *            -   a(a+1)...(a+k) b(b+1)...(b+k)   k+1
 *   =  1 +   >   -----------------------------  x   .
 *            -         c(c+1)...(c+k) (k+1)!
 *          k = 0
 *
 *  Cases addressed are
 *      Tests and escapes for negative integer a, b, or c
 *      Linear transformation if c - a or c - b negative integer
 *      Special case c = a or c = b
 *      Linear transformation for  x near +1
 *      Transformation for x < -0.5
 *      Psi function expansion if x > 0.5 and c - a - b integer
 *      Conditionally, a recurrence on c to make c-a-b > 0
 *
 * |x| > 1 is rejected.
 *
 * The parameters a, b, c are considered to be integer
 * valued if they are within 1.0e-14 of the nearest integer
 * (1.0e-13 for IEEE arithmetic).
 *
 * ACCURACY:
 *
 *
 *               Relative error (-1 < x < 1):
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -1,7        230000      1.2e-11     5.2e-14
 *
 * Several special cases also tested with a, b, c in
 * the range -7 to 7.
 *
 * ERROR MESSAGES:
 *
 * A "partial loss of precision" message is printed if
 * the internally estimated relative error exceeds 1^-12.
 * A "singularity" message is printed on overflow or
 * in cases not addressed (such as x < -1).
 */

/*                                                      hyp2f1  */
/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
#ifdef DEC
#define EPS 1.0e-14
#define EPS2 1.0e-11
#endif
#ifdef IBMPC
#define EPS 1.0e-13
#define EPS2 1.0e-10
#endif
#ifdef MIEEE
#define EPS 1.0e-13
#define EPS2 1.0e-10
#endif
#ifdef UNK
#define EPS 1.0e-13
#define EPS2 1.0e-10
#endif
#define ETHRESH 1.0e-12
#ifdef ANSIPROT
extern double md_fabs ( double );
extern double md_pow ( double, double );
extern double md_round ( double );
extern double md_gamma ( double );
extern double md_log ( double );
extern double md_exp ( double );
extern double psi ( double );
static double hyt2f1(double, double, double, double, double *);
static double hys2f1(double, double, double, double, double *);
double hyp2f1(double, double, double, double);
#else
double md_fabs(), md_pow(), md_round(), md_gamma(), md_log(), md_exp(), psi();
static double hyt2f1();
static double hys2f1();
double hyp2f1();
#endif
extern double MAXNUM, MACHEP;
double hyp2f1( a, b, c, x )
double a, b, c, x;
{
double d, d1, d2, e;
double p, q, r, s, y, ax;
double ia, ib, ic, id, err;
int flag, i, aid;
err = 0.0;
ax = md_fabs(x);
s = 1.0 - x;
flag = 0;
ia = md_round(a); /* nearest integer to a */
ib = md_round(b);
if( a <= 0 )
        {
        if( md_fabs(a-ia) < EPS )            /* a is a negative integer */
                flag |= 1;
        }
if( b <= 0 )
        {
        if( md_fabs(b-ib) < EPS )            /* b is a negative integer */
                flag |= 2;
        }
if( ax < 1.0 )
        {
        if( md_fabs(b-c) < EPS )             /* b = c */
                {
                y = md_pow( s, -a );    /* s to the -a power */
                goto hypdon;
                }
        if( md_fabs(a-c) < EPS )             /* a = c */
                {
                y = md_pow( s, -b );    /* s to the -b power */
                goto hypdon;
                }
        }
if( c <= 0.0 )
        {
        ic = md_round(c);       /* nearest integer to c */
        if( md_fabs(c-ic) < EPS )            /* c is a negative integer */
                {
                /* check if termination before explosion */
                if( (flag & 1) && (ia > ic) )
                        goto hypok;
                if( (flag & 2) && (ib > ic) )
                        goto hypok;
                goto hypdiv;
                }
        }
if( flag )                      /* function is a polynomial */
        goto hypok;
if( ax > 1.0 )                       /* series diverges      */
        goto hypdiv;
p = c - a;
ia = md_round(p); /* nearest integer to c-a */
if( (ia <= 0.0) && (md_fabs(p-ia) < EPS) )        /* negative int c - a */
        flag |= 4;
r = c - b;
ib = md_round(r); /* nearest integer to c-b */
if( (ib <= 0.0) && (md_fabs(r-ib) < EPS) )        /* negative int c - b */
        flag |= 8;
d = c - a - b;
id = md_round(d); /* nearest integer to d */
q = md_fabs(d-id);
/* Thanks to Christian Burger <BURGER@DMRHRZ11.HRZ.Uni-Marburg.DE>
 * for reporting a bug here.  */
if( md_fabs(ax-1.0) < EPS )                  /* |x| == 1.0   */
        {
        if( x > 0.0 )
                {
                if( flag & 12 ) /* negative int c-a or c-b */
                        {
                        if( d >= 0.0 )
                                goto hypf;
                        else
                                goto hypdiv;
                        }
                if( d <= 0.0 )
                        goto hypdiv;
                y = md_gamma(c)*md_gamma(d)/(md_gamma(p)*md_gamma(r));
                goto hypdon;
                }
        if( d <= -1.0 )
                goto hypdiv;
        }
/* Conditionally make d > 0 by recurrence on c
 * AMS55 #15.2.27
 */
if( d < 0.0 )
        {
/* Try the power series first */
        y = hyt2f1( a, b, c, x, &err );
        if( err < ETHRESH )
                goto hypdon;
/* Apply the recurrence if power series fails */
        err = 0.0;
        aid = 2 - id;
        e = c + aid;
        d2 = hyp2f1(a,b,e,x);
        d1 = hyp2f1(a,b,e+1.0,x);
        q = a + b + 1.0;
        for( i=0; i<aid; i++ )
                {
                r = e - 1.0;
                y = (e*(r-(2.0*e-q)*x)*d2 + (e-a)*(e-b)*x*d1)/(e*r*s);
                e = r;
                d1 = d2;
                d2 = y;
                }
        goto hypdon;
        }
if( flag & 12 )
        goto hypf; /* negative integer c-a or c-b */
hypok:
y = hyt2f1( a, b, c, x, &err );
hypdon:
if( err > ETHRESH )
        {
        mtherr( "hyp2f1", PLOSS );
/*      printf( "Estimated err = %.2e\n", err ); */
        }
return(y);
/* The transformation for c-a or c-b negative integer
 * AMS55 #15.3.3
 */
hypf:
y = md_pow( s, d ) * hys2f1( c-a, c-b, c, x, &err );
goto hypdon;
/* The alarm exit */
hypdiv:
mtherr( "hyp2f1", OVERFLOW );
return( MAXNUM );
}
/* Apply transformations for |x| near 1
 * then call the power series
 */
static double hyt2f1( a, b, c, x, loss )
double a, b, c, x;
double *loss;
{
double p, q, r, s, t, y, d, err, err1;
double ax, id, d1, d2, e, md_y1;
int i, aid;
err = 0.0;
s = 1.0 - x;
if( x < -0.5 )
        {
        if( b > a )
                y = md_pow( s, -a ) * hys2f1( a, c-b, c, -x/s, &err );
        else
                y = md_pow( s, -b ) * hys2f1( c-a, b, c, -x/s, &err );
        goto done;
        }
d = c - a - b;
id = md_round(d);       /* nearest integer to d */
if( x > 0.9 )
{
if( md_fabs(d-id) > EPS ) /* test for integer c-a-b */
        {
/* Try the power series first */
        y = hys2f1( a, b, c, x, &err );
        if( err < ETHRESH )
                goto done;
/* If power series fails, then apply AMS55 #15.3.6 */
        q = hys2f1( a, b, 1.0-d, s, &err ); 
        q *= md_gamma(d) /(md_gamma(c-a) * md_gamma(c-b));
        r = md_pow(s,d) * hys2f1( c-a, c-b, d+1.0, s, &err1 );
        r *= md_gamma(-d)/(md_gamma(a) * md_gamma(b));
        y = q + r;
        q = md_fabs(q); /* estimate cancellation error */
        r = md_fabs(r);
        if( q > r )
                r = q;
        err += err1 + (MACHEP*r)/y;
        y *= md_gamma(c);
        goto done;
        }
else
        {
/* Psi function expansion, AMS55 #15.3.10, #15.3.11, #15.3.12 */
        if( id >= 0.0 )
                {
                e = d;
                d1 = d;
                d2 = 0.0;
                aid = id;
                }
        else
                {
                e = -d;
                d1 = 0.0;
                d2 = d;
                aid = -id;
                }
        ax = md_log(s);
        /* sum for t = 0 */
        y = psi(1.0) + psi(1.0+e) - psi(a+d1) - psi(b+d1) - ax;
        y /= md_gamma(e+1.0);
        p = (a+d1) * (b+d1) * s / md_gamma(e+2.0);      /* Poch for t=1 */
        t = 1.0;
        do
                {
                r = psi(1.0+t) + psi(1.0+t+e) - psi(a+t+d1)
                        - psi(b+t+d1) - ax;
                q = p * r;
                y += q;
                p *= s * (a+t+d1) / (t+1.0);
                p *= (b+t+d1) / (t+1.0+e);
                t += 1.0;
                }
        while( md_fabs(q/y) > EPS );
        if( id == 0.0 )
                {
                y *= md_gamma(c)/(md_gamma(a)*md_gamma(b));
                goto psidon;
                }
        md_y1 = 1.0;
        if( aid == 1 )
                goto nosum;
        t = 0.0;
        p = 1.0;
        for( i=1; i<aid; i++ )
                {
                r = 1.0-e+t;
                p *= s * (a+t+d2) * (b+t+d2) / r;
                t += 1.0;
                p /= t;
                md_y1 += p;
                }
nosum:
        p = md_gamma(c);
        md_y1 *= md_gamma(e) * p / (md_gamma(a+d1) * md_gamma(b+d1));
        y *= p / (md_gamma(a+d2) * md_gamma(b+d2));
        if( (aid & 1) != 0 )
                y = -y;
        q = md_pow( s, id );    /* s to the id power */
        if( id > 0.0 )
                y *= q;
        else
                md_y1 *= q;
        y += md_y1;
psidon:
        goto done;
        }
}
/* Use defining power series if no special cases */
y = hys2f1( a, b, c, x, &err );
done:
*loss = err;
return(y);
}
/* Defining power series expansion of Gauss hypergeometric function */
static double hys2f1( a, b, c, x, loss )
double a, b, c, x;
double *loss; /* estimates loss of significance */
{
double f, g, h, k, m, s, u, umax;
int i;
i = 0;
umax = 0.0;
f = a;
g = b;
h = c;
s = 1.0;
u = 1.0;
k = 0.0;
do
        {
        if( md_fabs(h) < EPS )
                {
                *loss = 1.0;
                return( MAXNUM );
                }
        m = k + 1.0;
        u = u * ((f+k) * (g+k) * x / ((h+k) * m));
        s += u;
        k = md_fabs(u);  /* remember largest term summed */
        if( k > umax )
                umax = k;
        k = m;
        if( ++i > 10000 ) /* should never happen */
                {
                *loss = 1.0;
                return(s);
                }
        }
while( md_fabs(u/s) > MACHEP );
/* return estimated relative error */
*loss = (MACHEP*umax)/md_fabs(s) + (MACHEP*i);
return(s);
}