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+/* incbetl.c
+ *
+ * Incomplete beta integral
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double a, b, x, y, incbetl();
+ *
+ * y = incbetl( a, b, x );
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns incomplete beta integral of the arguments, evaluated
+ * from zero to x. The function is defined as
+ *
+ * x
+ * - -
+ * | (a+b) | | a-1 b-1
+ * ----------- | t (1-t) dt.
+ * - - | |
+ * | (a) | (b) -
+ * 0
+ *
+ * The domain of definition is 0 <= x <= 1. In this
+ * implementation a and b are restricted to positive values.
+ * The integral from x to 1 may be obtained by the symmetry
+ * relation
+ *
+ * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
+ *
+ * The integral is evaluated by a continued fraction expansion
+ * or, when b*x is small, by a power series.
+ *
+ * ACCURACY:
+ *
+ * Tested at random points (a,b,x) with x between 0 and 1.
+ * arithmetic domain # trials peak rms
+ * IEEE 0,5 20000 4.5e-18 2.4e-19
+ * IEEE 0,100 100000 3.9e-17 1.0e-17
+ * Half-integer a, b:
+ * IEEE .5,10000 100000 3.9e-14 4.4e-15
+ * Outputs smaller than the IEEE gradual underflow threshold
+ * were excluded from these statistics.
+ *
+ * ERROR MESSAGES:
+ *
+ * message condition value returned
+ * incbetl domain x<0, x>1 0.0
+ */
+
+
+/*
+Cephes Math Library, Release 2.3: January, 1995
+Copyright 1984, 1995 by Stephen L. Moshier
+*/
+
+#include <math.h>
+
+#define MAXGAML 1755.455L
+static long double big = 9.223372036854775808e18L;
+static long double biginv = 1.084202172485504434007e-19L;
+extern long double MACHEPL, MINLOGL, MAXLOGL;
+
+#ifdef ANSIPROT
+extern long double gammal ( long double );
+extern long double lgaml ( long double );
+extern long double expl ( long double );
+extern long double logl ( long double );
+extern long double fabsl ( long double );
+extern long double powl ( long double, long double );
+static long double incbcfl( long double, long double, long double );
+static long double incbdl( long double, long double, long double );
+static long double pseriesl( long double, long double, long double );
+#else
+long double gammal(), lgaml(), expl(), logl(), fabsl(), powl();
+static long double incbcfl(), incbdl(), pseriesl();
+#endif
+
+long double incbetl( aa, bb, xx )
+long double aa, bb, xx;
+{
+long double a, b, t, x, w, xc, y;
+int flag;
+
+if( aa <= 0.0L || bb <= 0.0L )
+ goto domerr;
+
+if( (xx <= 0.0L) || ( xx >= 1.0L) )
+ {
+ if( xx == 0.0L )
+ return( 0.0L );
+ if( xx == 1.0L )
+ return( 1.0L );
+domerr:
+ mtherr( "incbetl", DOMAIN );
+ return( 0.0L );
+ }
+
+flag = 0;
+if( (bb * xx) <= 1.0L && xx <= 0.95L)
+ {
+ t = pseriesl(aa, bb, xx);
+ goto done;
+ }
+
+w = 1.0L - xx;
+
+/* Reverse a and b if x is greater than the mean. */
+if( xx > (aa/(aa+bb)) )
+ {
+ flag = 1;
+ a = bb;
+ b = aa;
+ xc = xx;
+ x = w;
+ }
+else
+ {
+ a = aa;
+ b = bb;
+ xc = w;
+ x = xx;
+ }
+
+if( flag == 1 && (b * x) <= 1.0L && x <= 0.95L)
+ {
+ t = pseriesl(a, b, x);
+ goto done;
+ }
+
+/* Choose expansion for optimal convergence */
+y = x * (a+b-2.0L) - (a-1.0L);
+if( y < 0.0L )
+ w = incbcfl( a, b, x );
+else
+ w = incbdl( a, b, x ) / xc;
+
+/* Multiply w by the factor
+ a b _ _ _
+ x (1-x) | (a+b) / ( a | (a) | (b) ) . */
+
+y = a * logl(x);
+t = b * logl(xc);
+if( (a+b) < MAXGAML && fabsl(y) < MAXLOGL && fabsl(t) < MAXLOGL )
+ {
+ t = powl(xc,b);
+ t *= powl(x,a);
+ t /= a;
+ t *= w;
+ t *= gammal(a+b) / (gammal(a) * gammal(b));
+ goto done;
+ }
+else
+ {
+ /* Resort to logarithms. */
+ y += t + lgaml(a+b) - lgaml(a) - lgaml(b);
+ y += logl(w/a);
+ if( y < MINLOGL )
+ t = 0.0L;
+ else
+ t = expl(y);
+ }
+
+done:
+
+if( flag == 1 )
+ {
+ if( t <= MACHEPL )
+ t = 1.0L - MACHEPL;
+ else
+ t = 1.0L - t;
+ }
+return( t );
+}
+
+/* Continued fraction expansion #1
+ * for incomplete beta integral
+ */
+
+static long double incbcfl( a, b, x )
+long double a, b, x;
+{
+long double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
+long double k1, k2, k3, k4, k5, k6, k7, k8;
+long double r, t, ans, thresh;
+int n;
+
+k1 = a;
+k2 = a + b;
+k3 = a;
+k4 = a + 1.0L;
+k5 = 1.0L;
+k6 = b - 1.0L;
+k7 = k4;
+k8 = a + 2.0L;
+
+pkm2 = 0.0L;
+qkm2 = 1.0L;
+pkm1 = 1.0L;
+qkm1 = 1.0L;
+ans = 1.0L;
+r = 1.0L;
+n = 0;
+thresh = 3.0L * MACHEPL;
+do
+ {
+
+ xk = -( x * k1 * k2 )/( k3 * k4 );
+ pk = pkm1 + pkm2 * xk;
+ qk = qkm1 + qkm2 * xk;
+ pkm2 = pkm1;
+ pkm1 = pk;
+ qkm2 = qkm1;
+ qkm1 = qk;
+
+ xk = ( x * k5 * k6 )/( k7 * k8 );
+ pk = pkm1 + pkm2 * xk;
+ qk = qkm1 + qkm2 * xk;
+ pkm2 = pkm1;
+ pkm1 = pk;
+ qkm2 = qkm1;
+ qkm1 = qk;
+
+ if( qk != 0.0L )
+ r = pk/qk;
+ if( r != 0.0L )
+ {
+ t = fabsl( (ans - r)/r );
+ ans = r;
+ }
+ else
+ t = 1.0L;
+
+ if( t < thresh )
+ goto cdone;
+
+ k1 += 1.0L;
+ k2 += 1.0L;
+ k3 += 2.0L;
+ k4 += 2.0L;
+ k5 += 1.0L;
+ k6 -= 1.0L;
+ k7 += 2.0L;
+ k8 += 2.0L;
+
+ if( (fabsl(qk) + fabsl(pk)) > big )
+ {
+ pkm2 *= biginv;
+ pkm1 *= biginv;
+ qkm2 *= biginv;
+ qkm1 *= biginv;
+ }
+ if( (fabsl(qk) < biginv) || (fabsl(pk) < biginv) )
+ {
+ pkm2 *= big;
+ pkm1 *= big;
+ qkm2 *= big;
+ qkm1 *= big;
+ }
+ }
+while( ++n < 400 );
+mtherr( "incbetl", PLOSS );
+
+cdone:
+return(ans);
+}
+
+
+/* Continued fraction expansion #2
+ * for incomplete beta integral
+ */
+
+static long double incbdl( a, b, x )
+long double a, b, x;
+{
+long double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
+long double k1, k2, k3, k4, k5, k6, k7, k8;
+long double r, t, ans, z, thresh;
+int n;
+
+k1 = a;
+k2 = b - 1.0L;
+k3 = a;
+k4 = a + 1.0L;
+k5 = 1.0L;
+k6 = a + b;
+k7 = a + 1.0L;
+k8 = a + 2.0L;
+
+pkm2 = 0.0L;
+qkm2 = 1.0L;
+pkm1 = 1.0L;
+qkm1 = 1.0L;
+z = x / (1.0L-x);
+ans = 1.0L;
+r = 1.0L;
+n = 0;
+thresh = 3.0L * MACHEPL;
+do
+ {
+
+ xk = -( z * k1 * k2 )/( k3 * k4 );
+ pk = pkm1 + pkm2 * xk;
+ qk = qkm1 + qkm2 * xk;
+ pkm2 = pkm1;
+ pkm1 = pk;
+ qkm2 = qkm1;
+ qkm1 = qk;
+
+ xk = ( z * k5 * k6 )/( k7 * k8 );
+ pk = pkm1 + pkm2 * xk;
+ qk = qkm1 + qkm2 * xk;
+ pkm2 = pkm1;
+ pkm1 = pk;
+ qkm2 = qkm1;
+ qkm1 = qk;
+
+ if( qk != 0.0L )
+ r = pk/qk;
+ if( r != 0.0L )
+ {
+ t = fabsl( (ans - r)/r );
+ ans = r;
+ }
+ else
+ t = 1.0L;
+
+ if( t < thresh )
+ goto cdone;
+
+ k1 += 1.0L;
+ k2 -= 1.0L;
+ k3 += 2.0L;
+ k4 += 2.0L;
+ k5 += 1.0L;
+ k6 += 1.0L;
+ k7 += 2.0L;
+ k8 += 2.0L;
+
+ if( (fabsl(qk) + fabsl(pk)) > big )
+ {
+ pkm2 *= biginv;
+ pkm1 *= biginv;
+ qkm2 *= biginv;
+ qkm1 *= biginv;
+ }
+ if( (fabsl(qk) < biginv) || (fabsl(pk) < biginv) )
+ {
+ pkm2 *= big;
+ pkm1 *= big;
+ qkm2 *= big;
+ qkm1 *= big;
+ }
+ }
+while( ++n < 400 );
+mtherr( "incbetl", PLOSS );
+
+cdone:
+return(ans);
+}
+
+/* Power series for incomplete gamma integral.
+ Use when b*x is small. */
+
+static long double pseriesl( a, b, x )
+long double a, b, x;
+{
+long double s, t, u, v, n, t1, z, ai;
+
+ai = 1.0L / a;
+u = (1.0L - b) * x;
+v = u / (a + 1.0L);
+t1 = v;
+t = u;
+n = 2.0L;
+s = 0.0L;
+z = MACHEPL * ai;
+while( fabsl(v) > z )
+ {
+ u = (n - b) * x / n;
+ t *= u;
+ v = t / (a + n);
+ s += v;
+ n += 1.0L;
+ }
+s += t1;
+s += ai;
+
+u = a * logl(x);
+if( (a+b) < MAXGAML && fabsl(u) < MAXLOGL )
+ {
+ t = gammal(a+b)/(gammal(a)*gammal(b));
+ s = s * t * powl(x,a);
+ }
+else
+ {
+ t = lgaml(a+b) - lgaml(a) - lgaml(b) + u + logl(s);
+ if( t < MINLOGL )
+ s = 0.0L;
+ else
+ s = expl(t);
+ }
+return(s);
+}
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