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+/* incbetf.c
+ *
+ * Incomplete beta integral
+ *
+ *
+ * SYNOPSIS:
+ *
+ * float a, b, x, y, incbetf();
+ *
+ * y = incbetf( 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.
+ * If a < 1, the function calls itself recursively after a
+ * transformation to increase a to a+1.
+ *
+ * ACCURACY:
+ *
+ * Tested at random points (a,b,x) with a and b in the indicated
+ * interval and x between 0 and 1.
+ *
+ * arithmetic domain # trials peak rms
+ * Relative error:
+ * IEEE 0,30 10000 3.7e-5 5.1e-6
+ * IEEE 0,100 10000 1.7e-4 2.5e-5
+ * The useful domain for relative error is limited by underflow
+ * of the single precision exponential function.
+ * Absolute error:
+ * IEEE 0,30 100000 2.2e-5 9.6e-7
+ * IEEE 0,100 10000 6.5e-5 3.7e-6
+ *
+ * Larger errors may occur for extreme ratios of a and b.
+ *
+ * ERROR MESSAGES:
+ * message condition value returned
+ * incbetf domain x<0, x>1 0.0
+ */
+
+
+/*
+Cephes Math Library, Release 2.2: July, 1992
+Copyright 1984, 1987, 1992 by Stephen L. Moshier
+Direct inquiries to 30 Frost Street, Cambridge, MA 02140
+*/
+
+#include <math.h>
+
+#ifdef ANSIC
+float lgamf(float), expf(float), logf(float);
+static float incbdf(float, float, float);
+static float incbcff(float, float, float);
+float incbpsf(float, float, float);
+#else
+float lgamf(), expf(), logf();
+float incbpsf();
+static float incbcff(), incbdf();
+#endif
+
+#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
+
+/* BIG = 1/MACHEPF */
+#define BIG 16777216.
+extern float MACHEPF, MAXLOGF;
+#define MINLOGF (-MAXLOGF)
+
+float incbetf( float aaa, float bbb, float xxx )
+{
+float aa, bb, xx, ans, a, b, t, x, onemx;
+int flag;
+
+aa = aaa;
+bb = bbb;
+xx = xxx;
+if( (xx <= 0.0) || ( xx >= 1.0) )
+ {
+ if( xx == 0.0 )
+ return(0.0);
+ if( xx == 1.0 )
+ return( 1.0 );
+ mtherr( "incbetf", DOMAIN );
+ return( 0.0 );
+ }
+
+onemx = 1.0 - xx;
+
+
+/* transformation for small aa */
+
+if( aa <= 1.0 )
+ {
+ ans = incbetf( aa+1.0, bb, xx );
+ t = aa*logf(xx) + bb*logf( 1.0-xx )
+ + lgamf(aa+bb) - lgamf(aa+1.0) - lgamf(bb);
+ if( t > MINLOGF )
+ ans += expf(t);
+ return( ans );
+ }
+
+
+/* see if x is greater than the mean */
+
+if( xx > (aa/(aa+bb)) )
+ {
+ flag = 1;
+ a = bb;
+ b = aa;
+ t = xx;
+ x = onemx;
+ }
+else
+ {
+ flag = 0;
+ a = aa;
+ b = bb;
+ t = onemx;
+ x = xx;
+ }
+
+/* transformation for small aa */
+/*
+if( a <= 1.0 )
+ {
+ ans = a*logf(x) + b*logf( onemx )
+ + lgamf(a+b) - lgamf(a+1.0) - lgamf(b);
+ t = incbetf( a+1.0, b, x );
+ if( ans > MINLOGF )
+ t += expf(ans);
+ goto bdone;
+ }
+*/
+/* Choose expansion for optimal convergence */
+
+
+if( b > 10.0 )
+ {
+if( fabsf(b*x/a) < 0.3 )
+ {
+ t = incbpsf( a, b, x );
+ goto bdone;
+ }
+ }
+
+ans = x * (a+b-2.0)/(a-1.0);
+if( ans < 1.0 )
+ {
+ ans = incbcff( a, b, x );
+ t = b * logf( t );
+ }
+else
+ {
+ ans = incbdf( a, b, x );
+ t = (b-1.0) * logf(t);
+ }
+
+t += a*logf(x) + lgamf(a+b) - lgamf(a) - lgamf(b);
+t += logf( ans/a );
+
+if( t < MINLOGF )
+ {
+ t = 0.0;
+ if( flag == 0 )
+ {
+ mtherr( "incbetf", UNDERFLOW );
+ }
+ }
+else
+ {
+ t = expf(t);
+ }
+bdone:
+
+if( flag )
+ t = 1.0 - t;
+
+return( t );
+}
+
+/* Continued fraction expansion #1
+ * for incomplete beta integral
+ */
+
+static float incbcff( float aa, float bb, float xx )
+{
+float a, b, x, xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
+float k1, k2, k3, k4, k5, k6, k7, k8;
+float r, t, ans;
+static float big = BIG;
+int n;
+
+a = aa;
+b = bb;
+x = xx;
+k1 = a;
+k2 = a + b;
+k3 = a;
+k4 = a + 1.0;
+k5 = 1.0;
+k6 = b - 1.0;
+k7 = k4;
+k8 = a + 2.0;
+
+pkm2 = 0.0;
+qkm2 = 1.0;
+pkm1 = 1.0;
+qkm1 = 1.0;
+ans = 1.0;
+r = 0.0;
+n = 0;
+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 )
+ r = pk/qk;
+ if( r != 0 )
+ {
+ t = fabsf( (ans - r)/r );
+ ans = r;
+ }
+ else
+ t = 1.0;
+
+ if( t < MACHEPF )
+ goto cdone;
+
+ k1 += 1.0;
+ k2 += 1.0;
+ k3 += 2.0;
+ k4 += 2.0;
+ k5 += 1.0;
+ k6 -= 1.0;
+ k7 += 2.0;
+ k8 += 2.0;
+
+ if( (fabsf(qk) + fabsf(pk)) > big )
+ {
+ pkm2 *= MACHEPF;
+ pkm1 *= MACHEPF;
+ qkm2 *= MACHEPF;
+ qkm1 *= MACHEPF;
+ }
+ if( (fabsf(qk) < MACHEPF) || (fabsf(pk) < MACHEPF) )
+ {
+ pkm2 *= big;
+ pkm1 *= big;
+ qkm2 *= big;
+ qkm1 *= big;
+ }
+ }
+while( ++n < 100 );
+
+cdone:
+return(ans);
+}
+
+
+/* Continued fraction expansion #2
+ * for incomplete beta integral
+ */
+
+static float incbdf( float aa, float bb, float xx )
+{
+float a, b, x, xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
+float k1, k2, k3, k4, k5, k6, k7, k8;
+float r, t, ans, z;
+static float big = BIG;
+int n;
+
+a = aa;
+b = bb;
+x = xx;
+k1 = a;
+k2 = b - 1.0;
+k3 = a;
+k4 = a + 1.0;
+k5 = 1.0;
+k6 = a + b;
+k7 = a + 1.0;;
+k8 = a + 2.0;
+
+pkm2 = 0.0;
+qkm2 = 1.0;
+pkm1 = 1.0;
+qkm1 = 1.0;
+z = x / (1.0-x);
+ans = 1.0;
+r = 0.0;
+n = 0;
+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 )
+ r = pk/qk;
+ if( r != 0 )
+ {
+ t = fabsf( (ans - r)/r );
+ ans = r;
+ }
+ else
+ t = 1.0;
+
+ if( t < MACHEPF )
+ goto cdone;
+
+ k1 += 1.0;
+ k2 -= 1.0;
+ k3 += 2.0;
+ k4 += 2.0;
+ k5 += 1.0;
+ k6 += 1.0;
+ k7 += 2.0;
+ k8 += 2.0;
+
+ if( (fabsf(qk) + fabsf(pk)) > big )
+ {
+ pkm2 *= MACHEPF;
+ pkm1 *= MACHEPF;
+ qkm2 *= MACHEPF;
+ qkm1 *= MACHEPF;
+ }
+ if( (fabsf(qk) < MACHEPF) || (fabsf(pk) < MACHEPF) )
+ {
+ pkm2 *= big;
+ pkm1 *= big;
+ qkm2 *= big;
+ qkm1 *= big;
+ }
+ }
+while( ++n < 100 );
+
+cdone:
+return(ans);
+}
+
+
+/* power series */
+float incbpsf( float aa, float bb, float xx )
+{
+float a, b, x, t, u, y, s;
+
+a = aa;
+b = bb;
+x = xx;
+
+y = a * logf(x) + (b-1.0)*logf(1.0-x) - logf(a);
+y -= lgamf(a) + lgamf(b);
+y += lgamf(a+b);
+
+
+t = x / (1.0 - x);
+s = 0.0;
+u = 1.0;
+do
+ {
+ b -= 1.0;
+ if( b == 0.0 )
+ break;
+ a += 1.0;
+ u *= t*b/a;
+ s += u;
+ }
+while( fabsf(u) > MACHEPF );
+
+if( y < MINLOGF )
+ {
+ mtherr( "incbetf", UNDERFLOW );
+ s = 0.0;
+ }
+else
+ s = expf(y) * (1.0 + s);
+/*printf( "incbpsf: %.4e\n", s );*/
+return(s);
+}
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