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authorEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
committerEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
commit1077fa4d772832f77a677ce7fb7c2d513b959e3f (patch)
tree579bee13fb0b58d2800206366ec2caecbb15f3fc /libm/double/incbi.c
parent22358dd7ce7bb49792204b698f01a6f69b9c8e08 (diff)
uClibc now has a math library. muahahahaha!
-Erik
Diffstat (limited to 'libm/double/incbi.c')
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1 files changed, 313 insertions, 0 deletions
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+/* incbi()
+ *
+ * Inverse of imcomplete beta integral
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * double a, b, x, y, incbi();
+ *
+ * x = incbi( a, b, y );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Given y, the function finds x such that
+ *
+ * incbet( a, b, x ) = y .
+ *
+ * The routine performs interval halving or Newton iterations to find the
+ * root of incbet(a,b,x) - y = 0.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * x a,b
+ * arithmetic domain domain # trials peak rms
+ * IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13
+ * IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15
+ * IEEE 0,1 0,5 50000 1.1e-12 5.5e-15
+ * VAX 0,1 .5,100 25000 3.5e-14 1.1e-15
+ * With a and b constrained to half-integer or integer values:
+ * IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13
+ * IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16
+ * With a = .5, b constrained to half-integer or integer values:
+ * IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11
+ */
+
+
+/*
+Cephes Math Library Release 2.8: June, 2000
+Copyright 1984, 1996, 2000 by Stephen L. Moshier
+*/
+
+#include <math.h>
+
+extern double MACHEP, MAXNUM, MAXLOG, MINLOG;
+#ifdef ANSIPROT
+extern double ndtri ( double );
+extern double exp ( double );
+extern double fabs ( double );
+extern double log ( double );
+extern double sqrt ( double );
+extern double lgam ( double );
+extern double incbet ( double, double, double );
+#else
+double ndtri(), exp(), fabs(), log(), sqrt(), lgam(), incbet();
+#endif
+
+double incbi( aa, bb, yy0 )
+double aa, bb, yy0;
+{
+double a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
+int i, rflg, dir, nflg;
+
+
+i = 0;
+if( yy0 <= 0 )
+ return(0.0);
+if( yy0 >= 1.0 )
+ return(1.0);
+x0 = 0.0;
+yl = 0.0;
+x1 = 1.0;
+yh = 1.0;
+nflg = 0;
+
+if( aa <= 1.0 || bb <= 1.0 )
+ {
+ dithresh = 1.0e-6;
+ rflg = 0;
+ a = aa;
+ b = bb;
+ y0 = yy0;
+ x = a/(a+b);
+ y = incbet( a, b, x );
+ goto ihalve;
+ }
+else
+ {
+ dithresh = 1.0e-4;
+ }
+/* approximation to inverse function */
+
+yp = -ndtri(yy0);
+
+if( yy0 > 0.5 )
+ {
+ rflg = 1;
+ a = bb;
+ b = aa;
+ y0 = 1.0 - yy0;
+ yp = -yp;
+ }
+else
+ {
+ rflg = 0;
+ a = aa;
+ b = bb;
+ y0 = yy0;
+ }
+
+lgm = (yp * yp - 3.0)/6.0;
+x = 2.0/( 1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0) );
+d = yp * sqrt( x + lgm ) / x
+ - ( 1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0) )
+ * (lgm + 5.0/6.0 - 2.0/(3.0*x));
+d = 2.0 * d;
+if( d < MINLOG )
+ {
+ x = 1.0;
+ goto under;
+ }
+x = a/( a + b * exp(d) );
+y = incbet( a, b, x );
+yp = (y - y0)/y0;
+if( fabs(yp) < 0.2 )
+ goto newt;
+
+/* Resort to interval halving if not close enough. */
+ihalve:
+
+dir = 0;
+di = 0.5;
+for( i=0; i<100; i++ )
+ {
+ if( i != 0 )
+ {
+ x = x0 + di * (x1 - x0);
+ if( x == 1.0 )
+ x = 1.0 - MACHEP;
+ if( x == 0.0 )
+ {
+ di = 0.5;
+ x = x0 + di * (x1 - x0);
+ if( x == 0.0 )
+ goto under;
+ }
+ y = incbet( a, b, x );
+ yp = (x1 - x0)/(x1 + x0);
+ if( fabs(yp) < dithresh )
+ goto newt;
+ yp = (y-y0)/y0;
+ if( fabs(yp) < dithresh )
+ goto newt;
+ }
+ if( y < y0 )
+ {
+ x0 = x;
+ yl = y;
+ if( dir < 0 )
+ {
+ dir = 0;
+ di = 0.5;
+ }
+ else if( dir > 3 )
+ di = 1.0 - (1.0 - di) * (1.0 - di);
+ else if( dir > 1 )
+ di = 0.5 * di + 0.5;
+ else
+ di = (y0 - y)/(yh - yl);
+ dir += 1;
+ if( x0 > 0.75 )
+ {
+ if( rflg == 1 )
+ {
+ rflg = 0;
+ a = aa;
+ b = bb;
+ y0 = yy0;
+ }
+ else
+ {
+ rflg = 1;
+ a = bb;
+ b = aa;
+ y0 = 1.0 - yy0;
+ }
+ x = 1.0 - x;
+ y = incbet( a, b, x );
+ x0 = 0.0;
+ yl = 0.0;
+ x1 = 1.0;
+ yh = 1.0;
+ goto ihalve;
+ }
+ }
+ else
+ {
+ x1 = x;
+ if( rflg == 1 && x1 < MACHEP )
+ {
+ x = 0.0;
+ goto done;
+ }
+ yh = y;
+ if( dir > 0 )
+ {
+ dir = 0;
+ di = 0.5;
+ }
+ else if( dir < -3 )
+ di = di * di;
+ else if( dir < -1 )
+ di = 0.5 * di;
+ else
+ di = (y - y0)/(yh - yl);
+ dir -= 1;
+ }
+ }
+mtherr( "incbi", PLOSS );
+if( x0 >= 1.0 )
+ {
+ x = 1.0 - MACHEP;
+ goto done;
+ }
+if( x <= 0.0 )
+ {
+under:
+ mtherr( "incbi", UNDERFLOW );
+ x = 0.0;
+ goto done;
+ }
+
+newt:
+
+if( nflg )
+ goto done;
+nflg = 1;
+lgm = lgam(a+b) - lgam(a) - lgam(b);
+
+for( i=0; i<8; i++ )
+ {
+ /* Compute the function at this point. */
+ if( i != 0 )
+ y = incbet(a,b,x);
+ if( y < yl )
+ {
+ x = x0;
+ y = yl;
+ }
+ else if( y > yh )
+ {
+ x = x1;
+ y = yh;
+ }
+ else if( y < y0 )
+ {
+ x0 = x;
+ yl = y;
+ }
+ else
+ {
+ x1 = x;
+ yh = y;
+ }
+ if( x == 1.0 || x == 0.0 )
+ break;
+ /* Compute the derivative of the function at this point. */
+ d = (a - 1.0) * log(x) + (b - 1.0) * log(1.0-x) + lgm;
+ if( d < MINLOG )
+ goto done;
+ if( d > MAXLOG )
+ break;
+ d = exp(d);
+ /* Compute the step to the next approximation of x. */
+ d = (y - y0)/d;
+ xt = x - d;
+ if( xt <= x0 )
+ {
+ y = (x - x0) / (x1 - x0);
+ xt = x0 + 0.5 * y * (x - x0);
+ if( xt <= 0.0 )
+ break;
+ }
+ if( xt >= x1 )
+ {
+ y = (x1 - x) / (x1 - x0);
+ xt = x1 - 0.5 * y * (x1 - x);
+ if( xt >= 1.0 )
+ break;
+ }
+ x = xt;
+ if( fabs(d/x) < 128.0 * MACHEP )
+ goto done;
+ }
+/* Did not converge. */
+dithresh = 256.0 * MACHEP;
+goto ihalve;
+
+done:
+
+if( rflg )
+ {
+ if( x <= MACHEP )
+ x = 1.0 - MACHEP;
+ else
+ x = 1.0 - x;
+ }
+return( x );
+}