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Diffstat (limited to 'libm/ldouble/igamil.c')
| -rw-r--r-- | libm/ldouble/igamil.c | 193 |
1 files changed, 193 insertions, 0 deletions
diff --git a/libm/ldouble/igamil.c b/libm/ldouble/igamil.c new file mode 100644 index 000000000..1abe503e9 --- /dev/null +++ b/libm/ldouble/igamil.c @@ -0,0 +1,193 @@ +/* igamil() + * + * Inverse of complemented imcomplete gamma integral + * + * + * + * SYNOPSIS: + * + * long double a, x, y, igamil(); + * + * x = igamil( a, y ); + * + * + * + * DESCRIPTION: + * + * Given y, the function finds x such that + * + * igamc( a, x ) = y. + * + * Starting with the approximate value + * + * 3 + * x = a t + * + * where + * + * t = 1 - d - ndtri(y) sqrt(d) + * + * and + * + * d = 1/9a, + * + * the routine performs up to 10 Newton iterations to find the + * root of igamc(a,x) - y = 0. + * + * + * ACCURACY: + * + * Tested for a ranging from 0.5 to 30 and x from 0 to 0.5. + * + * Relative error: + * arithmetic domain # trials peak rms + * DEC 0,0.5 3400 8.8e-16 1.3e-16 + * IEEE 0,0.5 10000 1.1e-14 1.0e-15 + * + */ + +/* +Cephes Math Library Release 2.3: March, 1995 +Copyright 1984, 1995 by Stephen L. Moshier +*/ + +#include <math.h> + +extern long double MACHEPL, MAXNUML, MAXLOGL, MINLOGL; +#ifdef ANSIPROT +extern long double ndtril ( long double ); +extern long double expl ( long double ); +extern long double fabsl ( long double ); +extern long double logl ( long double ); +extern long double sqrtl ( long double ); +extern long double lgaml ( long double ); +extern long double igamcl ( long double, long double ); +#else +long double ndtril(), expl(), fabsl(), logl(), sqrtl(), lgaml(); +long double igamcl(); +#endif + +long double igamil( a, y0 ) +long double a, y0; +{ +long double x0, x1, x, yl, yh, y, d, lgm, dithresh; +int i, dir; + +/* bound the solution */ +x0 = MAXNUML; +yl = 0.0L; +x1 = 0.0L; +yh = 1.0L; +dithresh = 4.0 * MACHEPL; + +/* approximation to inverse function */ +d = 1.0L/(9.0L*a); +y = ( 1.0L - d - ndtril(y0) * sqrtl(d) ); +x = a * y * y * y; + +lgm = lgaml(a); + +for( i=0; i<10; i++ ) + { + if( x > x0 || x < x1 ) + goto ihalve; + y = igamcl(a,x); + if( y < yl || y > yh ) + goto ihalve; + if( y < y0 ) + { + x0 = x; + yl = y; + } + else + { + x1 = x; + yh = y; + } +/* compute the derivative of the function at this point */ + d = (a - 1.0L) * logl(x0) - x0 - lgm; + if( d < -MAXLOGL ) + goto ihalve; + d = -expl(d); +/* compute the step to the next approximation of x */ + d = (y - y0)/d; + x = x - d; + if( i < 3 ) + continue; + if( fabsl(d/x) < dithresh ) + goto done; + } + +/* Resort to interval halving if Newton iteration did not converge. */ +ihalve: + +d = 0.0625L; +if( x0 == MAXNUML ) + { + if( x <= 0.0L ) + x = 1.0L; + while( x0 == MAXNUML ) + { + x = (1.0L + d) * x; + y = igamcl( a, x ); + if( y < y0 ) + { + x0 = x; + yl = y; + break; + } + d = d + d; + } + } +d = 0.5L; +dir = 0; + +for( i=0; i<400; i++ ) + { + x = x1 + d * (x0 - x1); + y = igamcl( a, x ); + lgm = (x0 - x1)/(x1 + x0); + if( fabsl(lgm) < dithresh ) + break; + lgm = (y - y0)/y0; + if( fabsl(lgm) < dithresh ) + break; + if( x <= 0.0L ) + break; + if( y > y0 ) + { + x1 = x; + yh = y; + if( dir < 0 ) + { + dir = 0; + d = 0.5L; + } + else if( dir > 1 ) + d = 0.5L * d + 0.5L; + else + d = (y0 - yl)/(yh - yl); + dir += 1; + } + else + { + x0 = x; + yl = y; + if( dir > 0 ) + { + dir = 0; + d = 0.5L; + } + else if( dir < -1 ) + d = 0.5L * d; + else + d = (y0 - yl)/(yh - yl); + dir -= 1; + } + } +if( x == 0.0L ) + mtherr( "igamil", UNDERFLOW ); + +done: +return( x ); +} |