double igami( double a, double y0 )
{
double x0, x1, x, yl, yh, y, d, lgm, dithresh;
int i, dir;
/* bound the solution */
x0 = MAXNUM;
yl = 0;
x1 = 0;
yh = 1.0;
dithresh = 5.0 * MACHEP;
/* approximation to inverse function */
d = 1.0/(9.0*a);
y = ( 1.0 - d - ndtri(y0) * sqrt(d) );
x = a * y * y * y;
lgm = lgam(a);
for( i=0; i<10; i++ )
{
if( x > x0 || x < x1 )
goto ihalve;
y = igamc(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 - 1.0) * log(x) - x - lgm;
if( d < -MAXLOG )
goto ihalve;
d = -exp(d);
/* compute the step to the next approximation of x */
d = (y - y0)/d;
if( fabs(d/x) < MACHEP )
goto done;
x = x - d;
}
/* Resort to interval halving if Newton iteration did not converge. */
ihalve:
d = 0.0625;
if( x0 == MAXNUM )
{
if( x <= 0.0 )
x = 1.0;
while( x0 == MAXNUM )
{
x = (1.0 + d) * x;
y = igamc( a, x );
if( y < y0 )
{
x0 = x;
yl = y;
break;
}
d = d + d;
}
}
d = 0.5;
dir = 0;
for( i=0; i<400; i++ )
{
x = x1 + d * (x0 - x1);
y = igamc( a, x );
lgm = (x0 - x1)/(x1 + x0);
if( fabs(lgm) < dithresh )
break;
lgm = (y - y0)/y0;
if( fabs(lgm) < dithresh )
break;
if( x <= 0.0 )
break;
if( y >= y0 )
{
x1 = x;
yh = y;
if( dir < 0 )
{
dir = 0;
d = 0.5;
}
else if( dir > 1 )
d = 0.5 * d + 0.5;
else
d = (y0 - yl)/(yh - yl);
dir += 1;
}
else
{
x0 = x;
yl = y;
if( dir > 0 )
{
dir = 0;
d = 0.5;
}
else if( dir < -1 )
d = 0.5 * d;
else
d = (y0 - yl)/(yh - yl);
dir -= 1;
}
}
if( x == 0.0 )
mtherr( "igami", UNDERFLOW );
done:
return( x );
}
static DBL igami(DBL a, DBL y0)
{
DBL d, y, x0, lgm;
int i;
int sgngam = 0;
/* approximation to inverse function */
d = 1.0 / (9.0 * a);
y = (1.0 - d - ndtri(y0) * sqrt(d));
x0 = a * y * y * y;
lgm = lgam(a, &sgngam);
for (i = 0; i < 10; i++)
{
if (x0 <= 0.0)
{
/*
mtherr("igami", UNDERFLOW);
*/
return (0.0);
}
y = igamc(a, x0);
/* compute the derivative of the function at this point */
d = (a - 1.0) * log(x0) - x0 - lgm;
if (d < -MAXLOG)
{
/*
mtherr("igami", UNDERFLOW);
*/
goto done;
}
d = -exp(d);
/* compute the step to the next approximation of x */
if (d == 0.0)
{
goto done;
}
d = (y - y0) / d;
x0 = x0 - d;
if (i < 3)
{
continue;
}
if (fabs(d / x0) < 2.0 * MACHEP)
{
goto done;
}
}
done:
return (x0);
}
