/// <summary>
/// Inverse complemented error function (<see cref="Erfc(double)"/>.
/// </summary>
///
public static double Ierfc(double y)
{
double s = Normal.Inverse(-0.5 * y + 1);
double r = s * Math.Sqrt(2) / 2.0;
return(r);
}
private static double inverse(double a, double y)
{
// bound the solution
double x0 = Double.MaxValue;
double yl = 0;
double x1 = 0;
double yh = 1.0;
double dithresh = 5.0 * Constants.DoubleEpsilon;
// approximation to inverse function
double d = 1.0 / (9.0 * a);
double yy = (1.0 - d - Normal.Inverse(y) * Math.Sqrt(d));
double x = a * yy * yy * yy;
double lgm = Gamma.Log(a);
for (int i = 0; i < 10; i++)
{
if (x > x0 || x < x1)
{
goto ihalve;
}
yy = Gamma.UpperIncomplete(a, x);
if (yy < yl || yy > yh)
{
goto ihalve;
}
if (yy < y)
{
x0 = x;
yl = yy;
}
else
{
x1 = x;
yh = yy;
}
// compute the derivative of the function at this point
d = (a - 1.0) * Math.Log(x) - x - lgm;
if (d < -Constants.LogMax)
{
goto ihalve;
}
d = -Math.Exp(d);
// compute the step to the next approximation of x
d = (yy - y) / d;
if (Math.Abs(d / x) < Constants.DoubleEpsilon)
{
return(x);
}
x = x - d;
}
// Resort to interval halving if Newton iteration did not converge.
ihalve:
d = 0.0625;
if (x0 == Double.MaxValue)
{
if (x <= 0.0)
{
x = 1.0;
}
while (x0 == Double.MaxValue && !Double.IsNaN(x))
{
x = (1.0 + d) * x;
yy = Gamma.UpperIncomplete(a, x);
if (yy < y)
{
x0 = x;
yl = yy;
break;
}
d = d + d;
}
}
d = 0.5;
double dir = 0;
for (int i = 0; i < 400; i++)
{
double t = x1 + d * (x0 - x1);
if (Double.IsNaN(t))
{
break;
}
x = t;
yy = Gamma.UpperIncomplete(a, x);
lgm = (x0 - x1) / (x1 + x0);
if (Math.Abs(lgm) < dithresh)
{
break;
}
lgm = (yy - y) / y;
if (Math.Abs(lgm) < dithresh)
{
break;
}
if (x <= 0.0)
{
break;
}
if (yy >= y)
{
x1 = x;
yh = yy;
if (dir < 0)
{
dir = 0;
d = 0.5;
}
else if (dir > 1)
{
d = 0.5 * d + 0.5;
}
else
{
d = (y - yl) / (yh - yl);
}
dir += 1;
}
else
{
x0 = x;
yl = yy;
if (dir > 0)
{
dir = 0;
d = 0.5;
}
else if (dir < -1)
{
d = 0.5 * d;
}
else
{
d = (y - yl) / (yh - yl);
}
dir -= 1;
}
}
if (x == 0.0 || Double.IsNaN(x))
{
throw new ArithmeticException();
}
return(x);
}
/// <summary>
/// Inverse of incomplete beta integral.
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double IncompleteInverse(double aa, double bb, double yy0)
{
double a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh;
int i, dir;
bool nflg;
bool rflg;
if (yy0 <= 0)
{
return(0.0);
}
if (yy0 >= 1.0)
{
return(1.0);
}
if (aa <= 1.0 || bb <= 1.0)
{
nflg = true;
dithresh = 4.0 * Constants.DoubleEpsilon;
rflg = false;
a = aa;
b = bb;
y0 = yy0;
x = a / (a + b);
y = Incomplete(a, b, x);
goto ihalve;
}
else
{
nflg = false;
dithresh = 1.0e-4;
}
/* approximation to inverse function */
yp = -Normal.Inverse(yy0);
if (yy0 > 0.5)
{
rflg = true;
a = bb;
b = aa;
y0 = 1.0 - yy0;
yp = -yp;
}
else
{
rflg = false;
a = aa;
b = bb;
y0 = yy0;
}
lgm = (yp * yp - 3.0) / 6.0;
x0 = 2.0 / (1.0 / (2.0 * a - 1.0) + 1.0 / (2.0 * b - 1.0));
y = yp * Math.Sqrt(x0 + lgm) / x0
- (1.0 / (2.0 * b - 1.0) - 1.0 / (2.0 * a - 1.0))
* (lgm + 5.0 / 6.0 - 2.0 / (3.0 * x0));
y = 2.0 * y;
if (y < Constants.LogMin)
{
x0 = 1.0;
throw new ArithmeticException("underflow");
}
x = a / (a + b * Math.Exp(y));
y = Incomplete(a, b, x);
yp = (y - y0) / y0;
if (Math.Abs(yp) < 1.0e-2)
{
goto newt;
}
ihalve:
/* Resort to interval halving if not close enough */
x0 = 0.0;
yl = 0.0;
x1 = 1.0;
yh = 1.0;
di = 0.5;
dir = 0;
for (i = 0; i < 400; i++)
{
if (i != 0)
{
x = x0 + di * (x1 - x0);
if (x == 1.0)
{
x = 1.0 - Constants.DoubleEpsilon;
}
y = Incomplete(a, b, x);
yp = (x1 - x0) / (x1 + x0);
if (Math.Abs(yp) < dithresh)
{
x0 = x;
goto newt;
}
}
if (y < y0)
{
x0 = x;
yl = y;
if (dir < 0)
{
dir = 0;
di = 0.5;
}
else if (dir > 1)
{
di = 0.5 * di + 0.5;
}
else
{
di = (y0 - y) / (yh - yl);
}
dir += 1;
if (x0 > 0.75)
{
if (rflg)
{
rflg = false;
a = aa;
b = bb;
y0 = yy0;
}
else
{
rflg = true;
a = bb;
b = aa;
y0 = 1.0 - yy0;
}
x = 1.0 - x;
y = Incomplete(a, b, x);
goto ihalve;
}
}
else
{
x1 = x;
if (rflg && x1 < Constants.DoubleEpsilon)
{
x0 = 0.0;
goto done;
}
yh = y;
if (dir > 0)
{
dir = 0;
di = 0.5;
}
else if (dir < -1)
{
di = 0.5 * di;
}
else
{
di = (y - y0) / (yh - yl);
}
dir -= 1;
}
}
if (x0 >= 1.0)
{
x0 = 1.0 - Constants.DoubleEpsilon;
goto done;
}
if (x == 0.0)
{
throw new ArithmeticException("underflow");
}
newt:
if (nflg)
{
goto done;
}
x0 = x;
lgm = Gamma.Log(a + b) - Gamma.Log(a) - Gamma.Log(b);
for (i = 0; i < 10; i++)
{
/* Compute the function at this point. */
if (i != 0)
{
y = Incomplete(a, b, x0);
}
/* Compute the derivative of the function at this point. */
d = (a - 1.0) * Math.Log(x0) + (b - 1.0) * Math.Log(1.0 - x0) + lgm;
if (d < Constants.LogMin)
{
throw new ArithmeticException("underflow");
}
d = Math.Exp(d);
/* compute the step to the next approximation of x */
d = (y - y0) / d;
x = x0;
x0 = x0 - d;
if (x0 <= 0.0)
{
throw new ArithmeticException("underflow");
}
if (x0 >= 1.0)
{
x0 = 1.0 - Constants.DoubleEpsilon;
goto done;
}
if (Math.Abs(d / x0) < 64.0 * Constants.DoubleEpsilon)
{
goto done;
}
}
done:
if (rflg)
{
if (x0 <= Double.Epsilon)
{
x0 = 1.0 - Double.Epsilon;
}
else
{
x0 = 1.0 - x0;
}
}
return(x0);
}
