/// <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);
}
/// <summary>
/// Incomplete (regularized) Beta function Ix(a, b).
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double Incomplete(double a, double b, double x)
{
double aa, bb, t, xx, xc, w, y;
bool flag;
if (a <= 0.0)
{
throw new ArgumentOutOfRangeException("a", "Lower limit must be greater than zero.");
}
if (b <= 0.0)
{
throw new ArgumentOutOfRangeException("b", "Upper limit must be greater than zero.");
}
if ((x <= 0.0) || (x >= 1.0))
{
if (x == 0.0)
{
return(0.0);
}
if (x == 1.0)
{
return(1.0);
}
throw new ArgumentOutOfRangeException("x", "Value must be between 0 and 1.");
}
flag = false;
if ((b * x) <= 1.0 && x <= 0.95)
{
t = PowerSeries(a, b, x);
return(t);
}
w = 1.0 - x;
if (x > (a / (a + b)))
{
flag = true;
aa = b;
bb = a;
xc = x;
xx = w;
}
else
{
aa = a;
bb = b;
xc = w;
xx = x;
}
if (flag && (bb * xx) <= 1.0 && xx <= 0.95)
{
t = PowerSeries(aa, bb, xx);
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
return(t);
}
y = xx * (aa + bb - 2.0) - (aa - 1.0);
if (y < 0.0)
{
w = Incbcf(aa, bb, xx);
}
else
{
w = Incbd(aa, bb, xx) / xc;
}
y = aa * System.Math.Log(xx);
t = bb * System.Math.Log(xc);
if ((aa + bb) < Gamma.GammaMax && System.Math.Abs(y) < Constants.LogMax && System.Math.Abs(t) < Constants.LogMax)
{
t = System.Math.Pow(xc, bb);
t *= System.Math.Pow(xx, aa);
t /= aa;
t *= w;
t *= Gamma.Function(aa + bb) / (Gamma.Function(aa) * Gamma.Function(bb));
if (flag)
{
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
}
return(t);
}
y += t + Gamma.Log(aa + bb) - Gamma.Log(aa) - Gamma.Log(bb);
y += System.Math.Log(w / aa);
if (y < Constants.LogMin)
{
t = 0.0;
}
else
{
t = System.Math.Exp(y);
}
if (flag)
{
if (t <= Constants.DoubleEpsilon)
{
t = 1.0 - Constants.DoubleEpsilon;
}
else
{
t = 1.0 - t;
}
}
return(t);
}
/// <summary>
/// Returns the log factorial of a number (ln(n!))
/// </summary>
///
public static double LogFactorial(double n)
{
return(Gamma.Log(n + 1.0));
}
/// <summary>
/// Natural logarithm of the Beta function.
/// </summary>
///
/// <example>
/// Please see <see cref="Beta"/>
/// </example>
///
public static double Log(double a, double b)
{
return(Gamma.Log(a) + Gamma.Log(b) - Gamma.Log(a + b));
}
/// <summary>
/// Returns the extended factorial definition of a real number.
/// </summary>
///
public static double Factorial(double n)
{
return(Gamma.Function(n + 1.0));
}