/// <summary>
/// Normal (Gaussian) inverse cumulative distribution function.
/// </summary>
///
/// <remarks>
/// <para>
/// For small arguments <c>0 < y < exp(-2)</c>, the program computes <c>z =
/// sqrt( -2.0 * log(y) )</c>; then the approximation is <c>x = z - log(z)/z -
/// (1/z) P(1/z) / Q(1/z)</c>.</para>
/// <para>
/// There are two rational functions P/Q, one for <c>0 < y < exp(-32)</c> and
/// the other for <c>y</c> up to <c>exp(-2)</c>. For larger arguments, <c>w = y - 0.5</c>,
/// and <c>x/sqrt(2pi) = w + w^3 * R(w^2)/S(w^2))</c>.</para>
/// </remarks>
///
/// <returns>
/// Returns the value, <c>x</c>, for which the area under the Normal (Gaussian)
/// probability density function (integrated from minus infinity to <c>x</c>) is
/// equal to the argument <c>y</c> (assumes mean is zero, variance is one).
/// </returns>
///
public static double Inverse(double y0)
{
if (y0 <= 0.0)
{
if (y0 == 0)
{
return(Double.NegativeInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
if (y0 >= 1.0)
{
if (y0 == 1)
{
return(Double.PositiveInfinity);
}
throw new ArgumentOutOfRangeException("y0");
}
double s2pi = Math.Sqrt(2.0 * Math.PI);
int code = 1;
double y = y0;
double x;
double[] P0 =
{
-59.963350101410789,
98.001075418599967,
-56.676285746907027,
13.931260938727968,
-1.2391658386738125
};
double[] Q0 =
{
1.9544885833814176,
4.6762791289888153,
86.360242139089053,
-225.46268785411937,
200.26021238006066,
-82.037225616833339,
15.90562251262117,
-1.1833162112133
};
double[] P1 =
{
4.0554489230596245,
31.525109459989388,
57.162819224642128,
44.080507389320083,
14.684956192885803,
2.1866330685079025,
-0.14025607917135449,
-0.035042462682784818,
-0.00085745678515468545
};
double[] Q1 =
{
15.779988325646675,
45.390763512887922,
41.317203825467203,
15.04253856929075,
2.5046494620830941,
-0.14218292285478779,
-0.038080640769157827,
-0.00093325948089545744
};
double[] P2 =
{
3.2377489177694603,
6.9152288906898418,
3.9388102529247444,
1.3330346081580755,
0.20148538954917908,
0.012371663481782003,
0.00030158155350823543,
2.6580697468673755E-06,
6.2397453918498331E-09
};
double[] Q2 =
{
6.02427039364742,
3.6798356385616087,
1.3770209948908132,
0.21623699359449663,
0.013420400608854318,
0.00032801446468212774,
2.8924786474538068E-06,
6.7901940800998127E-09
};
if (y > 0.8646647167633873)
{
y = 1.0 - y;
code = 0;
}
if (y > 0.1353352832366127)
{
y -= 0.5;
double y2 = y * y;
x = y + y * ((y2 * Special.Polevl(y2, P0, 4)) / Special.P1evl(y2, Q0, 8));
x *= s2pi;
return(x);
}
x = Math.Sqrt(-2.0 * Math.Log(y));
double x0 = x - Math.Log(x) / x;
double z = 1.0 / x;
double x1;
if (x < 8.0)
{
x1 = (z * Special.Polevl(z, P1, 8)) / Special.P1evl(z, Q1, 8);
}
else
{
x1 = (z * Special.Polevl(z, P2, 8)) / Special.P1evl(z, Q2, 8);
}
x = x0 - x1;
if (code != 0)
{
x = -x;
}
return(x);
}
/// <summary>
/// Normal cumulative distribution function.
/// </summary>
///
/// <returns>
/// The area under the Gaussian p.d.f. integrated
/// from minus infinity to the given value.
/// </returns>
///
public static double Function(double value)
{
return(0.5 + 0.5 * Special.Erf(value / Constants.Sqrt2));
}
/// <summary>
/// Complemented cumulative distribution function.
/// </summary>
///
/// <returns>
/// The area under the Gaussian p.d.f. integrated
/// from the given value to positive infinity.
/// </returns>
///
public static double Complemented(double value)
{
return(0.5 * Special.Erfc(value / Constants.Sqrt2));
}