/// <summary>
/// Run example
/// </summary>
/// <seealso cref="http://en.wikipedia.org/wiki/Beta_function">Beta function</seealso>
public void Run()
{
// 1. Compute the Beta function at z = 1.0, w = 3.0
Console.WriteLine(@"1. Compute the Beta function at z = 1.0, w = 3.0");
Console.WriteLine(SpecialFunctions.Beta(1.0, 3.0));
Console.WriteLine();
// 2. Compute the logarithm of the Beta function at z = 1.0, w = 3.0
Console.WriteLine(@"2. Compute the logarithm of the Beta function at z = 1.0, w = 3.0");
Console.WriteLine(SpecialFunctions.BetaLn(1.0, 3.0));
Console.WriteLine();
// 3. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 0.7
Console.WriteLine(@"3. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 0.7");
Console.WriteLine(SpecialFunctions.BetaIncomplete(1.0, 3.0, 0.7));
Console.WriteLine();
// 4. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 1.0
Console.WriteLine(@"4. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 1.0");
Console.WriteLine(SpecialFunctions.BetaIncomplete(1.0, 3.0, 1.0));
Console.WriteLine();
// 5. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 0.7
Console.WriteLine(@"5. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 0.7");
Console.WriteLine(SpecialFunctions.BetaRegularized(1.0, 3.0, 0.7));
Console.WriteLine();
// 6. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 1.0
Console.WriteLine(@"6. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 1.0");
Console.WriteLine(SpecialFunctions.BetaRegularized(1.0, 3.0, 1.0));
Console.WriteLine();
}
/// <summary>
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
/// </summary>
/// <param name="d1">The first degree of freedom (d1) of the distribution. Range: d1 > 0.</param>
/// <param name="d2">The second degree of freedom (d2) of the distribution. Range: d2 > 0.</param>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the density at <paramref name="x"/>.</returns>
/// <seealso cref="Density"/>
public static double PDF(double d1, double d2, double x)
{
if (d1 <= 0.0 || d2 <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
return(Math.Sqrt(Math.Pow(d1 * x, d1) * Math.Pow(d2, d2) / Math.Pow((d1 * x) + d2, d1 + d2)) / (x * SpecialFunctions.Beta(d1 / 2.0, d2 / 2.0)));
}
/// <summary>
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
/// </summary>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the density at <paramref name="x"/>.</returns>
/// <seealso cref="PDF"/>
public double Density(double x)
{
return(Math.Sqrt(Math.Pow(_freedom1 * x, _freedom1) * Math.Pow(_freedom2, _freedom2) / Math.Pow((_freedom1 * x) + _freedom2, _freedom1 + _freedom2)) / (x * SpecialFunctions.Beta(_freedom1 / 2.0, _freedom2 / 2.0)));
}
public void ValidateDensityLn(double d1, double d2, double x)
{
var n = new FisherSnedecor(d1, d2);
double expected = Math.Log(Math.Sqrt(Math.Pow(d1 * x, d1) * Math.Pow(d2, d2) / Math.Pow((d1 * x) + d2, d1 + d2)) / (x * SpecialFunctions.Beta(d1 / 2.0, d2 / 2.0)));
Assert.AreEqual(expected, n.DensityLn(x));
Assert.AreEqual(expected, FisherSnedecor.PDFLn(d1, d2, x));
}
public void Beta()
{
AssertHelpers.AlmostEqual(0.5, SpecialFunctions.Beta(1.0, 2.0), 14);
AssertHelpers.AlmostEqual(1.0, SpecialFunctions.Beta(1.0, 1.0), 14);
}
public void Beta()
{
AssertHelpers.AlmostEqualRelative(0.5, SpecialFunctions.Beta(1.0, 2.0), 13);
AssertHelpers.AlmostEqualRelative(1.0, SpecialFunctions.Beta(1.0, 1.0), 13);
}
public void ValidateDensity(
[Values(0.1, 1.0, 10.0, 100.0, 0.1, 1.0, 10.0, 100.0, 0.1, 1.0, 10.0, 100.0, 0.1, 1.0, 10.0, 100.0, 0.1, 1.0, 10.0, 100.0, 0.1, 1.0, 10.0, 100.0)] double d1,
[Values(0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 100.0, 100.0, 100.0, 100.0, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 100.0, 100.0, 100.0, 100.0)] double d2,
[Values(1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 10.0)] double x)
{
var n = new FisherSnedecor(d1, d2);
Assert.AreEqual(Math.Sqrt(Math.Pow(d1 * x, d1) * Math.Pow(d2, d2) / Math.Pow((d1 * x) + d2, d1 + d2)) / (x * SpecialFunctions.Beta(d1 / 2.0, d2 / 2.0)), n.Density(x));
}
/// <summary>
/// 분포의 확률 밀도
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public double Density(double x)
{
return(Math.Sqrt(Math.Pow(_d1 * x, _d1) * Math.Pow(_d2, _d2) / Math.Pow((_d1 * x) + _d2, _d1 + _d2)) /
(x * SpecialFunctions.Beta(_d1 / 2.0, _d2 / 2.0)));
}
public void ValidateDensity(double d1, double d2, double x)
{
var n = new FisherSnedecor(d1, d2);
Assert.AreEqual <double>(Math.Sqrt(Math.Pow(d1 * x, d1) * Math.Pow(d2, d2) / (Math.Pow(d1 * x + d2, d1 + d2))) / (x * SpecialFunctions.Beta(d1 / 2.0, d2 / 2.0)), n.Density(x));
}
/// <summary>
/// Executes the example.
/// </summary>
public override void ExecuteExample()
{
// <seealso cref="http://en.wikipedia.org/wiki/Beta_function">Beta function</seealso>
MathDisplay.WriteLine("<b>Beta fuction</b>");
// 1. Compute the Beta function at z = 1.0, w = 3.0
MathDisplay.WriteLine(@"1. Compute the Beta function at z = 1.0, w = 3.0");
MathDisplay.WriteLine(SpecialFunctions.Beta(1.0, 3.0).ToString());
MathDisplay.WriteLine();
// 2. Compute the logarithm of the Beta function at z = 1.0, w = 3.0
MathDisplay.WriteLine(@"2. Compute the logarithm of the Beta function at z = 1.0, w = 3.0");
MathDisplay.WriteLine(SpecialFunctions.BetaLn(1.0, 3.0).ToString());
MathDisplay.WriteLine();
// 3. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 0.7
MathDisplay.WriteLine(@"3. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 0.7");
MathDisplay.WriteLine(SpecialFunctions.BetaIncomplete(1.0, 3.0, 0.7).ToString());
MathDisplay.WriteLine();
// 4. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 1.0
MathDisplay.WriteLine(@"4. Compute the Beta incomplete function at z = 1.0, w = 3.0, x = 1.0");
MathDisplay.WriteLine(SpecialFunctions.BetaIncomplete(1.0, 3.0, 1.0).ToString());
MathDisplay.WriteLine();
// 5. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 0.7
MathDisplay.WriteLine(@"5. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 0.7");
MathDisplay.WriteLine(SpecialFunctions.BetaRegularized(1.0, 3.0, 0.7).ToString());
MathDisplay.WriteLine();
// 6. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 1.0
MathDisplay.WriteLine(@"6. Compute the Beta regularized function at z = 1.0, w = 3.0, x = 1.0");
MathDisplay.WriteLine(SpecialFunctions.BetaRegularized(1.0, 3.0, 1.0).ToString());
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Common functions</b>");
// 1. Calculate the Digamma function at point 5.0
// <seealso cref="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</seealso>
MathDisplay.WriteLine(@"1. Calculate the Digamma function at point 5.0");
MathDisplay.WriteLine(SpecialFunctions.DiGamma(5.0).ToString());
MathDisplay.WriteLine();
// 2. Calculate the inverse Digamma function at point 1.5
MathDisplay.WriteLine(@"2. Calculate the inverse Digamma function at point 1.5");
MathDisplay.WriteLine(SpecialFunctions.DiGammaInv(1.5).ToString());
MathDisplay.WriteLine();
// 3. Calculate the 10'th Harmonic number
// <seealso cref="http://en.wikipedia.org/wiki/Harmonic_number">Harmonic number</seealso>
MathDisplay.WriteLine(@"3. Calculate the 10'th Harmonic number");
MathDisplay.WriteLine(SpecialFunctions.Harmonic(10).ToString());
MathDisplay.WriteLine();
// 4. Calculate the generalized harmonic number of order 10 of 3.0.
// <seealso cref="http://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers">Generalized harmonic numbers</seealso>
MathDisplay.WriteLine(@"4. Calculate the generalized harmonic number of order 10 of 3.0");
MathDisplay.WriteLine(SpecialFunctions.GeneralHarmonic(10, 3.0).ToString());
MathDisplay.WriteLine();
// 5. Calculate the logistic function of 3.0
// <seealso cref="http://en.wikipedia.org/wiki/Logistic_function">Logistic function</seealso>
MathDisplay.WriteLine(@"5. Calculate the logistic function of 3.0");
MathDisplay.WriteLine(SpecialFunctions.Logistic(3.0).ToString());
MathDisplay.WriteLine();
// 6. Calculate the logit function of 0.3
// <seealso cref="http://en.wikipedia.org/wiki/Logit">Logit function</seealso>
MathDisplay.WriteLine(@"6. Calculate the logit function of 0.3");
MathDisplay.WriteLine(SpecialFunctions.Logit(0.3).ToString());
MathDisplay.WriteLine();
// <seealso cref="http://en.wikipedia.org/wiki/Error_function">Error function</seealso>
MathDisplay.WriteLine("<b>Error function</b>");
// 1. Calculate the error function at point 2
MathDisplay.WriteLine(@"1. Calculate the error function at point 2");
MathDisplay.WriteLine(SpecialFunctions.Erf(2).ToString());
MathDisplay.WriteLine();
// 2. Sample 10 values of the error function in [-1.0; 1.0]
MathDisplay.WriteLine(@"2. Sample 10 values of the error function in [-1.0; 1.0]");
var data = Generate.LinearSpacedMap <double>(10, -1.0, 1.0, SpecialFunctions.Erf);
for (var i = 0; i < data.Length; i++)
{
MathDisplay.Write(data[i].ToString("N") + @" ");
}
MathDisplay.WriteLine();
MathDisplay.WriteLine();
// 3. Calculate the complementary error function at point 2
MathDisplay.WriteLine(@"3. Calculate the complementary error function at point 2");
MathDisplay.WriteLine(SpecialFunctions.Erfc(2).ToString());
MathDisplay.WriteLine();
// 4. Sample 10 values of the complementary error function in [-1.0; 1.0]
MathDisplay.WriteLine(@"4. Sample 10 values of the complementary error function in [-1.0; 1.0]");
data = Generate.LinearSpacedMap <double>(10, -1.0, 1.0, SpecialFunctions.Erfc);
for (var i = 0; i < data.Length; i++)
{
MathDisplay.Write(data[i].ToString("N") + @" ");
}
MathDisplay.WriteLine();
MathDisplay.WriteLine();
// 5. Calculate the inverse error function at point z=0.5
MathDisplay.WriteLine(@"5. Calculate the inverse error function at point z=0.5");
MathDisplay.WriteLine(SpecialFunctions.ErfInv(0.5).ToString());
MathDisplay.WriteLine();
// 6. Sample 10 values of the inverse error function in [-1.0; 1.0]
MathDisplay.WriteLine(@"6. Sample 10 values of the inverse error function in [-1.0; 1.0]");
data = Generate.LinearSpacedMap <double>(10, -1.0, 1.0, SpecialFunctions.ErfInv);
for (var i = 0; i < data.Length; i++)
{
MathDisplay.Write(data[i].ToString("N") + @" ");
}
MathDisplay.WriteLine();
MathDisplay.WriteLine();
// 7. Calculate the complementary inverse error function at point z=0.5
MathDisplay.WriteLine(@"7. Calculate the complementary inverse error function at point z=0.5");
MathDisplay.WriteLine(SpecialFunctions.ErfcInv(0.5).ToString());
MathDisplay.WriteLine();
// 8. Sample 10 values of the complementary inverse error function in [-1.0; 1.0]
MathDisplay.WriteLine(@"8. Sample 10 values of the complementary inverse error function in [-1.0; 1.0]");
data = Generate.LinearSpacedMap <double>(10, -1.0, 1.0, SpecialFunctions.ErfcInv);
for (var i = 0; i < data.Length; i++)
{
MathDisplay.Write(data[i].ToString("N") + @" ");
}
MathDisplay.WriteLine();
// <seealso cref="http://en.wikipedia.org/wiki/Factorial">Factorial</seealso>
MathDisplay.WriteLine("<b>Factorial</b>");
// 1. Compute the factorial of 5
MathDisplay.WriteLine(@"1. Compute the factorial of 5");
MathDisplay.WriteLine(SpecialFunctions.Factorial(5).ToString("N"));
MathDisplay.WriteLine();
// 2. Compute the logarithm of the factorial of 5
MathDisplay.WriteLine(@"2. Compute the logarithm of the factorial of 5");
MathDisplay.WriteLine(SpecialFunctions.FactorialLn(5).ToString("N"));
MathDisplay.WriteLine();
// <seealso cref="http://en.wikipedia.org/wiki/Binomial_coefficient">Binomial coefficient</seealso>
MathDisplay.WriteLine("<b>Binomial coefficient</b>");
// 3. Compute the binomial coefficient: 10 choose 8
MathDisplay.WriteLine(@"3. Compute the binomial coefficient: 10 choose 8");
MathDisplay.WriteLine(SpecialFunctions.Binomial(10, 8).ToString("N"));
MathDisplay.WriteLine();
// 4. Compute the logarithm of the binomial coefficient: 10 choose 8
MathDisplay.WriteLine(@"4. Compute the logarithm of the binomial coefficient: 10 choose 8");
MathDisplay.WriteLine(SpecialFunctions.BinomialLn(10, 8).ToString("N"));
MathDisplay.WriteLine();
// <seealso cref="http://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</seealso>
MathDisplay.WriteLine("<b>Multinomial coefficient</b>");
// 5. Compute the multinomial coefficient: 10 choose 2, 3, 5
MathDisplay.WriteLine(@"5. Compute the multinomial coefficient: 10 choose 2, 3, 5");
MathDisplay.WriteLine(SpecialFunctions.Multinomial(10, new[] { 2, 3, 5 }).ToString("N"));
MathDisplay.WriteLine();
// <seealso cref="http://en.wikipedia.org/wiki/Gamma_function">Gamma function</seealso>
MathDisplay.WriteLine("<b>Gamma function</b>");
// 1. Compute the Gamma function of 10
MathDisplay.WriteLine(@"1. Compute the Gamma function of 10");
MathDisplay.WriteLine(SpecialFunctions.Gamma(10).ToString("N"));
MathDisplay.WriteLine();
// 2. Compute the logarithm of the Gamma function of 10
MathDisplay.WriteLine(@"2. Compute the logarithm of the Gamma function of 10");
MathDisplay.WriteLine(SpecialFunctions.GammaLn(10).ToString("N"));
MathDisplay.WriteLine();
// 3. Compute the lower incomplete gamma(a, x) function at a = 10, x = 14
MathDisplay.WriteLine(@"3. Compute the lower incomplete gamma(a, x) function at a = 10, x = 14");
MathDisplay.WriteLine(SpecialFunctions.GammaLowerIncomplete(10, 14).ToString("N"));
MathDisplay.WriteLine();
// 4. Compute the lower incomplete gamma(a, x) function at a = 10, x = 100
MathDisplay.WriteLine(@"4. Compute the lower incomplete gamma(a, x) function at a = 10, x = 100");
MathDisplay.WriteLine(SpecialFunctions.GammaLowerIncomplete(10, 100).ToString("N"));
MathDisplay.WriteLine();
// 5. Compute the upper incomplete gamma(a, x) function at a = 10, x = 0
MathDisplay.WriteLine(@"5. Compute the upper incomplete gamma(a, x) function at a = 10, x = 0");
MathDisplay.WriteLine(SpecialFunctions.GammaUpperIncomplete(10, 0).ToString("N"));
MathDisplay.WriteLine();
// 6. Compute the upper incomplete gamma(a, x) function at a = 10, x = 10
MathDisplay.WriteLine(@"6. Compute the upper incomplete gamma(a, x) function at a = 10, x = 100");
MathDisplay.WriteLine(SpecialFunctions.GammaLowerIncomplete(10, 10).ToString("N"));
MathDisplay.WriteLine();
// 7. Compute the lower regularized gamma(a, x) function at a = 10, x = 14
MathDisplay.WriteLine(@"7. Compute the lower regularized gamma(a, x) function at a = 10, x = 14");
MathDisplay.WriteLine(SpecialFunctions.GammaLowerRegularized(10, 14).ToString("N"));
MathDisplay.WriteLine();
// 8. Compute the lower regularized gamma(a, x) function at a = 10, x = 100
MathDisplay.WriteLine(@"8. Compute the lower regularized gamma(a, x) function at a = 10, x = 100");
MathDisplay.WriteLine(SpecialFunctions.GammaLowerRegularized(10, 100).ToString("N"));
MathDisplay.WriteLine();
// 9. Compute the upper regularized gamma(a, x) function at a = 10, x = 0
MathDisplay.WriteLine(@"9. Compute the upper regularized gamma(a, x) function at a = 10, x = 0");
MathDisplay.WriteLine(SpecialFunctions.GammaUpperRegularized(10, 0).ToString("N"));
MathDisplay.WriteLine();
// 10. Compute the upper regularized gamma(a, x) function at a = 10, x = 10
MathDisplay.WriteLine(@"10. Compute the upper regularized gamma(a, x) function at a = 10, x = 100");
MathDisplay.WriteLine(SpecialFunctions.GammaUpperRegularized(10, 10).ToString("N"));
MathDisplay.WriteLine();
MathDisplay.WriteLine("<b>Numerical stability</b>");
// 1. Compute numerically stable exponential of 10 minus one
MathDisplay.WriteLine(@"1. Compute numerically stable exponential of 4.2876 minus one");
MathDisplay.WriteLine(SpecialFunctions.ExponentialMinusOne(4.2876).ToString());
MathDisplay.WriteLine();
// 2. Compute regular System.Math exponential of 15.28 minus one
MathDisplay.WriteLine(@"2. Compute regular System.Math exponential of 4.2876 minus one ");
MathDisplay.WriteLine((Math.Exp(4.2876) - 1).ToString());
MathDisplay.WriteLine();
// 3. Compute numerically stable hypotenuse of a right angle triangle with a = 5, b = 3
MathDisplay.WriteLine(@"3. Compute numerically stable hypotenuse of a right angle triangle with a = 5, b = 3");
MathDisplay.WriteLine(SpecialFunctions.Hypotenuse(5, 3).ToString());
MathDisplay.WriteLine();
}
