public void TestDoubleExponentialTransformationAlgorithm()
{
DoubleExponentialTransformation det = new DoubleExponentialTransformation();
Assert.That(
det.Integrate(TargetFunctionA, StartA, StopA, 1e-5),
NumericIs.AlmostEqualTo(TargetAreaA, 1e-5),
"Adaptive Target 1e-5");
Assert.That(
det.Integrate(TargetFunctionA, StartA, StopA, 1e-10),
NumericIs.AlmostEqualTo(TargetAreaA, 1e-10),
"Adaptive Target 1e-10");
}
public void TestDoubleExponentialTransformationAlgorithm()
{
DoubleExponentialTransformation det = new DoubleExponentialTransformation();
Assert.That(
det.Integrate(TargetFunctionA, StartA, StopA, 1e-5),
NumericIs.AlmostEqualTo(TargetAreaA, 1e-5),
"Adaptive Target 1e-5");
Assert.That(
det.Integrate(TargetFunctionA, StartA, StopA, 1e-10),
NumericIs.AlmostEqualTo(TargetAreaA, 1e-10),
"Adaptive Target 1e-10");
}
public static double ForwardStart(string cpflg, double S0, double t1, double t2, double r, double b, double vol, double a)
{
double return_value = double.NaN;
return_value = DoubleExponentialTransformation.Integrate((y) => { return(Normal.PDF(0, 1, y) * BlackScholesMethod.BlackScholes(cpflg, S0 * Math.Exp((b - 0.5 * vol * vol) * t1 + y * vol * Math.Sqrt(t1)), Math.Max(S0 * Math.Exp((b - 0.5 * vol * vol) * t1 + y * vol * Math.Sqrt(t1)) + a, 0), t2 - t1, r, b, vol)); }, -200, 200, 1e-4);
return(return_value);
}
/// <summary>
/// Approximation of the definite integral of an analytic smooth function by double-exponential quadrature. When either or both limits are infinite, the integrand is assumed rapidly decayed to zero as x -> infinity.
/// </summary>
/// <param name="f">The analytic smooth function to integrate.</param>
/// <param name="intervalBegin">Where the interval starts.</param>
/// <param name="intervalEnd">Where the interval stops.</param>
/// <param name="targetAbsoluteError">The expected relative accuracy of the approximation.</param>
/// <returns>Approximation of the finite integral in the given interval.</returns>
public static double DoubleExponential(Func <double, double> f, double intervalBegin, double intervalEnd, double targetAbsoluteError = 1E-8)
{
// Reference:
// Formula used for variable subsitution from
// 1. Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131-140.
// 2. quadgk.m, GNU Octave
if (intervalBegin > intervalEnd)
{
return(-DoubleExponential(f, intervalEnd, intervalBegin, targetAbsoluteError));
}
// (-oo, oo) => [-1, 1]
//
// integral_(-oo)^(oo) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
// g(t) = t / (1 - t^2)
// g'(t) = (1 + t^2) / (1 - t^2)^2
if (double.IsInfinity(intervalBegin) && double.IsInfinity(intervalEnd))
{
Func <double, double> u = (t) =>
{
return(f(t / (1 - t * t)) * (1 + t * t) / ((1 - t * t) * (1 - t * t)));
};
return(DoubleExponentialTransformation.Integrate(u, -1, 1, targetAbsoluteError));
}
// [a, oo) => [0, 1]
//
// integral_(a)^(oo) f(x) dx = integral_(0)^(oo) f(a + t^2) 2 t dt
// = integral_(0)^(1) f(a + g(s)^2) 2 g(s) g'(s) ds
// g(s) = s / (1 - s)
// g'(s) = 1 / (1 - s)^2
else if (double.IsInfinity(intervalEnd))
{
Func <double, double> u = (s) =>
{
return(2 * s * f(intervalBegin + (s / (1 - s)) * (s / (1 - s))) / ((1 - s) * (1 - s) * (1 - s)));
};
return(DoubleExponentialTransformation.Integrate(u, 0, 1, targetAbsoluteError));
}
// (-oo, b] => [-1, 0]
//
// integral_(-oo)^(b) f(x) dx = -integral_(-oo)^(0) f(b - t^2) 2 t dt
// = -integral_(-1)^(0) f(b - g(s)^2) 2 g(s) g'(s) ds
// g(s) = s / (1 + s)
// g'(s) = 1 / (1 + s)^2
else if (double.IsInfinity(intervalBegin))
{
Func <double, double> u = (s) =>
{
return(-2 * s * f(intervalEnd - s / (1 + s) * (s / (1 + s))) / ((1 + s) * (1 + s) * (1 + s)));
};
return(DoubleExponentialTransformation.Integrate(u, -1, 0, targetAbsoluteError));
}
else
{
return(DoubleExponentialTransformation.Integrate(f, intervalBegin, intervalEnd, targetAbsoluteError));
}
}
public void TestDoubleExponentialTransformationAlgorithm(double targetRelativeError)
{
Assert.AreEqual(
TargetAreaA,
DoubleExponentialTransformation.Integrate(TargetFunctionA, StartA, StopA, targetRelativeError),
targetRelativeError * TargetAreaA,
"DET Adaptive {0}",
targetRelativeError);
}
public void TestDoubleExponentialTransformationAlgorithm(double targetRelativeError)
{
var algorithm = new DoubleExponentialTransformation();
Assert.AreApproximatelyEqual(
TargetAreaA,
algorithm.Integrate(TargetFunctionA, StartA, StopA, targetRelativeError),
targetRelativeError * TargetAreaA,
"DET Adaptive {0}",
targetRelativeError);
}
private static void Integrate()
{
// perform a one-dimensional numerical integration
const double x0 = 0.0;
const double x1 = 1.0;
const double eps = 1e-6;
double integral = 2 / Constants.SqrtPi * DoubleExponentialTransformation.Integrate(x => Math.Exp(-x * x), x0, x1, eps);
double exact = SpecialFunctions.Erf(x1);
Console.WriteLine($"erf({x1}) = {exact}");
Console.WriteLine($"erf({x1}) = {integral} (num approx.)");
}
public void FindClassificationErrorFullPossibility()
{
double step = 0.00001;
Class1ErrorAssignmentChance = 0;
Class1ErrorNotAssignedChance = 0;
DoubleRange range1 = NormalDistributionClass1.GetRange(0.99);
DoubleRange range2 = NormalDistributionClass2.GetRange(0.99);
double begin = range2.Min;
double end = FindDistributionEqualityPoint();
double f1(double x) => Chance2 / Math.Sqrt(Variance2 * 2 * Math.PI)
* Math.Exp(Math.Pow((x - Mean2), 2) / (-2 * Variance2));
Class1ErrorAssignmentChance = DoubleExponentialTransformation.Integrate(f1, begin, end, 1e-5);
begin = end;
end = range1.Max;
double f2(double x) => Chance1 / Math.Sqrt(Variance1 * 2 * Math.PI)
* Math.Exp(Math.Pow((x - Mean1), 2) / (-2 * Variance1));
Class1ErrorNotAssignedChance = DoubleExponentialTransformation.Integrate(f2, begin, end, 1e-5);
//double x = Math.Min(range1.Min, range2.Min);
//double max = Math.Max(range2.Max, range1.Max);
//bool equalityPointPassed = false;
//while (x < max)
//{
// double dens1 = NormalDistributionClass1.ProbabilityDensityFunction(x);
// double dens2 = NormalDistributionClass2.ProbabilityDensityFunction(x);
// if (!equalityPointPassed)
// Class1ErrorAssignmentChance += dens2 * chance2;
// else
// Class1ErrorNotAssignedChance += dens1 * chance1;
// if (Math.Abs(dens1 - dens2) < 0.00003)
// equalityPointPassed = true;
// x += step;
//}
}
/// <summary>
/// Approximation of the definite integral of an analytic smooth function on a closed interval.
/// </summary>
/// <param name="f">The analytic smooth function to integrate.</param>
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
/// <returns>Approximation of the finite integral in the given interval.</returns>
public static double OnClosedInterval(Func <double, double> f, double intervalBegin, double intervalEnd)
{
return(DoubleExponentialTransformation.Integrate(f, intervalBegin, intervalEnd, 1e-8));
}
/// <summary>
/// Approximation of the definite integral of an analytic smooth function on a closed interval.
/// </summary>
/// <param name="f">The analytic smooth function to integrate.</param>
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
/// <param name="targetAbsoluteError">The expected relative accuracy of the approximation.</param>
/// <returns>Approximation of the finite integral in the given interval.</returns>
public static double OnClosedInterval(Func <double, double> f, double intervalBegin, double intervalEnd, double targetAbsoluteError)
{
return(DoubleExponentialTransformation.Integrate(f, intervalBegin, intervalEnd, targetAbsoluteError));
}
public void TestDoubleExponentialTransformationAlgorithm(double targetRelativeError)
{
var algorithm = new DoubleExponentialTransformation();
Assert.AreApproximatelyEqual(
TargetAreaA,
algorithm.Integrate(TargetFunctionA, StartA, StopA, targetRelativeError),
targetRelativeError * TargetAreaA,
"DET Adaptive {0}",
targetRelativeError);
}
private double Trigamma(double z)
{
Func <double, double> func = x => Math.Pow(x, z - 1) * Math.Log(x) / (1 - x);
return(-DoubleExponentialTransformation.Integrate(func, 0, 1, 1E-5));
}