public void SeperableIntegrals()
{
// Integrates \Pi_{j=0}^{d-1} \int_0^1 \! dx \, x_j^j = \Pi_{j=0}^{d-1} \frac{1}{j+1} = \frac{1}{d!}
// This is a simple test of a seperable integral
Func <IList <double>, double> f = delegate(IList <double> x) {
double y = 1.0;
for (int j = 0; j < x.Count; j++)
{
y *= MoreMath.Pow(x[j], j);
}
return(y);
};
for (int d = 1; d <= 10; d++)
{
Console.WriteLine(d);
IntegrationResult r = MultiFunctionMath.Integrate(f, UnitCube(d));
Console.WriteLine("{0}: {1} ({2}) ?= {3}", r.EvaluationCount, r.Value, r.Precision, 1.0 / AdvancedIntegerMath.Factorial(d));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
r.Value,
1.0 / AdvancedIntegerMath.Factorial(d),
new EvaluationSettings()
{
AbsolutePrecision = 2.0 * r.Precision
}
));
}
}
public void ZetaIntegrals()
{
// By expanding 1/(1-x) = 1 + x + x^2 + ... and integrating term-by-term
// it's easy to show that this integral is \zeta(d)
Func <IList <double>, double> f = delegate(IList <double> x) {
double p = 1.0;
for (int i = 0; i < x.Count; i++)
{
p *= x[i];
}
return(1.0 / (1.0 - p));
};
// \zeta(1) is infinite, so skip d=1
for (int d = 2; d <= 10; d++)
{
Console.WriteLine(d);
IntegrationResult result = MultiFunctionMath.Integrate(f, UnitCube(d));
Console.WriteLine("{0} ({1}) {2}", result.Value, result.Precision, AdvancedMath.RiemannZeta(d));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Value, AdvancedMath.RiemannZeta(d), new EvaluationSettings()
{
AbsolutePrecision = 2.0 * result.Precision
}));
}
}
public void SteinmetzVolume()
{
// Steinmetz solid is intersection of unit cylinders along all axes. This is another hard-edged integral. Analytic values are known for d=2-5.
// http://www.math.illinois.edu/~hildebr/ugresearch/cylinder-spring2013report.pdf
// http://www.math.uiuc.edu/~hildebr/igl/nvolumes-fall2012report.pdf
EvaluationSettings settings = new EvaluationSettings()
{
EvaluationBudget = 1000000, RelativePrecision = 1.0E-2
};
for (int d = 2; d <= 5; d++)
{
IntegrationResult v1 = MultiFunctionMath.Integrate((IList <double> x) => {
for (int i = 0; i < d; i++)
{
double s = 0.0;
for (int j = 0; j < d; j++)
{
if (j != i)
{
s += x[j] * x[j];
}
}
if (s > 1.0)
{
return(0.0);
}
}
return(1.0);
}, SymmetricUnitCube(d), settings);
double v2 = 0.0;
switch (d)
{
case 2:
// trivial square
v2 = 4.0;
break;
case 3:
v2 = 16.0 - 8.0 * Math.Sqrt(2.0);
break;
case 4:
v2 = 48.0 * (Math.PI / 4.0 - Math.Atan(Math.Sqrt(2.0)) / Math.Sqrt(2.0));
break;
case 5:
v2 = 256.0 * (Math.PI / 12.0 - Math.Atan(1.0 / (2.0 * Math.Sqrt(2.0))) / Math.Sqrt(2.0));
break;
}
Console.WriteLine("{0} {1} {2}", d, v1.Value, v2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(v1.Value, v2, new EvaluationSettings()
{
AbsolutePrecision = 4.0 * v1.Precision
}));
}
}
public void SeperableIntegrals()
{
// Integrates \Pi_{j=0}^{d-1} \int_0^1 \! dx \, x_j^j = \Pi_{j=0}^{d-1} \frac{1}{j+1} = \frac{1}{d!}
// This is a simple test of a seperable integral
Func <IReadOnlyList <double>, double> f = delegate(IReadOnlyList <double> x) {
double y = 1.0;
for (int j = 0; j < x.Count; j++)
{
y *= MoreMath.Pow(x[j], j);
}
return(y);
};
for (int d = 1; d <= 10; d++)
{
Console.WriteLine(d);
// Result gets very small, so rely on relative rather than absolute precision.
IntegrationSettings s = new IntegrationSettings()
{
AbsolutePrecision = 0, RelativePrecision = Math.Pow(10.0, -(6.0 - d / 2.0))
};
IntegrationResult r = MultiFunctionMath.Integrate(f, UnitCube(d), s);
Assert.IsTrue(r.Estimate.ConfidenceInterval(0.95).ClosedContains(1.0 / AdvancedIntegerMath.Factorial(d)));
}
}
public void WatsonIntegrals()
{
// Watson defined and analytically integrated three complicated triple integrals related to random walks in three dimension
// See http://mathworld.wolfram.com/WatsonsTripleIntegrals.html
Interval watsonWidth = Interval.FromEndpoints(0.0, Math.PI);
Interval[] watsonBox = new Interval[] { watsonWidth, watsonWidth, watsonWidth };
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (1.0 - Math.Cos(x[0]) * Math.Cos(x[1]) * Math.Cos(x[2])), watsonBox
).Estimate.ConfidenceInterval(0.99).ClosedContains(
MoreMath.Pow(AdvancedMath.Gamma(1.0 / 4.0), 4) / 4.0
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (3.0 - Math.Cos(x[0]) * Math.Cos(x[1]) - Math.Cos(x[1]) * Math.Cos(x[2]) - Math.Cos(x[0]) * Math.Cos(x[2])), watsonBox
).Estimate.ConfidenceInterval(0.99).ClosedContains(
3.0 * MoreMath.Pow(AdvancedMath.Gamma(1.0 / 3.0), 6) / Math.Pow(2.0, 14.0 / 3.0) / Math.PI
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (3.0 - Math.Cos(x[0]) - Math.Cos(x[1]) - Math.Cos(x[2])), watsonBox
).Estimate.ConfidenceInterval(0.99).ClosedContains(
Math.Sqrt(6.0) / 96.0 * AdvancedMath.Gamma(1.0 / 24.0) * AdvancedMath.Gamma(5.0 / 24.0) * AdvancedMath.Gamma(7.0 / 24.0) * AdvancedMath.Gamma(11.0 / 24.0)
)
);
}
private IntegrationResult RambleIntegral(int d, int s, IntegrationSettings settings)
{
return(MultiFunctionMath.Integrate((IReadOnlyList <double> x) => {
Complex z = 0.0;
for (int k = 0; k < d; k++)
{
z += ComplexMath.Exp(2.0 * Math.PI * Complex.I * x[k]);
}
return (MoreMath.Pow(ComplexMath.Abs(z), s));
}, UnitCube(d), settings));
}
public double BoxIntegralB(int d, int r, IntegrationSettings settings)
{
return(MultiFunctionMath.Integrate((IReadOnlyList <double> x) => {
double s = 0.0;
for (int k = 0; k < d; k++)
{
s += x[k] * x[k];
}
return (Math.Pow(s, r / 2.0));
}, UnitCube(d), settings).Value);
}
public void UnitSquareIntegrals()
{
// http://mathworld.wolfram.com/UnitSquareIntegral.html has a long list of integrals over the unit square.
// Many are taken from Guillera and Sondow,
// "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent.",
// 16 June 2005. (http://arxiv.org/abs/math.NT/0506319)
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (1.0 - x[0] * x[1]),
UnitCube(2)
).Estimate.ConfidenceInterval(0.95).ClosedContains(
AdvancedMath.RiemannZeta(2.0)
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => - Math.Log(x[0] * x[1]) / (1.0 - x[0] * x[1]),
UnitCube(2)
).Estimate.ConfidenceInterval(0.95).ClosedContains(
2.0 * AdvancedMath.RiemannZeta(3.0)
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => (x[0] - 1.0) / (1.0 - x[0] * x[1]) / Math.Log(x[0] * x[1]),
UnitCube(2)
).Estimate.ConfidenceInterval(0.95).ClosedContains(
AdvancedMath.EulerGamma
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => (x[0] - 1.0) / (1.0 + x[0] * x[1]) / Math.Log(x[0] * x[1]),
UnitCube(2)
).Estimate.ConfidenceInterval(0.95).ClosedContains(
Math.Log(4.0 / Math.PI)
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (1.0 + MoreMath.Sqr(x[0] * x[1])),
UnitCube(2)
).Estimate.ConfidenceInterval(0.95).ClosedContains(
AdvancedMath.Catalan
)
);
}
public double BoxIntegralD(int d, int r, EvaluationSettings settings)
{
return(MultiFunctionMath.Integrate((IList <double> x) => {
double s = 0.0;
for (int k = 0; k < d; k++)
{
double z = x[k] - x[k + d];
s += z * z;
}
return (Math.Pow(s, r / 2.0));
}, UnitCube(2 * d), settings).Value);
}
public void GaussianIntegrals()
{
Random rng = new Random(1);
for (int d = 2; d < 4; d++)
{
if (d == 4 || d == 5 || d == 6)
{
continue;
}
Console.WriteLine(d);
// Create a symmetric matrix
SymmetricMatrix A = new SymmetricMatrix(d);
for (int r = 0; r < d; r++)
{
for (int c = 0; c < r; c++)
{
A[r, c] = rng.NextDouble();
}
// Ensure it is positive definite by diagonal dominance
A[r, r] = r + 1.0;
}
// Compute its determinant, which appears in the analytic value of the integral
CholeskyDecomposition CD = A.CholeskyDecomposition();
double detA = CD.Determinant();
// Compute the integral
Func <IList <double>, double> f = (IList <double> x) => {
ColumnVector v = new ColumnVector(x);
double s = v.Transpose() * (A * v);
return(Math.Exp(-s));
};
Interval[] volume = new Interval[d];
for (int i = 0; i < d; i++)
{
volume[i] = Interval.FromEndpoints(Double.NegativeInfinity, Double.PositiveInfinity);
}
IntegrationResult I = MultiFunctionMath.Integrate(f, volume);
// Compare to the analytic result
Console.WriteLine("{0} ({1}) {2}", I.Value, I.Precision, Math.Sqrt(MoreMath.Pow(Math.PI, d) / detA));
Assert.IsTrue(TestUtilities.IsNearlyEqual(I.Value, Math.Sqrt(MoreMath.Pow(Math.PI, d) / detA), new EvaluationSettings()
{
AbsolutePrecision = 2.0 * I.Precision
}));
}
}
public void DoubleIntegrals()
{
// At http://mathworld.wolfram.com/DoubleIntegral.html, Mathworld
// lists a few double integrals with known values that we take as
// tests.
// One of the integrals listed there is just the zeta integral for d=2
// which we do above, so we omit it here.
// Because these are non-ocsilatory and have relatively low
// dimension, we can demand fairly high accuracy.
IntegrationResult i1 = MultiFunctionMath.Integrate(
(IList <double> x) => 1.0 / (1.0 - x[0] * x[0] * x[1] * x[1]),
UnitCube(2)
);
Console.WriteLine("{0} ({1})", i1.Value, i1.Precision);
Assert.IsTrue(TestUtilities.IsNearlyEqual(i1.Value, Math.PI * Math.PI / 8.0, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * i1.Precision
}));
//Assert.IsTrue(i1.Estimate.ConfidenceInterval(0.99).ClosedContains(Math.PI * Math.PI / 8.0));
IntegrationResult i2 = MultiFunctionMath.Integrate(
(IList <double> x) => 1.0 / (x[0] + x[1]) / Math.Sqrt((1.0 - x[0]) * (1.0 - x[1])),
UnitCube(2),
new EvaluationSettings()
{
RelativePrecision = 1.0E-5, EvaluationBudget = 1000000
}
);
Console.WriteLine("{0} ({1})", i2.Value, i2.Precision);
Assert.IsTrue(TestUtilities.IsNearlyEqual(i2.Value, 4.0 * AdvancedMath.Catalan, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * i2.Precision
}));
// For higher precision demands on this integral, we start getting Infinity +/- NaN for estimate and never terminate, look into this.
IntegrationResult i3 = MultiFunctionMath.Integrate(
(IList <double> x) => (x[0] - 1.0) / (1.0 - x[0] * x[1]) / Math.Log(x[0] * x[1]),
UnitCube(2)
);
Console.WriteLine("{0} ({1})", i3.Value, i3.Precision);
Assert.IsTrue(TestUtilities.IsNearlyEqual(i3.Value, AdvancedMath.EulerGamma, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * i3.Precision
}));
}
public void GaussianIntegrals()
{
Random rng = new Random(1);
for (int d = 2; d < 8; d++)
{
Console.WriteLine(d);
// Create a symmetric matrix
SymmetricMatrix A = new SymmetricMatrix(d);
for (int r = 0; r < d; r++)
{
for (int c = 0; c < r; c++)
{
A[r, c] = rng.NextDouble();
}
// Ensure it is positive definite by diagonal dominance
A[r, r] = r + 1.0;
}
// Compute its determinant, which appears in the analytic value of the integral
CholeskyDecomposition CD = A.CholeskyDecomposition();
double detA = CD.Determinant();
// Compute the integral
Func <IReadOnlyList <double>, double> f = (IReadOnlyList <double> x) => {
ColumnVector v = new ColumnVector(x);
double s = v.Transpose * (A * v);
return(Math.Exp(-s));
};
Interval[] volume = new Interval[d];
for (int i = 0; i < d; i++)
{
volume[i] = Interval.FromEndpoints(Double.NegativeInfinity, Double.PositiveInfinity);
}
// These are difficult integrals; demand reduced precision.
IntegrationSettings settings = new IntegrationSettings()
{
RelativePrecision = Math.Pow(10.0, -(4.0 - d / 2.0))
};
IntegrationResult I = MultiFunctionMath.Integrate(f, volume, settings);
// Compare to the analytic result
Assert.IsTrue(I.Estimate.ConfidenceInterval(0.95).ClosedContains(Math.Sqrt(MoreMath.Pow(Math.PI, d) / detA)));
}
}
public void BallVolumeIntegrals()
{
// The volume of a d-sphere is \frac{\pi^{d/2}}{\Gamma(d/2 + 1)}
// and the fraction in the first quadrant is 1/2^d of that
// This is a simple test of the integration of a discontinuous function
Func <IList <double>, double> f = delegate(IList <double> x) {
double r2 = 0.0;
for (int j = 0; j < x.Count; j++)
{
r2 += x[j] * x[j];
}
if (r2 <= 1.0)
{
return(1.0);
}
else
{
return(0.0);
}
};
for (int d = 1; d <= 8; d++)
{
if (d == 6)
{
continue; //. For d=6, integral returns after just ~300 evaluation with an underestimated error; look into this
}
Console.WriteLine(d);
IntegrationResult result = MultiFunctionMath.Integrate(f, UnitCube(d), new EvaluationSettings()
{
AbsolutePrecision = 0.0, RelativePrecision = 5.0E-3, EvaluationBudget = 1000000
});
double V = Math.Pow(Math.PI, d / 2.0) / AdvancedMath.Gamma(d / 2.0 + 1.0) * MoreMath.Pow(2.0, -d);
Console.WriteLine("{0} ({1}) {2}: {3}", result.Value, result.Precision, result.Value - V, result.EvaluationCount);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Value, V, new EvaluationSettings()
{
AbsolutePrecision = 4.0 * result.Precision
}));
}
}
public void ExponentialBox()
{
// Mentioned in passing in http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/BoxII.pdf
// This is easy because the integrand is seperable and smooth.
for (int d = 2; d <= 10; d++)
{
IntegrationResult result = MultiFunctionMath.Integrate((IReadOnlyList <double> r) => {
double s = 0.0;
foreach (double x in r)
{
s += x * x;
}
return(Math.Exp(-s));
}, UnitCube(d));
Assert.IsTrue(result.Estimate.ConfidenceInterval(0.95).ClosedContains(
MoreMath.Pow(Math.Sqrt(Math.PI) / 2.0 * AdvancedMath.Erf(1.0), d)
));
}
}
public void BallVolumeIntegrals()
{
// The volume of a d-sphere is \frac{\pi^{d/2}}{\Gamma(d/2 + 1)}
// and the fraction in the first quadrant is 1/2^d of that.
// This is a simple test of the integration of a discontinuous function.
Func <IReadOnlyList <double>, double> f = delegate(IReadOnlyList <double> x) {
double r2 = 0.0;
for (int j = 0; j < x.Count; j++)
{
r2 += x[j] * x[j];
}
if (r2 <= 1.0)
{
return(1.0);
}
else
{
return(0.0);
}
};
for (int d = 1; d <= 8; d++)
{
if (d == 6)
{
continue; //. For d=6, integral returns after just ~300 evaluation with an underestimated error; look into this
}
Console.WriteLine(d);
IntegrationSettings settings = new IntegrationSettings()
{
AbsolutePrecision = 0.0, RelativePrecision = 1.0E-2
};
IntegrationResult result = MultiFunctionMath.Integrate(f, UnitCube(d), settings);
double V = Math.Pow(Math.PI, d / 2.0) / AdvancedMath.Gamma(d / 2.0 + 1.0) * MoreMath.Pow(2.0, -d);
Assert.IsTrue(result.Estimate.ConfidenceInterval(0.95).ClosedContains(V));
}
}
public void WatsonIntegrals()
{
// Watson defined and analytically integrated three complicated triple integrals related to random walks in three dimension
// See http://mathworld.wolfram.com/WatsonsTripleIntegrals.html
// These integrals are difficult, so up the budget to about 1,000,000 and reduce the target accuracy to about 10^{-4}
EvaluationSettings settings = new EvaluationSettings()
{
RelativePrecision = 1.0E-4, EvaluationBudget = 1000000
};
Interval watsonWidth = Interval.FromEndpoints(0.0, Math.PI);
Interval[] watsonBox = new Interval[] { watsonWidth, watsonWidth, watsonWidth };
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList <double> x) => 1.0 / (1.0 - Math.Cos(x[0]) * Math.Cos(x[1]) * Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
MoreMath.Pow(AdvancedMath.Gamma(1.0 / 4.0), 4) / 4.0
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList <double> x) => 1.0 / (3.0 - Math.Cos(x[0]) * Math.Cos(x[1]) - Math.Cos(x[1]) * Math.Cos(x[2]) - Math.Cos(x[0]) * Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
3.0 * MoreMath.Pow(AdvancedMath.Gamma(1.0 / 3.0), 6) / Math.Pow(2.0, 14.0 / 3.0) / Math.PI
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList <double> x) => 1.0 / (3.0 - Math.Cos(x[0]) - Math.Cos(x[1]) - Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
Math.Sqrt(6.0) / 96.0 * AdvancedMath.Gamma(1.0 / 24.0) * AdvancedMath.Gamma(5.0 / 24.0) * AdvancedMath.Gamma(7.0 / 24.0) * AdvancedMath.Gamma(11.0 / 24.0)
)
);
}
public void ZetaIntegrals()
{
// By expanding 1/(1-x) = 1 + x + x^2 + ... and integrating term-by-term
// it's easy to show that this integral is \zeta(d)
Func <IReadOnlyList <double>, double> f = delegate(IReadOnlyList <double> x) {
double p = 1.0;
for (int i = 0; i < x.Count; i++)
{
p *= x[i];
}
return(1.0 / (1.0 - p));
};
// \zeta(1) is infinite, so skip d=1
for (int d = 2; d <= 15; d++)
{
Console.WriteLine(d);
IntegrationResult result = MultiFunctionMath.Integrate(f, UnitCube(d));
Assert.IsTrue(result.Estimate.ConfidenceInterval(0.95).ClosedContains(AdvancedMath.RiemannZeta(d)));
}
}
public void DoubleIntegrals()
{
// At http://mathworld.wolfram.com/DoubleIntegral.html, Mathworld
// lists a few double integrals with known values that we take as
// tests.
// One of the integrals listed there is just the zeta integral for d=2
// which we do above, so we omit it here.
// Because these are non-ocsilatory and have relatively low
// dimension, we can demand fairly high accuracy.
IntegrationResult i1 = MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (1.0 - x[0] * x[0] * x[1] * x[1]),
UnitCube(2)
);
Assert.IsTrue(i1.Estimate.ConfidenceInterval(0.95).ClosedContains(Math.PI * Math.PI / 8.0));
IntegrationResult i2 = MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => 1.0 / (x[0] + x[1]) / Math.Sqrt((1.0 - x[0]) * (1.0 - x[1])),
UnitCube(2),
new IntegrationSettings()
{
RelativePrecision = 1.0E-6
}
);
Assert.IsTrue(i2.Estimate.ConfidenceInterval(0.95).ClosedContains(4.0 * AdvancedMath.Catalan));
// For higher precision demands on this integral, we start getting Infinity +/- NaN for estimate and never terminate, look into this.
IntegrationResult i3 = MultiFunctionMath.Integrate(
(IReadOnlyList <double> x) => (x[0] - 1.0) / (1.0 - x[0] * x[1]) / Math.Log(x[0] * x[1]),
UnitCube(2)
);
Assert.IsTrue(i3.Estimate.ConfidenceInterval(0.95).ClosedContains(AdvancedMath.EulerGamma));
}
public void IsingIntegrals()
{
// See http://www.davidhbailey.com/dhbpapers/ising.pdf.
IntegrationSettings settings = new IntegrationSettings()
{
RelativePrecision = 1.0E-3, EvaluationBudget = 1000000
};
int d = 4;
Interval[] volume = new Interval[d];
for (int i = 0; i < volume.Length; i++)
{
volume[i] = Interval.FromEndpoints(0.0, Double.PositiveInfinity);
}
IntegrationResult r = MultiFunctionMath.Integrate((IReadOnlyList <double> x) => {
double p = 1.0;
double q = 0.0;
for (int i = 0; i < x.Count; i++)
{
double u = x[i];
double v = 1.0 / u;
q += (u + v);
p *= v;
}
return(p / MoreMath.Sqr(q));
}, volume, settings);
double c = 4.0 / AdvancedIntegerMath.Factorial(d);
Console.WriteLine("{0} {1}", c * r.Estimate, r.EvaluationCount);
Console.WriteLine(7.0 * AdvancedMath.RiemannZeta(3.0) / 12.0);
Assert.IsTrue((c * r.Estimate).ConfidenceInterval(0.99).ClosedContains(7.0 * AdvancedMath.RiemannZeta(3.0) / 12.0));
}
public static void Integration()
{
// This is the area of the upper half-circle, so the value should be pi/2.
IntegrationResult i = FunctionMath.Integrate(x => Math.Sqrt(1.0 - x * x), -1.0, +1.0);
Console.WriteLine($"i = {i.Estimate} after {i.EvaluationCount} evaluations.");
Interval zeroToPi = Interval.FromEndpoints(0.0, Math.PI);
IntegrationResult watson = MultiFunctionMath.Integrate(
x => 1.0 / (1.0 - Math.Cos(x[0]) * Math.Cos(x[1]) * Math.Cos(x[2])),
new Interval[] { zeroToPi, zeroToPi, zeroToPi }
);
IntegrationResult i2 = FunctionMath.Integrate(
x => Math.Sqrt(1.0 - x * x), -1.0, +1.0,
new IntegrationSettings()
{
RelativePrecision = 1.0E-4
}
);
// This integrand has a log singularity at x = 0.
// The value of this integral is the Catalan constant.
IntegrationResult soft = FunctionMath.Integrate(
x => - Math.Log(x) / (1.0 + x * x), 0.0, 1.0
);
// This integral has a power law singularity at x = 0.
// The value of this integral is 4.
IntegrationResult hard = FunctionMath.Integrate(
x => Math.Pow(x, -3.0 / 4.0), 0.0, 1.0,
new IntegrationSettings()
{
RelativePrecision = 1.0E-6
}
);
// The value of this infinite integral is sqrt(pi)
IntegrationResult infinite = FunctionMath.Integrate(
x => Math.Exp(-x * x), Double.NegativeInfinity, Double.PositiveInfinity
);
Func <IReadOnlyList <double>, double> distance = z => {
ColumnVector x = new ColumnVector(z[0], z[1], z[2]);
ColumnVector y = new ColumnVector(z[3], z[4], z[5]);
ColumnVector d = x - y;
return(d.Norm());
};
Interval oneBox = Interval.FromEndpoints(0.0, 1.0);
Interval[] sixBox = new Interval[] { oneBox, oneBox, oneBox, oneBox, oneBox, oneBox };
IntegrationSettings settings = new IntegrationSettings()
{
RelativePrecision = 1.0E-4,
AbsolutePrecision = 0.0,
Listener = r => {
Console.WriteLine($"Estimate {r.Estimate} after {r.EvaluationCount} evaluations.");
}
};
IntegrationResult numeric = MultiFunctionMath.Integrate(distance, sixBox, settings);
Console.WriteLine($"The numeric result is {numeric.Estimate}.");
double analytic = 4.0 / 105.0 + 17.0 / 105.0 * Math.Sqrt(2.0) - 2.0 / 35.0 * Math.Sqrt(3.0)
+ Math.Log(1.0 + Math.Sqrt(2.0)) / 5.0 + 2.0 / 5.0 * Math.Log(2.0 + Math.Sqrt(3.0))
- Math.PI / 15.0;
Console.WriteLine($"The analytic result is {analytic}.");
}