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 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 CatalanHankelMatrixDeterminant()
{
for (int d = 1; d <= 8; d++)
{
SymmetricMatrix S = new SymmetricMatrix(d);
for (int r = 0; r < d; r++)
{
for (int c = 0; c <= r; c++)
{
int n = r + c;
S[r, c] = AdvancedIntegerMath.BinomialCoefficient(2 * n, n) / (n + 1);
}
}
CholeskyDecomposition CD = S.CholeskyDecomposition();
Assert.IsTrue(TestUtilities.IsNearlyEqual(CD.Determinant(), 1.0));
}
}