public static GaussLegendreCoefficients GaussLegendre(int N)
{
var beta = from i in 1.To(N - 1)
select 0.5 / Math.Sqrt(1.0 - (2.0 * i).Pow(-2));
SymmetricMatrix T = new SymmetricMatrix(N); // how to use TridiagonalMatrix?
beta.ForEach((b, i) =>
{
T[i + 1, i] = b;
T[i, i + 1] = b;
});
RealEigensystem e = T.Eigensystem();
var indices = from i in 0.To(N - 1)
orderby e.Eigenvalue(i)
select i;
return(new GaussLegendreCoefficients
{
mu = indices.Select(i => e.Eigenvalue(i)).ToArray(),
wt = indices.Select(i => 2 * e.Eigenvector(i)[0].Pow(2)).ToArray()
});
}
public void SymmetricRandomMatrixEigenvectors()
{
for (int d = 1; d <= 100; d = d + 11)
{
Console.WriteLine("d={0}", d);
SymmetricMatrix M = CreateSymmetricRandomMatrix(d, 1);
double tr = M.Trace();
RealEigensystem E = M.Eigensystem();
Assert.IsTrue(E.Dimension == M.Dimension);
SquareMatrix D = new SquareMatrix(E.Dimension);
double[] es = new double[E.Dimension];
for (int i = 0; i < E.Dimension; i++)
{
double e = E.Eigenvalue(i);
ColumnVector v = E.Eigenvector(i);
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(M, v, e));
D[i, i] = e;
es[i] = e;
}
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(es, tr));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
D, E.TransformMatrix.Transpose() * M * E.TransformMatrix
));
}
}
/// <summary>
/// Converts conic representation of an ellipse (given the coefficients vector) to the parametric representation.
/// </summary>
/// <param name="conic">The implicit equation coefficients</param>
/// <returns>Parametric representation</returns>
private static EllipseParams Conic2Parametric(double[] conic)
{
var A = new SymmetricMatrix(2);
A[0, 0] = conic[0]; // a
A[1, 1] = conic[2]; // c
A[0, 1] = A[1, 0] = 0.5 * conic[1]; // half b
var B = new ColumnVector(conic[3], conic[4]);
var C = conic[5];
var eig = A.Eigensystem();
Debug.Assert(eig.Eigenvalue(0) * eig.Eigenvalue(1) > 0);
var D = new double[] { eig.Eigenvalue(0), eig.Eigenvalue(1) };
var Q = eig.Eigentransformation().Transpose();
var t = -0.5 * A.Inverse() * B;
var c_h = t.Transpose() * A * t + B.Transpose() * t + C;
return(new EllipseParams
{
Center = new Point(t[0], t[1]),
XRadius = Math.Sqrt(-c_h / D[0]),
YRadius = Math.Sqrt(-c_h / D[1]),
Degrees = 180 * Math.Atan2(Q[0, 1], Q[0, 0]) / Math.PI,
});
}
public void RealEigenvalueOrdering()
{
int d = 10;
Random rng = new Random(d + 1);
SymmetricMatrix A = new SymmetricMatrix(d);
A.Fill((int r, int c) => - 1.0 + 2.0 * rng.NextDouble());
RealEigensystem E = A.Eigensystem();
for (int i = 0; i < E.Dimension; i++)
{
Console.WriteLine(E.Eigenvalue(i));
}
E.Sort(OrderBy.ValueAscending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(E.Eigenvalue(i - 1) <= E.Eigenvalue(i));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.ValueDescending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(E.Eigenvalue(i - 1) >= E.Eigenvalue(i));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.MagnitudeAscending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(Math.Abs(E.Eigenvalue(i - 1)) <= Math.Abs(E.Eigenvalue(i)));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.MagnitudeDescending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(Math.Abs(E.Eigenvalue(i - 1)) >= Math.Abs(E.Eigenvalue(i)));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
}