public void SquareUnitMatrixEigensystem()
{
int d = 3;
SquareMatrix I = UnitMatrix.OfDimension(d).ToSquareMatrix();
ComplexEigendecomposition E = I.Eigendecomposition();
Assert.IsTrue(E.Dimension == d);
for (int i = 0; i < d; i++)
{
Complex val = E.Eigenpairs[i].Eigenvalue;
Assert.IsTrue(val == 1.0);
ComplexColumnVector vec = E.Eigenpairs[i].Eigenvector;
for (int j = 0; j < d; j++)
{
if (i == j)
{
Assert.IsTrue(vec[j] == 1.0);
}
else
{
Assert.IsTrue(vec[j] == 0.0);
}
}
}
}
public void ComplexColumnVectorArithmetic()
{
ComplexColumnVector u = new ComplexColumnVector(Complex.I, Complex.Zero, Complex.One);
Assert.IsTrue(u.Dimension == 3);
Assert.IsTrue(1.0 * u == u);
}
public void ComplexColumnVectorManipulations()
{
ComplexColumnVector u = new ComplexColumnVector(3);
u[0] = Complex.I;
u[2] = Complex.One;
Assert.IsTrue(u.Dimension == 3);
ComplexColumnVector v = u.Copy();
Assert.IsTrue(v == u);
v[0] += 1.0;
Assert.IsTrue(v != u);
}
public void ComplexColunVectorAsMatrix()
{
ComplexColumnVector u = new ComplexColumnVector(Complex.I, Complex.Zero, Complex.One);
Assert.IsTrue(u.ColumnCount == 1);
Assert.IsTrue(u.RowCount == u.Dimension);
for (int r = 0; r < u.Dimension; r++)
{
Assert.IsTrue(u[r, 0] == u[r]);
}
for (int r = 0; r < u.Dimension; r++)
{
u[r, 0] = 0.0;
}
}
public void ComplexVector()
{
ComplexColumnVector v = new ComplexColumnVector(
Complex.One, Complex.Zero, Complex.I
);
Assert.IsTrue(v.Dimension == 3);
Assert.IsTrue(v[2] == Complex.I);
ComplexColumnVector v2 = 2.0 * v;
Assert.IsTrue(v2[1] == 0.0);
ComplexColumnVector vi = Complex.I * v;
Assert.IsTrue(vi[2] == -1.0);
v[1] += 1.0;
Assert.IsTrue(v[1] == 1.0);
}
public void SquareRandomMatrixEigenvalues()
{
for (int d = 1; d <= 67; d = d + 11)
{
SquareMatrix M = CreateSquareRandomMatrix(d, d);
double tr = M.Trace();
DateTime start = DateTime.Now;
ComplexEigendecomposition E = M.Eigendecomposition();
DateTime finish = DateTime.Now;
Console.WriteLine("d={0} t={1} ms", d, (finish - start).Milliseconds);
Assert.IsTrue(E.Dimension == d);
Complex[] es = new Complex[d];
for (int i = 0; i < d; i++)
{
es[i] = E.Eigenpairs[i].Eigenvalue;
ComplexColumnVector v = E.Eigenpairs[i].Eigenvector;
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(M, v, es[i]));
}
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(es, tr));
}
}
public static bool IsNearlyEigenpair(AnySquareMatrix A, ComplexColumnVector v, Complex a)
{
int d = A.Dimension;
// compute products
/*
* Complex[] Av = new Complex[d];
* for (int i=0; i<d; i++) {
* Av[i] = 0.0;
* for (int j=0; j<d; j++) {
* Av[i] += A[i,j]*v[j];
* }
* }
*/
ComplexColumnVector Av = A * v;
ComplexColumnVector av = a * v;
// compute tolerance
double N = A.MaxNorm();
double n = 0.0;
for (int i = 0; i < d; i++)
{
n += ComplexMath.Abs(v[i]);
}
double ep = TargetPrecision * (N * n / d + ComplexMath.Abs(a) * n);
// compare elements within tolerance
for (int i = 0; i < d; i++)
{
if (ComplexMath.Abs(Av[i] - av[i]) > ep)
{
return(false);
}
}
return(true);
}
public void SquareVandermondeMatrixEigenvalues()
{
for (int d = 1; d <= 8; d++)
{
SquareMatrix H = CreateVandermondeMatrix(d);
double tr = H.Trace();
ComplexEigendecomposition E = H.Eigendecomposition();
double sum = 0.0;
foreach (ComplexEigenpair pair in E.Eigenpairs)
{
double e = pair.Eigenvalue.Re;
sum += e;
ComplexColumnVector vc = pair.Eigenvector;
ColumnVector v = new ColumnVector(d);
for (int j = 0; j < d; j++)
{
v[j] = vc[j].Re;
}
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(H, v, e));
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(tr, sum));
}
}
