private static Func <double, SquareMatrix> GetJordanFormExpM(SquareMatrix jordanForm)
{
var eigenvalues = jordanForm.Eigenvalues();
var l1 = eigenvalues[0].Re;
var l2 = eigenvalues[1].Re;
// Если l1 и l2 разные.
if ((l1 - l2).IsNotZero())
{
return(t => { return new SquareMatrix(new [, ]
{
{ Math.Exp(l1 * t), 0.0 },
{ 0.0, Math.Exp(l2 * t) }
}); });
}
if (jordanForm.GetGeometricMultiplicity(l1) == 2)
{
return(t => { return new SquareMatrix(new [, ]
{
{ Math.Exp(l1 * t), 0.0 },
{ 0.0, Math.Exp(l1 * t) }
}); });
}
return(t => { return new SquareMatrix(new [, ]
{
{ Math.Exp(l1 * t), t * Math.Exp(l1 * t) },
{ 0.0, Math.Exp(l1 * t) }
}); });
}
/// <summary>
/// Получить собственные (и присоединённые) векторы матрицы.
/// </summary>
/// <param name="originMatrix">Входная матрица.</param>
/// <returns>Матрица, состоящая из комплексных чисел.</returns>
public static SquareMatrix GetEigenvectors(this SquareMatrix originMatrix)
{
var eigenvalues = originMatrix.Eigenvalues();
if (eigenvalues.Any(v => v.Im != 0))
{
throw new ComplexEigenvaluesException("Комплексные собственные значения не поддерживаются.");
}
const double max = 1.0;
const double min = 0.0;
var eigenvectors = new SquareMatrix(originMatrix.Dimension);
var identityMatrix = SquareMatrixHelper.GetIdentityMatrix(originMatrix.Dimension);
// Два различных собственных значения.
if (!(eigenvalues[0] - eigenvalues[1]).Re.IsZero())
{
var eigenSol1 = originMatrix - eigenvalues[0].Re * identityMatrix;
var eigenSol2 = originMatrix - eigenvalues[1].Re * identityMatrix;
if (!eigenSol1.Row(0).IsZero())
{
eigenvectors = new SquareMatrix(new [, ]
{
{ eigenSol1[0, 1], eigenSol2[0, 1] },
{ -eigenSol1[0, 0], -eigenSol2[0, 0] }
});
}
else
{
eigenvectors = new SquareMatrix(new [, ]
{
{ eigenSol1[1, 1], eigenSol2[0, 1] },
{ -eigenSol1[1, 0], -eigenSol2[0, 0] }
});
}
}
else if ((eigenvalues[0] - eigenvalues[1]).Re.IsZero())
{
while (true)
{
if ((originMatrix - eigenvalues[1].Re * identityMatrix).GetRank() != 0)
{
var possibleAttachedVectorEntries = GetRandomColumnVector(max, min);
var possibleAttachedVector = new ColumnVector(possibleAttachedVectorEntries);
var possibleEigenvector = (originMatrix - eigenvalues[1].Re * identityMatrix) * possibleAttachedVector;
if (originMatrix * possibleAttachedVector == eigenvalues[1].Re * possibleAttachedVector)
{
continue;
}
if (possibleEigenvector.OneNorm().IsZero())
{
continue;
}
if (((originMatrix - eigenvalues[0].Re * identityMatrix) * possibleEigenvector).OneNorm().IsZero())
{
eigenvectors = new SquareMatrix(new [, ]
{
{ possibleEigenvector[0], possibleAttachedVector[0] },
{ possibleEigenvector[1], possibleAttachedVector[1] }
});
break;
}
}
eigenvectors = new SquareMatrix(new [, ]
{
{ 1.0, 0.0 },
{ 0.0, 1.0 }
});
break;
}
}
return(eigenvectors);
}
public void DifficultEigenvalue()
{
SquareMatrix A = new SquareMatrix(4);
A[0, 0] = 0.0; A[0, 1] = 2.0; A[0, 2] = 0.0; A[0, 3] = -1.0;
A[1, 0] = 1.0; A[1, 1] = 0.0; A[1, 2] = 0.0; A[1, 3] = 0.0;
A[2, 0] = 0.0; A[2, 1] = 1.0; A[2, 2] = 0.0; A[2, 3] = 0.0;
A[3, 0] = 0.0; A[3, 1] = 0.0; A[3, 2] = 1.0; A[3, 3] = 0.0;
Complex[] zs = A.Eigenvalues();
foreach (Complex z in zs) {
Console.WriteLine("{0} ({1} {2})", z, ComplexMath.Abs(z), ComplexMath.Arg(z));
}
}
public void TimedEigenvalues()
{
int n = 200;
Random rng = new Random(1);
SquareMatrix A = new SquareMatrix(n);
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
A[r, c] = rng.NextDouble();
}
}
Stopwatch s = Stopwatch.StartNew();
Complex[] eigenvalues = A.Eigenvalues();
s.Stop();
Console.WriteLine(s.ElapsedMilliseconds);
}
public void KnownEigenvalues()
{
// This example is presented in http://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter3.pdf
// It has eigenvalues 1 \pm 2i, 3, 4, 5 \pm 6i
double[,] souce = new double[,] {
{ 7, 3, 4, -11, -9, -2},
{-6, 4, -5, 7, 1, 12},
{-1, -9, 2, 2, 9, 1},
{-8, 0, -1, 5, 0, 8},
{-4, 3, -5, 7, 2, 10},
{6, 1, 4, -11, -7, -1}
};
SquareMatrix A = new SquareMatrix(6);
for (int r = 0; r < souce.GetLength(0); r++) {
for (int c = 0; c < souce.GetLength(1); c++) {
A[r, c] = souce[r, c];
}
}
Complex[] eigenvalues = A.Eigenvalues();
foreach (Complex eigenvalue in eigenvalues) {
Console.WriteLine(eigenvalue);
}
}