private static Vector2 Eigenvector(this SymmetricMatrix2 @this, double lambda)
{
var rowX = new Vector2(@this.XX - lambda, @this.XY);
var rowY = new Vector2(@this.XY, @this.YY - lambda);
var normX2 = rowX.Norm2;
var normY2 = rowY.Norm2;
if (normX2 > normY2)
{
if (normX2 > BasicMath.Epsilon)
{
return(rowX.Rotate90() / Math.Sqrt(normX2));
}
else
{
return(Vector2.UnitX);
}
}
else
{
if (normY2 > BasicMath.Epsilon)
{
return(rowY.Rotate90() / Math.Sqrt(normY2));
}
else
{
return(Vector2.UnitX);
}
}
}
public static void ComputeEigenvalues(this SymmetricMatrix2 @this, out double low, out double high)
{
var halfTrace = @this.Trace() / 2;
var disc = BasicMath.Sqrt(halfTrace * halfTrace - @this.Det());
low = halfTrace - disc;
high = halfTrace + disc;
}
// finds least-squares solution to a system of orthogonality equations (i.e. result is fitted to be orthogonal to vectors)
public static void FitOrthogonal(IEnumerable <Vector2> vectors, out Vector2 result, out double residual)
{
var mat = SymmetricMatrix2.TensorSquare(vectors.First());
foreach (var vector in vectors.Skip(1))
{
mat += SymmetricMatrix2.TensorSquare(vector);
}
mat.ComputeLowEigendata(out residual, out result);
}
public void Eigenvectors_and_eigenvalues_of_satisfy_defining_equations
([Values(0.577, 0.577, 0.577, 0.577)] double xx,
[Values(2.718, 0.577, 0.577, 0.577)] double yy,
[Values(3.141, 0, 1e-40, 1e-60)] double xy)
{
var mat = new SymmetricMatrix2(xx, yy, xy);
var loVal = mat.LowEigenvalue();
var loVec = mat.LowEigenvector();
Expect((mat - SymmetricMatrix2.Scalar(loVal)).Det(), Is.EqualTo(0).Within(_tolerance));
Expect(Vector2.Distance(mat * loVec, loVal * loVec), Is.LessThan(_tolerance));
var hiVal = mat.HighEigenvalue();
var hiVec = mat.HighEigenvector();
Expect((mat - SymmetricMatrix2.Scalar(hiVal)).Det(), Is.EqualTo(0).Within(_tolerance));
Expect(Vector2.Distance(mat * hiVec, hiVal * hiVec), Is.LessThan(_tolerance));
Expect(loVal, Is.LessThan(hiVal + _tolerance));
Expect(loVal + hiVal, Is.EqualTo(mat.Trace()).Within(_tolerance));
}
public static double HighEigenvalue(this SymmetricMatrix2 @this)
{
var halfTrace = @this.Trace() / 2;
return(halfTrace + BasicMath.Sqrt(halfTrace * halfTrace - @this.Det()));
}
public static double Trace(this SymmetricMatrix2 @this)
{
return(@this.XX + @this.YY);
}
public static double Det(this SymmetricMatrix2 @this)
{
return(@this.XX * @this.YY - @this.XY * @this.XY);
}
public static Vector2 RowY(this SymmetricMatrix2 @this)
{
return(new Vector2(@this.XY, @this.YY));
}
public static void ComputeEigendata(this SymmetricMatrix2 @this, out double lowEvalue, out double highEvalue, out Vector2 lowEvector)
{
@this.ComputeEigenvalues(out lowEvalue, out highEvalue);
lowEvector = @this.Eigenvector(lowEvalue);
}
public static void ComputeHighEigendata(this SymmetricMatrix2 @this, out double eigenvalue, out Vector2 eigenvector)
{
eigenvalue = @this.HighEigenvalue();
eigenvector = @this.Eigenvector(eigenvalue);
}
public static void ComputeEigenvectors(this SymmetricMatrix2 @this, out Vector2 low, out Vector2 high)
{
low = @this.LowEigenvector();
high = low.Rotate90();
}
public static Vector2 HighEigenvector(this SymmetricMatrix2 @this)
{
return(@this.Eigenvector(@this.HighEigenvalue()));
}