/// <summary>
/// Returns the diagonal matrix Σ of the decomposition.
/// <para>Σ is a diagonal matrix. The singular values are provided in
/// non-increasing order, for compatibility with Jama.</para>
/// </summary>
/// <returns>the Σ matrix</returns>
public RealMatrix getS()
{
if (cachedS == null)
{
// cache the matrix for subsequent calls
cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues);
}
return(cachedS);
}
/// <summary>
/// Gets the block diagonal matrix D of the decomposition.
/// D is a block diagonal matrix.
/// Real eigenvalues are on the diagonal while complex values are on
/// 2x2 blocks { {real +imaginary}, {-imaginary, real} }.
/// </summary>
/// <returns>the D matrix.</returns>
/// <remarks>
/// See <see cref="getRealEigenvalues()"/><para/>
/// See <see cref="getImagEigenvalues()"/>
/// </remarks>
public RealMatrix getD()
{
if (cachedD == null)
{
// cache the matrix for subsequent calls
cachedD = MatrixUtils.createRealDiagonalMatrix(realEigenvalues);
for (int i = 0; i < imagEigenvalues.Length; i++)
{
if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) > 0)
{
cachedD.setEntry(i, i + 1, imagEigenvalues[i]);
}
else if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) < 0)
{
cachedD.setEntry(i, i - 1, imagEigenvalues[i]);
}
}
}
return(cachedD);
}
/// <summary>
/// Computes the square-root of the matrix.
/// This implementation assumes that the matrix is symmetric and positive
/// definite.
/// </summary>
/// <returns>the square-root of the matrix.</returns>
/// <exception cref="MathUnsupportedOperationException"> if the matrix is not
/// symmetric or not positive definite.</exception>
public RealMatrix getSquareRoot()
{
if (!isSymmetric)
{
throw new MathUnsupportedOperationException();
}
double[] sqrtEigenValues = new double[realEigenvalues.Length];
for (int i = 0; i < realEigenvalues.Length; i++)
{
double eigen = realEigenvalues[i];
if (eigen <= 0)
{
throw new MathUnsupportedOperationException();
}
sqrtEigenValues[i] = FastMath.sqrt(eigen);
}
RealMatrix sqrtEigen = MatrixUtils.createRealDiagonalMatrix(sqrtEigenValues);
RealMatrix v = getV();
RealMatrix vT = getVT();
return(v.multiply(sqrtEigen).multiply(vT));
}