/**
* <p>
* Given a set of polynomial coefficients, compute the roots of the polynomial. Depending on
* the polynomial being considered the roots may contain complex number. When complex numbers are
* present they will come in pairs of complex conjugates.
* </p>
*
* <p>
* Coefficients are ordered from least to most significant, e.g: y = c[0] + x*c[1] + x*x*c[2].
* </p>
*
* @param coefficients Coefficients of the polynomial.
* @return The roots of the polynomial
*/
public static Complex_F64[] findRoots(params double[] coefficients)
{
int N = coefficients.Length - 1;
// Construct the companion matrix
DMatrixRMaj c = new DMatrixRMaj(N, N);
double a = coefficients[N];
for (int i = 0; i < N; i++)
{
c.set(i, N - 1, -coefficients[i] / a);
}
for (int i = 1; i < N; i++)
{
c.set(i, i - 1, 1);
}
// use generalized eigenvalue decomposition to find the roots
EigenDecomposition_F64 <DMatrixRMaj> evd = DecompositionFactory_DDRM.eig(N, false);
evd.decompose(c);
Complex_F64[] roots = new Complex_F64[N];
for (int i = 0; i < N; i++)
{
roots[i] = evd.getEigenvalue(i);
}
return(roots);
}
/**
* <p>
* Puts all the real eigenvectors into the columns of a matrix. If an eigenvalue is imaginary
* then the corresponding eigenvector will have zeros in its column.
* </p>
*
* @param eig An eigenvalue decomposition which has already decomposed a matrix.
* @return An m by m matrix containing eigenvectors in its columns.
*/
public static DMatrixRMaj createMatrixV(EigenDecomposition_F64 <DMatrixRMaj> eig)
{
int N = eig.getNumberOfEigenvalues();
DMatrixRMaj V = new DMatrixRMaj(N, N);
for (int i = 0; i < N; i++)
{
Complex_F64 c = eig.getEigenvalue(i);
if (c.isReal())
{
DMatrixRMaj v = eig.getEigenVector(i);
if (v != null)
{
for (int j = 0; j < N; j++)
{
V.set(j, i, v.get(j, 0));
}
}
}
}
return(V);
}
/**
* <p>
* Checks to see if the matrix is positive semidefinite:
* </p>
* <p>
* x<sup>T</sup> A x ≥ 0<br>
* for all x where x is a non-zero vector and A is a symmetric matrix.
* </p>
*
* @param A square symmetric matrix. Not modified.
*
* @return True if it is positive semidefinite and false if it is not.
*/
public static bool isPositiveSemidefinite(DMatrixRMaj A)
{
if (!isSquare(A))
{
return(false);
}
EigenDecomposition_F64 <DMatrixRMaj> eig = DecompositionFactory_DDRM.eig(A.numCols, false);
if (eig.inputModified())
{
A = (DMatrixRMaj)A.copy();
}
eig.decompose(A);
for (int i = 0; i < A.numRows; i++)
{
Complex_F64 v = eig.getEigenvalue(i);
if (v.getReal() < 0)
{
return(false);
}
}
return(true);
}
/**
* <p>
* A diagonal matrix where real diagonal element contains a real eigenvalue. If an eigenvalue
* is imaginary then zero is stored in its place.
* </p>
*
* @param eig An eigenvalue decomposition which has already decomposed a matrix.
* @return A diagonal matrix containing the eigenvalues.
*/
public static DMatrixRMaj createMatrixD(EigenDecomposition_F64 <DMatrixRMaj> eig)
{
int N = eig.getNumberOfEigenvalues();
DMatrixRMaj D = new DMatrixRMaj(N, N);
for (int i = 0; i < N; i++)
{
Complex_F64 c = eig.getEigenvalue(i);
if (c.isReal())
{
D.set(i, i, c.real);
}
}
return(D);
}
public Complex_F64 getEigenvalue(int index)
{
return(symmetric ? symmetricAlg.getEigenvalue(index) :
generalAlg.getEigenvalue(index));
}