/**
* <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);
}
/**
* @param tol Tolerance for a matrix being symmetric
*/
//public SwitchingEigenDecomposition_DDRM(int matrixSize, bool computeVectors, double tol)
//{
// symmetricAlg = DecompositionFactory_DDRM.eig(matrixSize, computeVectors, true);
// generalAlg = DecompositionFactory_DDRM.eig(matrixSize, computeVectors, false);
// this.computeVectors = computeVectors;
// this.tol = tol;
//}
public SwitchingEigenDecomposition_DDRM(EigenDecomposition_F64 <DMatrixRMaj> symmetricAlg,
EigenDecomposition_F64 <DMatrixRMaj> generalAlg, double tol)
{
this.symmetricAlg = symmetricAlg;
this.generalAlg = generalAlg;
this.tol = tol;
}
/**
*
* @param computeVectors
* @param tol Tolerance for a matrix being symmetric
*/
public SwitchingEigenDecomposition_DDRM(int matrixSize, bool computeVectors, double tol)
{
symmetricAlg = DecompositionFactory_DDRM.eig(matrixSize, computeVectors, true);
generalAlg = DecompositionFactory_DDRM.eig(matrixSize, computeVectors, false);
this.computeVectors = computeVectors;
this.tol = tol;
}
/**
* <p>
* Computes a metric which measures the the quality of an eigen value decomposition. If a
* value is returned that is close to or smaller than 1e-15 then it is within machine precision.
* </p>
* <p>
* EVD quality is defined as:<br>
* <br>
* Quality = ||A*V - V*D|| / ||A*V||.
* </p>
*
* @param orig The original matrix. Not modified.
* @param eig EVD of the original matrix. Not modified.
* @return The quality of the decomposition.
*/
public static double quality(DMatrixRMaj orig, EigenDecomposition_F64 <DMatrixRMaj> eig)
{
DMatrixRMaj A = orig;
DMatrixRMaj V = EigenOps_DDRM.createMatrixV(eig);
DMatrixRMaj D = EigenOps_DDRM.createMatrixD(eig);
// L = A*V
DMatrixRMaj L = new DMatrixRMaj(A.numRows, V.numCols);
CommonOps_DDRM.mult(A, V, L);
// R = V*D
DMatrixRMaj R = new DMatrixRMaj(V.numRows, D.numCols);
CommonOps_DDRM.mult(V, D, R);
DMatrixRMaj diff = new DMatrixRMaj(L.numRows, L.numCols);
CommonOps_DDRM.subtract(L, R, diff);
double top = NormOps_DDRM.normF(diff);
double bottom = NormOps_DDRM.normF(L);
double error = top / bottom;
return(error);
}
/**
* <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>
* 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>
* 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);
}
