Package org.apache.commons.math3.linear

Class EigenDecomposition

  • public class EigenDecomposition
extends java.lang.Object
    Calculates the eigen decomposition of a real matrix.

    The eigen decomposition of matrix A is a set of two matrices: V and D such that A = V × D × VT. A, V and D are all m × m matrices.

    This class is similar in spirit to the EigenvalueDecomposition class from the JAMA library, with the following changes:

    As of 3.1, this class supports general real matrices (both symmetric and non-symmetric):

    If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal, i.e. A = V.multiply(D.multiply(V.transpose())) and V.multiply(V.transpose()) equals the identity matrix.

    If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks: 

        [lambda, mu    ]
    [   -mu, lambda]
    The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.multiply(V) equals V.multiply(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition of V.

    This implementation is based on the paper by A. Drubrulle, R.S. Martin and J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971) Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag, New-York

    Since:
    2.0 (changed to concrete class in 3.0)
    See Also:
    MathWorld, Wikipedia

    • Field Detail

      • EPSILON

        private static final double EPSILON
        Internally used epsilon criteria.
        See Also:
        Constant Field Values

      • maxIter

        private byte maxIter
        Maximum number of iterations accepted in the implicit QL transformation

      • main

        private double[] main
        Main diagonal of the tridiagonal matrix.

      • secondary

        private double[] secondary
        Secondary diagonal of the tridiagonal matrix.

      • transformer

        private TriDiagonalTransformer transformer
        Transformer to tridiagonal (may be null if matrix is already tridiagonal).

      • realEigenvalues

        private double[] realEigenvalues
        Real part of the realEigenvalues.

      • imagEigenvalues

        private double[] imagEigenvalues
        Imaginary part of the realEigenvalues.

      • cachedV

        private RealMatrix cachedV
        Cached value of V.

      • cachedD

        private RealMatrix cachedD
        Cached value of D.

      • cachedVt

        private RealMatrix cachedVt
        Cached value of Vt.

      • isSymmetric

        private final boolean isSymmetric
        Whether the matrix is symmetric.

    • Constructor Detail

      • EigenDecomposition

        public EigenDecomposition​(RealMatrix matrix)
                   throws MathArithmeticException
        Calculates the eigen decomposition of the given real matrix.

        Supports decomposition of a general matrix since 3.1.

        Parameters:
        matrix - Matrix to decompose.
        Throws:
        MaxCountExceededException - if the algorithm fails to converge.
        MathArithmeticException - if the decomposition of a general matrix results in a matrix with zero norm
        Since:
        3.1

      • EigenDecomposition

        @Deprecated
public EigenDecomposition​(RealMatrix matrix,
                          double splitTolerance)
                   throws MathArithmeticException
        Deprecated.
        in 3.1 (to be removed in 4.0) due to unused parameter
        Calculates the eigen decomposition of the given real matrix.
        Parameters:
        matrix - Matrix to decompose.
        splitTolerance - Dummy parameter (present for backward compatibility only).
        Throws:
        MathArithmeticException - if the decomposition of a general matrix results in a matrix with zero norm
        MaxCountExceededException - if the algorithm fails to converge.

      • EigenDecomposition

        public EigenDecomposition​(double[] main,
                          double[] secondary)
        Calculates the eigen decomposition of the symmetric tridiagonal matrix. The Householder matrix is assumed to be the identity matrix.
        Parameters:
        main - Main diagonal of the symmetric tridiagonal form.
        secondary - Secondary of the tridiagonal form.
        Throws:
        MaxCountExceededException - if the algorithm fails to converge.
        Since:
        3.1

      • EigenDecomposition

        @Deprecated
public EigenDecomposition​(double[] main,
                          double[] secondary,
                          double splitTolerance)
        Deprecated.
        in 3.1 (to be removed in 4.0) due to unused parameter
        Calculates the eigen decomposition of the symmetric tridiagonal matrix. The Householder matrix is assumed to be the identity matrix.
        Parameters:
        main - Main diagonal of the symmetric tridiagonal form.
        secondary - Secondary of the tridiagonal form.
        splitTolerance - Dummy parameter (present for backward compatibility only).
        Throws:
        MaxCountExceededException - if the algorithm fails to converge.

    • Method Detail

      • getV

        public RealMatrix getV()
        Gets the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).
        Returns:
        the V matrix.

      • getD

        public RealMatrix getD()
        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} }.
        Returns:
        the D matrix.
        See Also:
        getRealEigenvalues(), getImagEigenvalues()

      • getVT

        public RealMatrix getVT()
        Gets the transpose of the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).
        Returns:
        the transpose of the V matrix.

      • hasComplexEigenvalues

        public boolean hasComplexEigenvalues()
        Returns whether the calculated eigen values are complex or real.

        The method performs a zero check for each element of the getImagEigenvalues() array and returns true if any element is not equal to zero.

        Returns:
        true if the eigen values are complex, false otherwise
        Since:
        3.1

      • getRealEigenvalues

        public double[] getRealEigenvalues()
        Gets a copy of the real parts of the eigenvalues of the original matrix.
        Returns:
        a copy of the real parts of the eigenvalues of the original matrix.
        See Also:
        getD(), getRealEigenvalue(int), getImagEigenvalues()

      • getRealEigenvalue

        public double getRealEigenvalue​(int i)
        Returns the real part of the ith eigenvalue of the original matrix.
        Parameters:
        i - index of the eigenvalue (counting from 0)
        Returns:
        real part of the ith eigenvalue of the original matrix.
        See Also:
        getD(), getRealEigenvalues(), getImagEigenvalue(int)

      • getImagEigenvalues

        public double[] getImagEigenvalues()
        Gets a copy of the imaginary parts of the eigenvalues of the original matrix.
        Returns:
        a copy of the imaginary parts of the eigenvalues of the original matrix.
        See Also:
        getD(), getImagEigenvalue(int), getRealEigenvalues()

      • getImagEigenvalue

        public double getImagEigenvalue​(int i)
        Gets the imaginary part of the ith eigenvalue of the original matrix.
        Parameters:
        i - Index of the eigenvalue (counting from 0).
        Returns:
        the imaginary part of the ith eigenvalue of the original matrix.
        See Also:
        getD(), getImagEigenvalues(), getRealEigenvalue(int)

      • getEigenvector

        public RealVector getEigenvector​(int i)
        Gets a copy of the ith eigenvector of the original matrix.
        Parameters:
        i - Index of the eigenvector (counting from 0).
        Returns:
        a copy of the ith eigenvector of the original matrix.
        See Also:
        getD()

      • getDeterminant

        public double getDeterminant()
        Computes the determinant of the matrix.
        Returns:
        the determinant of the matrix.

      • getSquareRoot

        public RealMatrix getSquareRoot()
        Computes the square-root of the matrix. This implementation assumes that the matrix is symmetric and positive definite.
        Returns:
        the square-root of the matrix.
        Throws:
        MathUnsupportedOperationException - if the matrix is not symmetric or not positive definite.
        Since:
        3.1

      • getSolver

        public DecompositionSolver getSolver()
        Gets a solver for finding the A × X = B solution in exact linear sense.

        Since 3.1, eigen decomposition of a general matrix is supported, but the DecompositionSolver only supports real eigenvalues.

        Returns:
        a solver
        Throws:
        MathUnsupportedOperationException - if the decomposition resulted in complex eigenvalues

      • transformToTridiagonal

        private void transformToTridiagonal​(RealMatrix matrix)
        Transforms the matrix to tridiagonal form.
        Parameters:
        matrix - Matrix to transform.

      • findEigenVectors

        private void findEigenVectors​(double[][] householderMatrix)
        Find eigenvalues and eigenvectors (Dubrulle et al., 1971)
        Parameters:
        householderMatrix - Householder matrix of the transformation to tridiagonal form.

      • transformToSchur

        private SchurTransformer transformToSchur​(RealMatrix matrix)
        Transforms the matrix to Schur form and calculates the eigenvalues.
        Parameters:
        matrix - Matrix to transform.
        Returns:
        the Shur transform for this matrix

      • cdiv

        private Complex cdiv​(double xr,
                     double xi,
                     double yr,
                     double yi)
        Performs a division of two complex numbers.
        Parameters:
        xr - real part of the first number
        xi - imaginary part of the first number
        yr - real part of the second number
        yi - imaginary part of the second number
        Returns:
        result of the complex division

      • findEigenVectorsFromSchur

        private void findEigenVectorsFromSchur​(SchurTransformer schur)
                                throws MathArithmeticException
        Find eigenvectors from a matrix transformed to Schur form.
        Parameters:
        schur - the schur transformation of the matrix
        Throws:
        MathArithmeticException - if the Schur form has a norm of zero