Class EigenDecomposition

java.lang.Object
org.apache.commons.math3.linear.EigenDecomposition

public class EigenDecomposition extends 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:
  • Field Details

    • EPSILON

      private static final double EPSILON
      Internally used epsilon criteria.
      See Also:
    • 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.
    • eigenvectors

      private ArrayRealVector[] eigenvectors
      Eigenvectors.
    • 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 Details

    • 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 Details

    • 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:
    • 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:
    • 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:
    • 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:
    • 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:
    • 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:
    • 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