## org.apache.commons.math.linear
Class EigenDecompositionImpl

    java.lang.Object
      org.apache.commons.math.linear.EigenDecompositionImpl

All Implemented Interfaces:
  EigenDecomposition

    public class EigenDecompositionImpl

extends java.lang.Object
implements EigenDecomposition

Calculates the eigen decomposition of a real symmetric 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.

As of 2.0, this class supports only symmetric matrices, and hence computes only real realEigenvalues. This implies the D matrix returned by `getD()` is always diagonal and the imaginary values returned `getImagEigenvalue(int)` and `getImagEigenvalues()` are always null.

When called with a `RealMatrix` argument, this implementation only uses the upper part of the matrix, the part below the diagonal is not accessed at all.

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
Version:
  $Revision: 912413 $ $Date: 2010-02-21 16:46:12 -0500 (Sun, 21 Feb 2010) $

Nested Class Summary
`private static class` `EigenDecompositionImpl.Solver` Specialized solver.

Field Summary
`private RealMatrix` `cachedD` Cached value of D.
`private RealMatrix` `cachedV` Cached value of V.
`private RealMatrix` `cachedVt` Cached value of Vt.
`private ArrayRealVector[]` `eigenvectors` Eigenvectors.
`private double[]` `imagEigenvalues` Imaginary part of the realEigenvalues.
`private double[]` `main` Main diagonal of the tridiagonal matrix.
`private byte` `maxIter` Maximum number of iterations accepted in the implicit QL transformation
`private double[]` `realEigenvalues` Real part of the realEigenvalues.
`private double[]` `secondary` Secondary diagonal of the tridiagonal matrix.
`private TriDiagonalTransformer` `transformer` Transformer to tridiagonal (may be null if matrix is already tridiagonal).

Constructor Summary
`EigenDecompositionImpl(double[] main, double[] secondary, double splitTolerance)` Calculates the eigen decomposition of the symmetric tridiagonal matrix.
`EigenDecompositionImpl(RealMatrix matrix, double splitTolerance)` Calculates the eigen decomposition of the given symmetric matrix.

Method Summary
`private void` `findEigenVectors(double[][] householderMatrix)` Find eigenvalues and eigenvectors (Dubrulle et al., 1971)
`RealMatrix` `getD()` Returns the block diagonal matrix D of the decomposition.
`double` `getDeterminant()` Return the determinant of the matrix
`RealVector` `getEigenvector(int i)` Returns a copy of the ith eigenvector of the original matrix.
`double` `getImagEigenvalue(int i)` Returns the imaginary part of the ith eigenvalue of the original matrix.
`double[]` `getImagEigenvalues()` Returns a copy of the imaginary parts of the eigenvalues of the original matrix.
`double` `getRealEigenvalue(int i)` Returns the real part of the ith eigenvalue of the original matrix.
`double[]` `getRealEigenvalues()` Returns a copy of the real parts of the eigenvalues of the original matrix.
`DecompositionSolver` `getSolver()` Get a solver for finding the A × X = B solution in exact linear sense.
`RealMatrix` `getV()` Returns the matrix V of the decomposition.
`RealMatrix` `getVT()` Returns the transpose of the matrix V of the decomposition.
`private boolean` `isSymmetric(RealMatrix matrix)` Check if a matrix is symmetric.
`private void` `transformToTridiagonal(RealMatrix matrix)` Transform matrix to tridiagonal.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Field Detail

### 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.

Constructor Detail

### EigenDecompositionImpl

    public EigenDecompositionImpl(RealMatrix matrix,
                                  double splitTolerance)
                           throws InvalidMatrixException

  Calculates the eigen decomposition of the given symmetric matrix.

  Parameters:
    `matrix` - The symmetric matrix to decompose.
    `splitTolerance` - dummy parameter, present for backward compatibility only.
  Throws:
    `InvalidMatrixException` - (wrapping a `ConvergenceException` if algorithm fails to converge

### EigenDecompositionImpl

    public EigenDecompositionImpl(double[] main,
                                  double[] secondary,
                                  double splitTolerance)
                           throws InvalidMatrixException

  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 triadiagonal form
    `secondary` - Secondary of the tridiagonal form
    `splitTolerance` - dummy parameter, present for backward compatibility only.
  Throws:
    `InvalidMatrixException` - (wrapping a `ConvergenceException` if algorithm fails to converge

Method Detail

### isSymmetric

    private boolean isSymmetric(RealMatrix matrix)

  Check if a matrix is symmetric.

  Parameters:
    `matrix` - matrix to check
  Returns:
    true if matrix is symmetric

### getV

    public RealMatrix getV()
                    throws InvalidMatrixException

  Returns 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.

  Specified by:
    `getV` in interface `EigenDecomposition`
  Returns:
    the V matrix
  Throws:
    `InvalidMatrixException`

### getD

    public RealMatrix getD()
                    throws InvalidMatrixException

  Returns 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} }.

  Specified by:
    `getD` in interface `EigenDecomposition`
  Returns:
    the D matrix
  Throws:
    `InvalidMatrixException`
  See Also:
    `EigenDecomposition.getRealEigenvalues()`, `EigenDecomposition.getImagEigenvalues()`

### getVT

    public RealMatrix getVT()
                     throws InvalidMatrixException

  Returns 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.

  Specified by:
    `getVT` in interface `EigenDecomposition`
  Returns:
    the transpose of the V matrix
  Throws:
    `InvalidMatrixException`

### getRealEigenvalues

    public double[] getRealEigenvalues()
                                throws InvalidMatrixException

  Returns a copy of the real parts of the eigenvalues of the original matrix.

  Specified by:
    `getRealEigenvalues` in interface `EigenDecomposition`
  Returns:
    a copy of the real parts of the eigenvalues of the original matrix
  Throws:
    `InvalidMatrixException`
  See Also:
    `EigenDecomposition.getD()`, `EigenDecomposition.getRealEigenvalue(int)`, `EigenDecomposition.getImagEigenvalues()`

### getRealEigenvalue

    public double getRealEigenvalue(int i)
                             throws InvalidMatrixException,
                                    java.lang.ArrayIndexOutOfBoundsException

  Returns the real part of the ith eigenvalue of the original matrix.

  Specified by:
    `getRealEigenvalue` in interface `EigenDecomposition`
  Parameters:
    `i` - index of the eigenvalue (counting from 0)
  Returns:
    real part of the ith eigenvalue of the original matrix
  Throws:
    `InvalidMatrixException`
    `java.lang.ArrayIndexOutOfBoundsException`
  See Also:
    `EigenDecomposition.getD()`, `EigenDecomposition.getRealEigenvalues()`, `EigenDecomposition.getImagEigenvalue(int)`

### getImagEigenvalues

    public double[] getImagEigenvalues()
                                throws InvalidMatrixException

  Returns a copy of the imaginary parts of the eigenvalues of the original matrix.

  Specified by:
    `getImagEigenvalues` in interface `EigenDecomposition`
  Returns:
    a copy of the imaginary parts of the eigenvalues of the original matrix
  Throws:
    `InvalidMatrixException`
  See Also:
    `EigenDecomposition.getD()`, `EigenDecomposition.getImagEigenvalue(int)`, `EigenDecomposition.getRealEigenvalues()`

### getImagEigenvalue

    public double getImagEigenvalue(int i)
                             throws InvalidMatrixException,
                                    java.lang.ArrayIndexOutOfBoundsException

  Returns the imaginary part of the ith eigenvalue of the original matrix.

  Specified by:
    `getImagEigenvalue` in interface `EigenDecomposition`
  Parameters:
    `i` - index of the eigenvalue (counting from 0)
  Returns:
    imaginary part of the ith eigenvalue of the original matrix
  Throws:
    `InvalidMatrixException`
    `java.lang.ArrayIndexOutOfBoundsException`
  See Also:
    `EigenDecomposition.getD()`, `EigenDecomposition.getImagEigenvalues()`, `EigenDecomposition.getRealEigenvalue(int)`

### getEigenvector

    public RealVector getEigenvector(int i)
                              throws InvalidMatrixException,
                                     java.lang.ArrayIndexOutOfBoundsException

  Returns a copy of the ith eigenvector of the original matrix.

  Specified by:
    `getEigenvector` in interface `EigenDecomposition`
  Parameters:
    `i` - index of the eigenvector (counting from 0)
  Returns:
    copy of the ith eigenvector of the original matrix
  Throws:
    `InvalidMatrixException`
    `java.lang.ArrayIndexOutOfBoundsException`
  See Also:
    `EigenDecomposition.getD()`

### getDeterminant

    public double getDeterminant()

  Return the determinant of the matrix

  Specified by:
    `getDeterminant` in interface `EigenDecomposition`
  Returns:
    determinant of the matrix

### getSolver

    public DecompositionSolver getSolver()

  Get a solver for finding the A × X = B solution in exact linear sense.

  Specified by:
    `getSolver` in interface `EigenDecomposition`
  Returns:
    a solver

### transformToTridiagonal

    private void transformToTridiagonal(RealMatrix matrix)

  Transform matrix to tridiagonal.

  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 tri-diagonal form.

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