Examples of EigenDecomposition

org.apache.commons.math.linear.EigenDecomposition
nist.gov/javanumerics/jama/">JAMA library, with the following changes:
- a {@link #getVT() getVt} method has been added,
- two {@link #getRealEigenvalue(int) getRealEigenvalue} and {@link #getImagEigenvalue(int) getImagEigenvalue} methods to pick up a single eigenvalue have been added,
- a {@link #getEigenvector(int) getEigenvector} method to pick up a singleeigenvector has been added,
- a {@link #getDeterminant() getDeterminant} method has been added.
- a {@link #getSolver() getSolver} method has been added.
@see MathWorld @see Wikipedia @version $Revision: 997726 $ $Date: 2010-09-16 14:39:51 +0200 (jeu. 16 sept. 2010) $ @since 2.0
org.apache.commons.math3.linear.EigenDecomposition
nist.gov/javanumerics/jama/">JAMA library, with the following changes:
- a {@link #getVT() getVt} method has been added,
- two {@link #getRealEigenvalue(int) getRealEigenvalue} and {@link #getImagEigenvalue(int) getImagEigenvalue} methods to pick up a single eigenvalue have been added,
- a {@link #getEigenvector(int) getEigenvector} method to pick up a singleeigenvector has been added,
- a {@link #getDeterminant() getDeterminant} method has been added.
- a {@link #getSolver() getSolver} method has been added.
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
@see MathWorld @see Wikipedia @since 2.0 (changed to concrete class in 3.0)
org.apache.mahout.math.solver.EigenDecomposition
Eigenvalues and eigenvectors of a real matrix.
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.times(D.times(V.transpose())) and V.times(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.times(V) equals V.times(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon V.cond().
org.ejml.factory.EigenDecomposition

This is a generic interface for computing the eigenvalues and eigenvectors of a matrix. Eigenvalues and eigenvectors have the following property:

A*v=λ*v

where A is a square matrix and v is an eigenvector associated with the eigenvalue λ.

In general, both eigenvalues and eigenvectors can be complex numbers. For symmetric matrices the eigenvalues and eigenvectors are always real numbers. EJML does not support complex matrices but it does have minimal support for complex numbers. As a result complex eigenvalues are found, but only the real eigenvectors are computed.

To create a new instance of {@link EigenDecomposition} use {@link DecompositionFactory}. If the matrix is known to be symmetric be sure to use the symmetric decomposition, which is much faster and more accurate than the general purpose one.
@author Peter Abeles

Examples of org.ejml.factory.EigenDecomposition

     * Sees if it correctly computed the eigenvalues.  Does not check eigenvectors.
     */
    public void checkKnownReal_JustValue() {
        DenseMatrix64F A = new DenseMatrix64F(3,3, true, 0.907265, 0.832472, 0.255310, 0.667810, 0.871323, 0.612657, 0.025059, 0.126475, 0.427002);


        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,A));


        testForEigenvalue(alg,A,1.686542,0,1);
        testForEigenvalue(alg,A,0.079014,0,1);

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Examples of org.ejml.factory.EigenDecomposition

    public void checkKnownSymmetric_JustValue() {
        DenseMatrix64F A = new DenseMatrix64F(3,3, true,
                0.98139,   0.78650,   0.78564,
                0.78650,   1.03207,   0.29794,
                0.78564,   0.29794,   0.91926);
        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,A));


        testForEigenvalue(alg,A,0.00426,0,1);
        testForEigenvalue(alg,A,0.67856,0,1);

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Examples of org.ejml.factory.EigenDecomposition

     * are real and some are complex.
     */
    public void checkKnownComplex() {
        DenseMatrix64F A = new DenseMatrix64F(3,3, true, -0.418284, 0.279875, 0.452912, -0.093748, -0.045179, 0.310949, 0.250513, -0.304077, -0.031414);


        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,A));
        performStandardTests(alg,A,-1);


        testForEigenpair(alg,-0.39996,0,0.87010,0.43425,-0.23314);

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Examples of org.ejml.factory.EigenDecomposition

    public void checkRandomSymmetric() {
        for( int N = 1; N <= 15; N++ ) {
            for( int i = 0; i < 20; i++ ) {
                DenseMatrix64F A = RandomMatrices.createSymmetric(N,-1,1,rand);


                EigenDecomposition alg = createDecomposition();


                assertTrue(safeDecomposition(alg,A));


                performStandardTests(alg,A,N);
            }

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Examples of org.ejml.factory.EigenDecomposition

     * check for.  If it fails that check it will either loop forever or exit before converging.
     */
    public void checkExceptional() {
        DenseMatrix64F A = new DenseMatrix64F(5,5, true, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0);


        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,A));


        performStandardTests(alg,A,1);
    }

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Examples of org.ejml.factory.EigenDecomposition

    }


    public void checkIdentity() {
        DenseMatrix64F I = CommonOps.identity(4);


        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,I));


        performStandardTests(alg,I,4);

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Examples of org.ejml.factory.EigenDecomposition

    }


    public void checkAllZeros() {
        DenseMatrix64F A = new DenseMatrix64F(5,5);


        EigenDecomposition alg = createDecomposition();


        assertTrue(safeDecomposition(alg,A));


        performStandardTests(alg,A,5);
        testEigenvalues(alg,0);

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Examples of org.ejml.factory.EigenDecomposition

        performStandardTests(alg,A,5);
        testEigenvalues(alg,0);
    }


    public void checkWithSomeRepeatedValuesSymm() {
        EigenDecomposition alg = createDecomposition();


        checkSymmetricMatrix(alg,2,-3,-3,-3);
        checkSymmetricMatrix(alg,2,-3,2,2);
        checkSymmetricMatrix(alg,1,1,1,2);
    }

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Examples of org.ejml.factory.EigenDecomposition

        checkSymmetricMatrix(alg,1,1,1,2);
    }


    public void checkWithSingularSymm() {


        EigenDecomposition alg = createDecomposition();


        checkSymmetricMatrix(alg,1,0,1,2);
    }

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Examples of org.ejml.factory.EigenDecomposition

    }


    public void checkSmallValue( boolean symmetric) {


//        System.out.println("Symmetric = "+symmetric);
        EigenDecomposition alg = createDecomposition();


        for( int i = 0; i < 20; i++ ) {
            DenseMatrix64F A = symmetric ?
                    RandomMatrices.createSymmetric(4,-1,1,rand) :
                    RandomMatrices.createRandom(4,4,-1,1,rand);

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