Examples of org.apache.commons.math3.linear.EigenDecomposition

org.apache.commons.math3.linear.Array2DRowRealMatrix
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)

                          double abs,
                          ConvergenceChecker<UnivariatePointValuePair> checker) {
        super(checker);


        if (rel < MIN_RELATIVE_TOLERANCE) {
            throw new NumberIsTooSmallException(rel, MIN_RELATIVE_TOLERANCE, true);
        }
        if (abs <= 0) {
            throw new NotStrictlyPositiveException(abs);
        }

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    @Override
    public double inverseCumulativeProbability(double p) throws OutOfRangeException {
        double ret;


        if (p < 0.0 || p > 1.0) {
            throw new OutOfRangeException(p, 0.0, 1.0);
        } else if (p == 1.0) {
            ret = Double.POSITIVE_INFINITY;
        } else {
            ret = -mean * FastMath.log(1.0 - p);
        }

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        if (lo >= hi) {
            throw new NumberIsTooLargeException(lo, hi, false);
        }
        if (init < lo ||
            init > hi) {
            throw new OutOfRangeException(init, lo, hi);
        }


        lower = lo;
        upper = hi;
        start = init;

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    @Override
    public double inverseCumulativeProbability(double p)
        throws OutOfRangeException {
        if (p < 0 || p > 1) {
            throw new OutOfRangeException(p, 0, 1);
        }
        if (p == 0) {
            return a;
        }
        if (p == 1) {

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        /**
         * {@inheritDoc}
         * @throws TooManyEvaluationsException.
         */
        public void trigger(int max) {
            throw new TooManyEvaluationsException(max);
        }

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        /**
         * {@inheritDoc}
         * @throws TooManyIterationsException.
         */
        public void trigger(int max) {
            throw new TooManyIterationsException(max);
        }

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                                              double threshold) {
        super(wrong, threshold, false);
        this.index = index;
        this.threshold = threshold;


        final ExceptionContext context = getContext();
        context.addMessage(LocalizedFormats.NOT_POSITIVE_DEFINITE_MATRIX);
        context.addMessage(LocalizedFormats.ARRAY_ELEMENT, wrong, index);
    }

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            // build the P matrix elements from Taylor series formulas
            final BigFraction[] pI = pData[i];
            final int factor = -(i + 1);
            int aj = factor;
            for (int j = 0; j < pI.length; ++j) {
                pI[j] = new BigFraction(aj * (j + 2));
                aj *= factor;
            }
        }


        return new Array2DRowFieldMatrix<BigFraction>(pData, false);

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      List<SiteWithPolynomial> nearestSites =
          nearestSiteMap.get(site);
      
      RealVector vector = new ArrayRealVector(SITES_FOR_APPROX);
      RealMatrix matrix = new Array2DRowRealMatrix(
          SITES_FOR_APPROX, DefaultPolynomial.NUM_COEFFS);
      
      for (int row = 0; row < SITES_FOR_APPROX; row++) {
        SiteWithPolynomial nearSite = nearestSites.get(row);
        DefaultPolynomial.populateMatrix(matrix, row, nearSite.pos.x, nearSite.pos.z);

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        }


        // solve the rectangular system in the least square sense
        // to get the best estimate of the Nordsieck vector [s2 ... sk]
        QRDecomposition decomposition;
        decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false));
        RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false));
        return new Array2DRowRealMatrix(x.getData(), false);
    }

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