import org.apache.commons.math3.linear.NonPositiveDefiniteMatrixException; //导入依赖的package包/类
@Override
public SlopeCoefficients estimateCoefficients(final DerivationEquation eq)
    throws EstimationException {
  final double[][] sourceTriangleMatrix = eq.getCovarianceLowerTriangularMatrix();
  // Copy matrix and enhance it to a full matrix as expected by CholeskyDecomposition
  // FIXME: Avoid copy job to speed-up the solving process e.g. by extending the CholeskyDecomposition constructor
  final int length = sourceTriangleMatrix.length;
  final double[][] matrix = new double[length][];
  for (int i = 0; i < length; i++) {
    matrix[i] = new double[length];
    final double[] s = sourceTriangleMatrix[i];
    final double[] t = matrix[i];
    for (int j = 0; j <= i; j++) {
      t[j] = s[j];
    }
    for (int j = i + 1; j < length; j++) {
      t[j] = sourceTriangleMatrix[j][i];
    }
  }
  final RealMatrix coefficients =
      new Array2DRowRealMatrix(matrix, false);
  try {
    final DecompositionSolver solver = new CholeskyDecomposition(coefficients).getSolver();
    final RealVector constants = new ArrayRealVector(eq.getConstraints(), true);
    final RealVector solution = solver.solve(constants);
    return new DefaultSlopeCoefficients(solution.toArray());
  } catch (final NonPositiveDefiniteMatrixException e) {
    throw new EstimationException("Matrix inversion error due to data is linearly dependent", e);
  }
}
 

import org.apache.commons.math3.linear.NonPositiveDefiniteMatrixException; //导入依赖的package包/类
@Override
protected RealVector solve(final RealMatrix jacobian,
                           final RealVector residuals) {
    try {
        final Pair<RealMatrix, RealVector> normalEquation =
                computeNormalMatrix(jacobian, residuals);
        final RealMatrix normal = normalEquation.getFirst();
        final RealVector jTr = normalEquation.getSecond();
        return new CholeskyDecomposition(
                normal, SINGULARITY_THRESHOLD, SINGULARITY_THRESHOLD)
                .getSolver()
                .solve(jTr);
    } catch (NonPositiveDefiniteMatrixException e) {
        throw new ConvergenceException(LocalizedFormats.UNABLE_TO_SOLVE_SINGULAR_PROBLEM, e);
    }
}
 

import org.apache.commons.math3.linear.NonPositiveDefiniteMatrixException; //导入依赖的package包/类
/**
 * Creates a multivariate normal distribution with the given mean vector and
 * covariance matrix.
 * <br/>
 * The number of dimensions is equal to the length of the mean vector
 * and to the number of rows and columns of the covariance matrix.
 * It is frequently written as "p" in formulae.
 *
 * @param rng Random Number Generator.
 * @param means Vector of means.
 * @param covariances Covariance matrix.
 * @throws DimensionMismatchException if the arrays length are
 * inconsistent.
 * @throws SingularMatrixException if the eigenvalue decomposition cannot
 * be performed on the provided covariance matrix.
 * @throws NonPositiveDefiniteMatrixException if any of the eigenvalues is
 * negative.
 */
public MultivariateNormalDistribution(RandomGenerator rng,
                                      final double[] means,
                                      final double[][] covariances)
        throws SingularMatrixException,
               DimensionMismatchException,
               NonPositiveDefiniteMatrixException {
    super(rng, means.length);

    final int dim = means.length;

    if (covariances.length != dim) {
        throw new DimensionMismatchException(covariances.length, dim);
    }

    for (int i = 0; i < dim; i++) {
        if (dim != covariances[i].length) {
            throw new DimensionMismatchException(covariances[i].length, dim);
        }
    }

    this.means = MathArrays.copyOf(means);

    covarianceMatrix = new Array2DRowRealMatrix(covariances);

    // Covariance matrix eigen decomposition.
    final EigenDecomposition covMatDec = new EigenDecomposition(covarianceMatrix);

    // Compute and store the inverse.
    covarianceMatrixInverse = covMatDec.getSolver().getInverse();
    // Compute and store the determinant.
    covarianceMatrixDeterminant = covMatDec.getDeterminant();

    // Eigenvalues of the covariance matrix.
    final double[] covMatEigenvalues = covMatDec.getRealEigenvalues();

    for (int i = 0; i < covMatEigenvalues.length; i++) {
        if (covMatEigenvalues[i] < 0) {
            throw new NonPositiveDefiniteMatrixException(covMatEigenvalues[i], i, 0);
        }
    }

    // Matrix where each column is an eigenvector of the covariance matrix.
    final Array2DRowRealMatrix covMatEigenvectors = new Array2DRowRealMatrix(dim, dim);
    for (int v = 0; v < dim; v++) {
        final double[] evec = covMatDec.getEigenvector(v).toArray();
        covMatEigenvectors.setColumn(v, evec);
    }

    final RealMatrix tmpMatrix = covMatEigenvectors.transpose();

    // Scale each eigenvector by the square root of its eigenvalue.
    for (int row = 0; row < dim; row++) {
        final double factor = FastMath.sqrt(covMatEigenvalues[row]);
        for (int col = 0; col < dim; col++) {
            tmpMatrix.multiplyEntry(row, col, factor);
        }
    }

    samplingMatrix = covMatEigenvectors.multiply(tmpMatrix);
}