Finding the matrix condition number using Hessenberg decomposition.

Here’s a Java program that calculates the condition number of a matrix using Hessenberg decomposition:

import Jama.Matrix;

public class MatrixConditionNumber {

    public static double calculateConditionNumber(Matrix matrix) {

        Matrix hessenberg = matrix.hessenberg();

        Matrix[] hessenbergDecomposition = hessenberg.qr();

        Matrix qMatrix = hessenbergDecomposition[0];

        Matrix rMatrix = hessenbergDecomposition[1];

        Matrix inverseQ = qMatrix.inverse();

        Matrix inverseR = rMatrix.inverse();

        Matrix inverseHessenberg = inverseR.times(inverseQ.transpose());

        double largestEigenvalue = inverseHessenberg.eig().getRealEigenvalues()[0];

        double smallestEigenvalue = inverseHessenberg.inverse().eig().getRealEigenvalues()[0];

        double conditionNumber = Math.abs(largestEigenvalue / smallestEigenvalue);

        return conditionNumber;

    }
    public static void main(String[] args) {

        // Example usage

        double[][] matrixData = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};

        Matrix matrix = new Matrix(matrixData);

        double conditionNumber = calculateConditionNumber(matrix);

        System.out.println("Condition Number: " + conditionNumber);

    }

}

In this program, we use the Jama library, which provides a set of linear algebra classes and methods. You need to have the Jama library added to your project’s dependencies for this program to work.

The `calculateConditionNumber` method takes a matrix as input and performs the Hessenberg decomposition on it. It then calculates the inverse of the Hessenberg matrix and finds its eigenvalues. The largest eigenvalue corresponds to the largest singular value of the original matrix, while the smallest eigenvalue corresponds to the smallest singular value. The condition number is calculated as the absolute value of the ratio between the largest and smallest eigenvalues.

In the `main` method, we create an example matrix and calculate its condition number using the `calculateConditionNumber` method. Finally, we print the result to the console.

Remember to include the Jama library in your project to run this program successfully.