Finding the matrix eigenvalues using Jacobi algorithm

Here’s a Java implementation of the Jacobi algorithm for finding the eigenvalues of a matrix:

import java.util.Arrays;

public class JacobiAlgorithm {

  // Function to find the maximum off-diagonal element

  private static double findMaxOffDiagonal(double[][] matrix) {

    int n = matrix.length;

    double maxVal = 0.0;

    int row = 0, col = 0;

    for (int i = 0; i < n; i++) {

      for (int j = i + 1; j < n; j++) {

        if (Math.abs(matrix[i][j]) > maxVal) {

          maxVal = Math.abs(matrix[i][j]);

          row = i;

          col = j;

        }

      }

    }

    return matrix[row][col];

  }

  // Function to perform the Jacobi rotation

  private static void performJacobiRotation(double[][] matrix, double[][] eigenVectors, int p, int q) {

    int n = matrix.length;

    double tau = (matrix[q][q] - matrix[p][p]) / (2 * matrix[p][q]);

    double t = Math.sqrt(1 + tau * tau);

    double sinTheta = 1 / t;

    double cosTheta = tau / t;

    for (int i = 0; i < n; i++) {

      double oldPi = matrix[p][i];

      double oldQi = matrix[q][i];     

      matrix[p][i] = oldPi * cosTheta + oldQi * sinTheta;

      matrix[q][i] = -oldPi * sinTheta + oldQi * cosTheta;

      double oldVi = eigenVectors[i][p];

      double oldWi = eigenVectors[i][q];

      eigenVectors[i][p] = oldVi * cosTheta + oldWi * sinTheta;

      eigenVectors[i][q] = -oldVi * sinTheta + oldWi * cosTheta;

    }

  }

  // Function to find the eigenvalues and eigenvectors of the matrix

  public static void findEigenvalues(double[][] matrix) {

    int n = matrix.length;

    double[][] eigenVectors = new double[n][n];

    // Initialize eigenVectors as an identity matrix

    for (int i = 0; i < n; i++) {

      eigenVectors[i][i] = 1.0;

    }  

    // Set the maximum number of iterations

    int maxIterations = 100;

    int iterations = 0;

    while (findMaxOffDiagonal(matrix) > 1e-8 && iterations < maxIterations) {

      int p = 0, q = 1;

      double maxOffDiagonal = findMaxOffDiagonal(matrix);    

      // Find indices p and q corresponding to the maximum off-diagonal element

      for (int i = 0; i < n; i++) {

        for (int j = i + 1; j < n; j++) {

          if (Math.abs(matrix[i][j]) == maxOffDiagonal) {

            p = i;

            q = j;

            break;

          }

        }

      }
      performJacobiRotation(matrix, eigenVectors, p, q);

      iterations++;

    }
    // Print the eigenvalues

    System.out.println("Eigenvalues:");

    for (int i = 0; i < n; i++) {

      System.out.println(matrix[i][i]);

    }
    // Print the eigenvectors

    System.out.println("Eigenvectors:");

    for (int i = 0; i

 < n; i++) {

      System.out.println(Arrays.toString(eigenVectors[i]));

    }

  }

  public static void main(String[] args) {

    double[][] matrix = {

      {4.0, -3.0, 1.0},

      {-3.0, 6.0, -4.0},

      {1.0, -4.0, 7.0}

    };
    findEigenvalues(matrix);

  }

}

This code takes a matrix as input and uses the Jacobi algorithm to find the eigenvalues and eigenvectors of the matrix. It iteratively performs Jacobi rotations until the maximum off-diagonal element becomes smaller than a threshold. The resulting eigenvalues and eigenvectors are then printed.

In the `main` method, a sample matrix is provided for demonstration purposes. You can modify the matrix with your desired values or pass it dynamically as per your requirements.