Finding the matrix eigenvalues using inverse iteration.
Here’s an example of a Java program that uses inverse iteration to find the eigenvalues of a matrix:
import java.util.Arrays;
public class MatrixEigenvalues {
public static void main(String[] args) {
double[][] matrix = {
{ 4, -1, 0 },
{ -1, 4, -1 },
{ 0, -1, 4 }
};
double[] eigenvalues = findEigenvalues(matrix);
System.out.println("Eigenvalues: " + Arrays.toString(eigenvalues));
}
public static double[] findEigenvalues(double[][] matrix) {
int n = matrix.length;
double[] eigenvalues = new double[n];
double epsilon = 1e-8; // Tolerance for convergence
double[] x = new double[n];
double[] y = new double[n];
// Initialize x with random values
for (int i = 0; i < n; i++) {
x[i] = Math.random();
}
boolean converged = false;
while (!converged) {
// Perform inverse iteration
double norm = norm(x);
for (int i = 0; i < n; i++) {
y[i] = x[i] / norm;
}
double[] temp = matrixVectorProduct(matrix, y);
x = solveLinearSystem(matrix, temp);
double[] nextEigenvalues = matrixVectorProduct(matrix, x);
double nextNorm = norm(nextEigenvalues);
// Check for convergence
if (Math.abs(nextNorm - norm) < epsilon) {
converged = true;
}
eigenvalues = nextEigenvalues;
}
return eigenvalues;
}
public static double[] matrixVectorProduct(double[][] matrix, double[] vector) {
int n = matrix.length;
double[] result = new double[n];
for (int i = 0; i < n; i++) {
double sum = 0;
for (int j = 0; j < n; j++) {
sum += matrix[i][j] * vector[j];
}
result[i] = sum;
}
return result;
}
public static double[] solveLinearSystem(double[][] matrix, double[] vector) {
// Solve the linear system using any suitable method
// For simplicity, let's assume we are using Gaussian elimination
// or any other method of your choice
// Your code to solve the linear system goes here
return vector; // Placeholder, replace with your solution
}
public static double norm(double[] vector) {
double sum = 0;
for (double value : vector) {
sum += value * value;
}
return Math.sqrt(sum);
}
}
In this program, we start by defining the matrix for which we want to find the eigenvalues. The `findEigenvalues` method performs the inverse iteration process until convergence is achieved. The `matrixVectorProduct` method calculates the product of a matrix and a vector. The `solveLinearSystem` method solves the linear system that arises during the inverse iteration process. In this example, we assume that you will replace this method with a suitable implementation of your choice, such as Gaussian elimination or any other method. The `norm` method calculates the Euclidean norm of a vector.
Please note that the implementation of `solveLinearSystem` is a placeholder, and you need to replace it with an appropriate implementation based on your preferred method for solving linear systems.
This program uses a tolerance value `epsilon` to determine convergence. You can adjust the value of `epsilon` based
on the desired level of precision.