Finding the matrix eigenvectors using power iteration
Here’s an example of a Java program that uses power iteration to find the eigenvectors of a matrix:
import java.util.Arrays;
public class PowerIteration {
public static void main(String[] args) {
double[][] matrix = {{2, 1}, {1, 3}}; // Example matrix
double[] eigenvector = findEigenvector(matrix);
System.out.println("Eigenvector: " + Arrays.toString(eigenvector));
}
public static double[] findEigenvector(double[][] matrix) {
int n = matrix.length;
// Initialize a random vector
double[] vector = new double[n];
for (int i = 0; i < n; i++) {
vector[i] = Math.random();
}
// Perform power iteration
double epsilon = 1e-8; // Tolerance for convergence
double error = Double.MAX_VALUE;
int maxIterations = 1000;
int iterations = 0;
while (error > epsilon && iterations < maxIterations) {
double[] newVector = multiplyMatrixVector(matrix, vector);
double eigenvalue = calculateEigenvalue(vector, newVector);
// Normalize the vector
double norm = calculateNorm(newVector);
for (int i = 0; i < n; i++) {
vector[i] = newVector[i] / norm;
}
// Calculate the error
error = Math.abs(eigenvalue - calculateEigenvalue(vector, newVector));
iterations++;
}
return vector;
}
public static double[] multiplyMatrixVector(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 calculateEigenvalue(double[] oldVector, double[] newVector) {
int n = oldVector.length;
double eigenvalue = 0;
for (int i = 0; i < n; i++) {
eigenvalue += newVector[i] / oldVector[i];
}
return eigenvalue / n;
}
public static double calculateNorm(double[] vector) {
int n = vector.length;
double sum = 0;
for (int i = 0; i < n; i++) {
sum += vector[i] * vector[i];
}
return Math.sqrt(sum);
}
}
In this program, the `findEigenvector` method implements the power iteration algorithm to find the eigenvector corresponding to the largest eigenvalue of the given matrix. The `multiplyMatrixVector` method multiplies a matrix by a vector, the `calculateEigenvalue` method calculates an estimate of the eigenvalue using the old and new vectors, and the `calculateNorm` method calculates the Euclidean norm of a vector.
In the `main` method, we define an example matrix and call the `findEigenvector` method to obtain the eigenvector. Finally, the program prints the resulting eigenvector.
Note that this is a simplified implementation and may not work correctly for all matrices or handle edge cases. It is intended to serve as a starting point for understanding and implementing the power iteration algorithm in Java.