Finding the Hessenberg decomposition of a matrix.
Here’s an example Java code for finding the Hessenberg decomposition of a matrix:
import java.util.Arrays;
public class HessenbergDecomposition {
public static void main(String[] args) {
double[][] matrix = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
double[][] hessenbergMatrix = computeHessenberg(matrix);
System.out.println("Original Matrix:");
printMatrix(matrix);
System.out.println("\nHessenberg Matrix:");
printMatrix(hessenbergMatrix);
}
public static double[][] computeHessenberg(double[][] matrix) {
int n = matrix.length;
double[][] hessenberg = Arrays.copyOf(matrix, n);
for (int k = 1; k < n - 1; k++) {
double[] column = new double[n - k];
for (int i = k; i < n; i++) {
column[i - k] = hessenberg[i][k - 1];
}
double norm = getNorm(column);
double[] u = new double[n - k];
u[0] = column[0] + Math.signum(column[0]) * norm;
double uNorm = getNorm(u);
if (uNorm != 0) {
for (int i = 0; i < n - k; i++) {
u[i] /= uNorm;
}
for (int i = k; i < n; i++) {
double[] v = new double[n - k];
for (int j = 0; j < n - k; j++) {
v[j] = u[j] * hessenberg[i][k - 1];
}
for (int j = 0; j < n - k; j++) {
hessenberg[i][k - 1] -= 2 * v[j];
}
for (int j = k; j < n; j++) {
double[] w = new double[n - k];
for (int l = 0; l < n - k; l++) {
w[l] = u[l] * hessenberg[i][j];
}
for (int l = 0; l < n - k; l++) {
hessenberg[i][j] -= 2 * w[l];
}
}
}
for (int i = 0; i < n; i++) {
double[] z = new double[n - k];
for (int j = 0; j < n - k; j++) {
z[j] = u[j] * hessenberg[i][k - 1];
}
for (int j = 0; j < n - k; j++) {
hessenberg[i][k - 1] -= 2 * z[j];
}
}
}
}
return hessenberg;
}
public static double getNorm(double[] vector) {
double norm = 0;
for (double element : vector) {
norm += element * element;
}
return Math.sqrt(norm);
}
public static void printMatrix(double[][] matrix) {
for (double[] row : matrix) {
for (double element : row) {
System.out.print(element + " ");
}
System.out.println();
}
}
}
import java.util.Arrays;
public class HessenbergDecomposition {
public static void main(String[] args) {
double[][] matrix = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
double[][] hessenbergMatrix = computeHessenberg(matrix);
System.out.println("Original Matrix:");
printMatrix(matrix);
System.out.println("\nHessenberg Matrix:");
printMatrix(hessenbergMatrix);
}
public static double[][] computeHessenberg(double[][] matrix) {
int n = matrix.length;
double[][] hessenberg = Arrays.copyOf(matrix, n);
for (int k = 1; k < n - 1; k++) {
double[] column = new double[n - k];
for (int i = k; i < n; i++) {
column[i - k] = hessenberg[i][k - 1];
}
double norm = getNorm(column);
double[] u = new double[n - k];
u[0] = column[0] + Math.signum(column[0]) * norm;
double uNorm = getNorm(u);
if (uNorm != 0) {
for (int i = 0; i < n - k; i++) {
u[i] /= uNorm;
}
for (int i = k; i < n; i++) {
double[] v = new double[n - k];
for (int j = 0; j < n - k; j++) {
v[j] = u[j] * hessenberg[i][k - 1];
}
for (int j = 0; j < n - k; j++) {
hessenberg[i][k - 1] -= 2 * v[j];
}
for (int j = k; j < n; j++) {
double[] w = new double[n - k];
for (int l = 0; l < n - k; l++) {
w[l] = u[l] * hessenberg[i][j];
}
for (int l = 0; l < n - k; l++) {
hessenberg[i][j] -= 2 * w[l];
}
}
}
for (int i = 0; i < n; i++) {
double[] z = new double[n - k];
for (int j = 0; j < n - k; j++) {
z[j] = u[j] * hessenberg[i][k - 1];
}
for (int j = 0; j < n - k; j++) {
hessenberg[i][k - 1] -= 2 * z[j];
}
}
}
}
return hessenberg;
}
public static double getNorm(double[] vector) {
double norm = 0;
for (double element : vector) {
norm += element * element;
}
return Math.sqrt(norm);
}
public static void printMatrix(double[][] matrix) {
for (double[] row : matrix) {
for (double element : row) {
System.out.print(element + " ");
}
System.out.println();
}
}
}
import java.util.Arrays; public class HessenbergDecomposition { public static void main(String[] args) { double[][] matrix = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}; double[][] hessenbergMatrix = computeHessenberg(matrix); System.out.println("Original Matrix:"); printMatrix(matrix); System.out.println("\nHessenberg Matrix:"); printMatrix(hessenbergMatrix); } public static double[][] computeHessenberg(double[][] matrix) { int n = matrix.length; double[][] hessenberg = Arrays.copyOf(matrix, n); for (int k = 1; k < n - 1; k++) { double[] column = new double[n - k]; for (int i = k; i < n; i++) { column[i - k] = hessenberg[i][k - 1]; } double norm = getNorm(column); double[] u = new double[n - k]; u[0] = column[0] + Math.signum(column[0]) * norm; double uNorm = getNorm(u); if (uNorm != 0) { for (int i = 0; i < n - k; i++) { u[i] /= uNorm; } for (int i = k; i < n; i++) { double[] v = new double[n - k]; for (int j = 0; j < n - k; j++) { v[j] = u[j] * hessenberg[i][k - 1]; } for (int j = 0; j < n - k; j++) { hessenberg[i][k - 1] -= 2 * v[j]; } for (int j = k; j < n; j++) { double[] w = new double[n - k]; for (int l = 0; l < n - k; l++) { w[l] = u[l] * hessenberg[i][j]; } for (int l = 0; l < n - k; l++) { hessenberg[i][j] -= 2 * w[l]; } } } for (int i = 0; i < n; i++) { double[] z = new double[n - k]; for (int j = 0; j < n - k; j++) { z[j] = u[j] * hessenberg[i][k - 1]; } for (int j = 0; j < n - k; j++) { hessenberg[i][k - 1] -= 2 * z[j]; } } } } return hessenberg; } public static double getNorm(double[] vector) { double norm = 0; for (double element : vector) { norm += element * element; } return Math.sqrt(norm); } public static void printMatrix(double[][] matrix) { for (double[] row : matrix) { for (double element : row) { System.out.print(element + " "); } System.out.println(); } } }
This code demonstrates how to compute the Hessenberg decomposition of a matrix. It uses the Householder transformation to eliminate
the subdiagonal elements in the matrix and transform it into Hessenberg form. The `computeHessenberg` method takes a matrix as input and returns its Hessenberg form as a new matrix. The `getNorm` method calculates the norm of a vector, and the `printMatrix` method prints the elements of a matrix. The example code applies the Hessenberg decomposition to a sample matrix and displays both the original matrix and its Hessenberg form.
No comments yet! You be the first to comment.