Finding the matrix condition number using Bidiagonalization.
I can help you with that. Here’s a Java program that calculates the condition number of a matrix using Bidiagonalization:
import java.util.Arrays;
public class BidiagonalizationConditionNumber {
public static void main(String[] args) {
double[][] matrix = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
};
double conditionNumber = calculateConditionNumber(matrix);
System.out.println("Condition Number: " + conditionNumber);
}
public static double calculateConditionNumber(double[][] matrix) {
double[][] bidiagonal = bidiagonalize(matrix);
double largestSingularValue = bidiagonal[bidiagonal.length - 1][bidiagonal.length - 1];
double smallestSingularValue = bidiagonal[0][0];
return largestSingularValue / smallestSingularValue;
}
public static double[][] bidiagonalize(double[][] matrix) {
int m = matrix.length;
int n = matrix[0].length;
double[][] bidiagonal = new double[m][n];
double[] beta = new double[n];
for (int j = 0; j < n; j++) {
double[] column = getColumn(matrix, j);
double norm = calculateNorm(column);
if (norm != 0.0) {
double sign = (column[0] >= 0) ? 1.0 : -1.0;
double alpha = sign * norm;
bidiagonal[0][j] = alpha;
column[0] += alpha;
double betaValue = calculateNorm(column);
beta[j] = betaValue / alpha;
normalizeColumn(column);
setColumn(matrix, column, j);
}
}
for (int i = 0; i < m; i++) {
double[] row = getRow(matrix, i);
double norm = calculateNorm(row);
if (norm != 0.0) {
double sign = (row[0] >= 0) ? 1.0 : -1.0;
double alpha = sign * norm;
bidiagonal[i][0] = alpha;
row[0] += alpha;
normalizeRow(row);
setRow(matrix, row, i);
}
}
for (int j = 1; j < n; j++) {
double[] column = getColumn(matrix, j);
double[] prevColumn = getColumn(matrix, j - 1);
for (int i = 0; i < m; i++) {
column[i] -= beta[j - 1] * prevColumn[i];
}
bidiagonal[j][j] = calculateNorm(column);
if (bidiagonal[j][j] != 0.0) {
normalizeColumn(column);
setColumn(matrix, column, j);
}
}
return bidiagonal;
}
public static double calculateNorm(double[] vector) {
double normSquared = 0.0;
for (double value : vector) {
normSquared += value * value;
}
return Math.sqrt(normSquared);
}
public static void normalizeColumn(double[] column) {
double norm = calculateNorm(column);
for (int i = 0; i < column.length; i++) {
column[i] /= norm;
}
}
public static void normalizeRow(double[] row) {
double norm = calculateNorm(row);
for (int i = 0; i < row.length; i++) {
row[i] /= norm;
}
}
public static double
[] getColumn(double[][] matrix, int columnIndex) {
return Arrays.stream(matrix)
.mapToDouble(row -> row[columnIndex])
.toArray();
}
public static void setColumn(double[][] matrix, double[] column, int columnIndex) {
for (int i = 0; i < matrix.length; i++) {
matrix[i][columnIndex] = column[i];
}
}
public static double[] getRow(double[][] matrix, int rowIndex) {
return matrix[rowIndex];
}
public static void setRow(double[][] matrix, double[] row, int rowIndex) {
matrix[rowIndex] = row;
}
}
This program defines the `calculateConditionNumber` method that takes a matrix as input and returns its condition number. The `bidiagonalize` method implements the Bidiagonalization algorithm, which transforms the given matrix into a bidiagonal matrix. The largest and smallest singular values are then extracted from the bidiagonal matrix to calculate the condition number. The remaining methods assist in performing the necessary matrix operations.
In the main method, we create a sample matrix and calculate its condition number using the `calculateConditionNumber` method. The result is then printed to the console.
Note: This program assumes that the input matrix is a square matrix. You may need to modify it if you want to handle rectangular matrices.