Finding the matrix determinant using Bidiagonalization.

Here’s a Java program that calculates the determinant of a matrix using bidiagonalization:

public class BidiagonalizationDeterminant {

    public static void main(String[] args) {

        double[][] matrix = {

            {1, 2, 3},

            {4, 5, 6},

            {7, 8, 9}

        };

        double determinant = calculateDeterminant(matrix);

        System.out.println("Determinant: " + determinant);

    }

    public static double calculateDeterminant(double[][] matrix) {

        int n = matrix.length;

        double[][] bidiagonal = bidiagonalize(matrix);

        double determinant = 1.0;

        for (int i = 0; i < n; i++) {

            determinant *= bidiagonal[i][i];

        }

        return determinant;

    }

    public static double[][] bidiagonalize(double[][] matrix) {

        int n = matrix.length;

        double[][] bidiagonal = new double[n][n];

        double[] u = new double[n];

        double[] v = new double[n];

        for (int k = 0; k < n - 1; k++) {

            double alpha = 0.0;

            double beta = 0.0;

            for (int i = k; i < n; i++) {

                alpha += matrix[i][k] * matrix[i][k];

            }

            alpha = Math.sqrt(alpha);

            if (matrix[k][k] < 0) {

                alpha = -alpha;

            }

            if (alpha != 0.0) {

                for (int i = k; i < n; i++) {

                    u[i] = matrix[i][k];

                }

                u[k] += alpha;

                double normU = 0.0;

                for (int i = k; i < n; i++) {

                    normU += u[i] * u[i];

                }

                normU = Math.sqrt(normU);

                for (int i = k; i < n; i++) {

                    u[i] /= normU;

                }

                for (int j = k; j < n; j++) {

                    beta = 0.0;

                    for (int i = k; i < n; i++) {

                        beta += matrix[i][j] * u[i];

                    }

                    for (int i = k; i < n; i++) {

                        v[i] = matrix[i][j] - beta * u[i];

                    }

                    for (int i = k; i < n; i++) {

                        matrix[i][j] = v[i];

                    }

                }

            }

            if (k < n - 2) {

                alpha = 0.0;

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

                    alpha += matrix[k][j] * matrix[k][j];

                }

                alpha = Math.sqrt(alpha);

                if (matrix[k][k + 1] < 0) {

                    alpha = -alpha;

                }

                if (alpha != 0.0) {

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

                        u[j] = matrix[k][j];

                    }

                    u[k + 1] += alpha;

                    double normU = 0.0;

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

                        normU += u[j] * u[j];

                    }

                    normU = Math.sqrt(normU);

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

                        u[j] /= normU;

                    }

 for (int i = k; i < n; i++) {

                        beta = 0.0;

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

                            beta += matrix[i][j] * u[j];

                        }

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

                            v[j] = matrix[i][j] - beta * u[j];

                        }

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

                            matrix[i][j] = v[j];

                        }

                    }

                }

            }

            for (int i = 0; i < n; i++) {

                bidiagonal[i][k] = matrix[i][k];

                bidiagonal[k][i] = matrix[k][i];

            }

        }

        bidiagonal[n - 1][n - 1] = matrix[n - 1][n - 1];

        return bidiagonal;

    }

}

In this program, the `calculateDeterminant` method takes a matrix as input and returns its determinant using the bidiagonalization technique. The `bidiagonalize` method performs the bidiagonalization process, transforming the input matrix into a bidiagonal matrix. Finally, the determinant is calculated by multiplying the diagonal elements of the bidiagonal matrix.

You can modify the `matrix` variable in the `main` method with your own matrix values. The program will output the determinant of the matrix.