Finding the LU decomposition of a matrix

Here’s an example Java program that performs LU decomposition of a matrix using the Doolittle algorithm:

   public class LUDecomposition {

    public static void main(String[] args) {

        double[][] matrix = {

            {1, 2, 3},

            {4, 5, 6},

            {7, 8, 10}

        };

        double[][][] lu = decompose(matrix);

        System.out.println("LU decomposition:");

        System.out.println("L = ");

        printMatrix(lu[0]);

        System.out.println("U = ");

        printMatrix(lu[1]);

    }

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

        int n = matrix.length;

        double[][] L = new double[n][n];

        double[][] U = new double[n][n];

        // Perform LU decomposition

        for (int i = 0; i < n; i++) {

            // Compute L

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

                if (j < i) {

                    L[j][i] = 0;

                } else if (j == i) {

                    L[j][i] = 1;

                } else {

                    double sum = 0;

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

                        sum += L[j][k] * U[k][i];

                    }

                    L[j][i] = matrix[j][i] - sum;

                }

            }

            // Compute U

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

                if (j < i) {

                    U[i][j] = 0;

                } else {

                    double sum = 0;

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

                        sum += L[i][k] * U[k][j];

                    }

                    U[i][j] = (matrix[i][j] - sum) / L[i][i];

                }

            }

        }

        return new double[][][] { L, U };

    }

    public static void printMatrix(double[][] matrix) {

        for (double[] row : matrix) {

            for (double value : row) {

                System.out.printf("%.2f ", value);

            }

            System.out.println();
        }
    }
}

This program decomposes a 3×3 matrix into its lower triangular matrix (L) and upper triangular matrix (U) components. The `decompose` method implements the Doolittle algorithm, which iteratively computes the entries of L and U based on the given matrix. The `printMatrix` method is used to display the matrices L and U.

When you run this program, you will see the L and U matrices printed in the console as output