feat: Add LU Decomposition algorithm

src/main/java/com/thealgorithms/matrix/LUDecomposition.java

@@ -0,0 +1,92 @@
+// File: src/main/java/com/thealgorithms/matrix/LUDecomposition.java
+package com.thealgorithms.matrix;
+
+import java.util.Arrays;
+
+/**
+ * Implements the LU Decomposition algorithm for a square matrix.
+ *
+ * LU decomposition factors a matrix as the product of a lower triangular matrix (L)
+ * and an upper triangular matrix (U). This implementation uses the Doolittle algorithm,
+ * where the diagonal elements of the lower triangular matrix L are all 1s.
+ */
+public class LUDecomposition {
+
+    /**
+     * Decomposes a given square matrix into Lower (L) and Upper (U) matrices.
+     *
+     * @param matrix The square matrix to be decomposed.
+     * @return An array containing L and U matrices. result[0] is L, result[1] is U.
+     * @throws IllegalArgumentException if the matrix is not square.
+     */
+    public static double[][][] decompose(double[][] matrix) {
+        int n = matrix.length;
+        if (n == 0 || matrix[0].length != n) {
+            throw new IllegalArgumentException("Matrix must be square.");
+        }
+
+        double[][] lower = new double[n][n];
+        double[][] upper = new double[n][n];
+
+        for (int i = 0; i < n; i++) {
+            // Upper Triangular Matrix
+            for (int k = i; k < n; k++) {
+                double sum = 0;
+                for (int j = 0; j < i; j++) {
+                    sum += (lower[i][j] * upper[j][k]);
+                }
+                upper[i][k] = matrix[i][k] - sum;
+            }
+
+            // Lower Triangular Matrix
+            for (int k = i; k < n; k++) {
+                if (i == k) {
+                    lower[i][i] = 1; // Diagonal of L is 1
+                } else {
+                    double sum = 0;
+                    for (int j = 0; j < i; j++) {
+                        sum += (lower[k][j] * upper[j][i]);
+                    }
+                    lower[k][i] = (matrix[k][i] - sum) / upper[i][i];
+                }
+            }
+        }
+
+        return new double[][][] {lower, upper};
+    }
+
+    /**
+     * A utility function to print a matrix.
+     * @param matrix The matrix to be printed.
+     */
+    public static void printMatrix(double[][] matrix) {
+        for (double[] row : matrix) {
+            System.out.println(Arrays.toString(row));
+        }
+    }
+
+    /**
+     * Main method to demonstrate the LU Decomposition.
+     * @param args Command-line arguments (not used).
+     */
+    public static void main(String[] args) {
+        double[][] matrix = {
+            {2, -1, -2},
+            {-4, 6, 3},
+            {-4, -2, 8}
+        };
+
+        System.out.println("Original Matrix:");
+        printMatrix(matrix);
+
+        double[][][] luMatrices = decompose(matrix);
+        double[][] lower = luMatrices[0];
+        double[][] upper = luMatrices[1];
+
+        System.out.println("\nLower Triangular Matrix (L):");
+        printMatrix(lower);
+
+        System.out.println("\nUpper Triangular Matrix (U):");
+        printMatrix(upper);
+    }
+}