Added LU Decomposition Algorithm for matrix (#6834)

* Added LU decomposition algorthm

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

* Added LU decomposition algorthim

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

@@ -0,0 +1,88 @@
+package com.thealgorithms.matrix;
+
+/**
+ * LU Decomposition algorithm
+ * --------------------------
+ * Decomposes a square matrix a into a product of two matrices:
+ * a = l * u
+ * where:
+ * - l is a lower triangular matrix with 1s on its diagonal
+ * - u is an upper triangular matrix
+ *
+ * Reference:
+ * https://en.wikipedia.org/wiki/lu_decomposition
+ */
+public final class LUDecomposition {
+
+    private LUDecomposition() {
+    }
+
+    /**
+     * A helper class to store both l and u matrices
+     */
+    public static class LU {
+        double[][] l;
+        double[][] u;
+
+        LU(double[][] l, double[][] u) {
+            this.l = l;
+            this.u = u;
+        }
+    }
+
+    /**
+     * Performs LU Decomposition on a square matrix a
+     *
+     * @param a input square matrix
+     * @return LU object containing l and u matrices
+     */
+    public static LU decompose(double[][] a) {
+        int n = a.length;
+        double[][] l = new double[n][n];
+        double[][] u = 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 += l[i][j] * u[j][k];
+                }
+                u[i][k] = a[i][k] - sum;
+            }
+
+            // lower triangular matrix
+            for (int k = i; k < n; k++) {
+                if (i == k) {
+                    l[i][i] = 1; // diagonal as 1
+                } else {
+                    double sum = 0;
+                    for (int j = 0; j < i; j++) {
+                        sum += l[k][j] * u[j][i];
+                    }
+                    l[k][i] = (a[k][i] - sum) / u[i][i];
+                }
+            }
+        }
+
+        return new LU(l, u);
+    }
+
+    /**
+     * Utility function to print a matrix
+     *
+     * @param m matrix to print
+     */
+    public static void printMatrix(double[][] m) {
+        for (double[] row : m) {
+            System.out.print("[");
+            for (int j = 0; j < row.length; j++) {
+                System.out.printf("%7.3f", row[j]);
+                if (j < row.length - 1) {
+                    System.out.print(", ");
+                }
+            }
+            System.out.println("]");
+        }
+    }
+}

src/test/java/com/thealgorithms/matrix/LUDecompositionTest.java

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+package com.thealgorithms.matrix;
+
+import static org.junit.jupiter.api.Assertions.assertArrayEquals;
+
+import org.junit.jupiter.api.Test;
+
+public class LUDecompositionTest {
+
+    @Test
+    public void testLUDecomposition() {
+        double[][] a = {{4, 3}, {6, 3}};
+
+        // Perform LU decomposition
+        LUDecomposition.LU lu = LUDecomposition.decompose(a);
+        double[][] l = lu.l;
+        double[][] u = lu.u;
+
+        // Reconstruct a from l and u
+        double[][] reconstructed = multiplyMatrices(l, u);
+
+        // Assert that reconstructed matrix matches original a
+        for (int i = 0; i < a.length; i++) {
+            assertArrayEquals(a[i], reconstructed[i], 1e-9);
+        }
+    }
+
+    // Helper method to multiply two matrices
+    private double[][] multiplyMatrices(double[][] a, double[][] b) {
+        int n = a.length;
+        double[][] c = new double[n][n];
+        for (int i = 0; i < n; i++) {
+            for (int j = 0; j < n; j++) {
+                for (int k = 0; k < n; k++) {
+                    c[i][j] += a[i][k] * b[k][j];
+                }
+            }
+        }
+        return c;
+    }
+}