feat: Add Neville's Algorithm (#6842)

* feat: Add Neville's algorithm for polynomial interpolation

* Update Neville.java

* style: Fix linter formatting issues.

* Handled Div by Zero Case

* Update NevilleTest.java

* Update Neville.java

* Update NevilleTest.java

* Update Neville.java

src/main/java/com/thealgorithms/maths/Neville.java

@@ -0,0 +1,61 @@
+package com.thealgorithms.maths;
+
+import java.util.HashSet;
+import java.util.Set;
+
+/**
+ * In numerical analysis, Neville's algorithm is an algorithm used for
+ * polynomial interpolation. Given n+1 points, there is a unique polynomial of
+ * degree at most n that passes through all the points. Neville's algorithm
+ * computes the value of this polynomial at a given point.
+ *
+ * <p>
+ * Wikipedia: https://en.wikipedia.org/wiki/Neville%27s_algorithm
+ *
+ * @author Mitrajit Ghorui(KeyKyrios)
+ */
+public final class Neville {
+
+    private Neville() {
+    }
+
+    /**
+     * Evaluates the polynomial that passes through the given points at a
+     * specific x-coordinate.
+     *
+     * @param x The x-coordinates of the points. Must be the same length as y.
+     * @param y The y-coordinates of the points. Must be the same length as x.
+     * @param target The x-coordinate at which to evaluate the polynomial.
+     * @return The interpolated y-value at the target x-coordinate.
+     * @throws IllegalArgumentException if the lengths of x and y arrays are
+     * different, if the arrays are empty, or if x-coordinates are not unique.
+     */
+    public static double interpolate(double[] x, double[] y, double target) {
+        if (x.length != y.length) {
+            throw new IllegalArgumentException("x and y arrays must have the same length.");
+        }
+        if (x.length == 0) {
+            throw new IllegalArgumentException("Input arrays cannot be empty.");
+        }
+
+        // Check for duplicate x-coordinates to prevent division by zero
+        Set<Double> seenX = new HashSet<>();
+        for (double val : x) {
+            if (!seenX.add(val)) {
+                throw new IllegalArgumentException("Input x-coordinates must be unique.");
+            }
+        }
+
+        int n = x.length;
+        double[] p = new double[n];
+        System.arraycopy(y, 0, p, 0, n); // Initialize p with y values
+
+        for (int k = 1; k < n; k++) {
+            for (int i = 0; i < n - k; i++) {
+                p[i] = ((target - x[i + k]) * p[i] + (x[i] - target) * p[i + 1]) / (x[i] - x[i + k]);
+            }
+        }
+
+        return p[0];
+    }
+}

src/test/java/com/thealgorithms/maths/NevilleTest.java

@@ -0,0 +1,69 @@
+package com.thealgorithms.maths;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import org.junit.jupiter.api.Test;
+
+public class NevilleTest {
+
+    @Test
+    public void testInterpolateLinear() {
+        // Test with a simple linear function y = 2x + 1
+        // Points (0, 1) and (2, 5)
+        double[] x = {0, 2};
+        double[] y = {1, 5};
+        // We want to find y when x = 1, which should be 3
+        double target = 1;
+        double expected = 3.0;
+        assertEquals(expected, Neville.interpolate(x, y, target), 1e-9);
+    }
+
+    @Test
+    public void testInterpolateQuadratic() {
+        // Test with a quadratic function y = x^2
+        // Points (0, 0), (1, 1), (3, 9)
+        double[] x = {0, 1, 3};
+        double[] y = {0, 1, 9};
+        // We want to find y when x = 2, which should be 4
+        double target = 2;
+        double expected = 4.0;
+        assertEquals(expected, Neville.interpolate(x, y, target), 1e-9);
+    }
+
+    @Test
+    public void testInterpolateWithNegativeNumbers() {
+        // Test with y = x^2 - 2x + 1
+        // Points (-1, 4), (0, 1), (2, 1)
+        double[] x = {-1, 0, 2};
+        double[] y = {4, 1, 1};
+        // We want to find y when x = 1, which should be 0
+        double target = 1;
+        double expected = 0.0;
+        assertEquals(expected, Neville.interpolate(x, y, target), 1e-9);
+    }
+
+    @Test
+    public void testMismatchedArrayLengths() {
+        double[] x = {1, 2};
+        double[] y = {1};
+        double target = 1.5;
+        assertThrows(IllegalArgumentException.class, () -> Neville.interpolate(x, y, target));
+    }
+
+    @Test
+    public void testEmptyArrays() {
+        double[] x = {};
+        double[] y = {};
+        double target = 1;
+        assertThrows(IllegalArgumentException.class, () -> Neville.interpolate(x, y, target));
+    }
+
+    @Test
+    public void testDuplicateXCoordinates() {
+        double[] x = {1, 2, 1};
+        double[] y = {5, 8, 3};
+        double target = 1.5;
+        assertThrows(IllegalArgumentException.class, () -> Neville.interpolate(x, y, target));
+    }
+}