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MuhammadEzzatHBKMuhammadEzzatHBK
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Add Monte Carlo's Integral Approximation (#6235)
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‎src/main/java/com/thealgorithms/randomized/MonteCarloIntegration.java

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package com.thealgorithms.randomized;
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import java.util.Random;
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import java.util.function.Function;
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/**
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* A demonstration of the Monte Carlo integration algorithm in Java.
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*
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* <p>This class estimates the value of definite integrals using randomized sampling,
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* also known as the Monte Carlo method. It is particularly effective for:
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* <ul>
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* <li>Functions that are difficult or impossible to integrate analytically</li>
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* <li>High-dimensional integrals where traditional methods are inefficient</li>
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* <li>Simulation and probabilistic analysis tasks</li>
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* </ul>
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*
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* <p>The core idea is to sample random points uniformly from the integration domain,
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* evaluate the function at those points, and compute the scaled average to estimate the integral.
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*
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* <p>For a one-dimensional integral over [a, b], the approximation is the function range (b-a),
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* multiplied by the function average result for a random sample.
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* See more: <a href="https://en.wikipedia.org/wiki/Monte_Carlo_integration">Monte Carlo Integration</a>
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*
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* @author: MuhammadEzzatHBK
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*/
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public final class MonteCarloIntegration {
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private MonteCarloIntegration() {
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}
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/**
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* Approximates the definite integral of a given function over a specified
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* interval using the Monte Carlo method with a fixed random seed for
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* reproducibility.
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*
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* @param fx the function to integrate
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* @param a the lower bound of the interval
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* @param b the upper bound of the interval
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* @param n the number of random samples to use
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* @param seed the seed for the random number generator
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* @return the approximate value of the integral
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*/
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public static double approximate(Function<Double, Double> fx, double a, double b, int n, long seed) {
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return doApproximate(fx, a, b, n, new Random(seed));
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}
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/**
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* Approximates the definite integral of a given function over a specified
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* interval using the Monte Carlo method with a random seed based on the
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* current system time for more randomness.
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*
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* @param fx the function to integrate
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* @param a the lower bound of the interval
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* @param b the upper bound of the interval
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* @param n the number of random samples to use
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* @return the approximate value of the integral
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*/
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public static double approximate(Function<Double, Double> fx, double a, double b, int n) {
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return doApproximate(fx, a, b, n, new Random(System.currentTimeMillis()));
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}
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private static double doApproximate(Function<Double, Double> fx, double a, double b, int n, Random generator) {
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if (!validate(fx, a, b, n)) {
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throw new IllegalArgumentException("Invalid input parameters");
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}
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double totalArea = 0.0;
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double interval = b - a;
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for (int i = 0; i < n; i++) {
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double x = a + generator.nextDouble() * interval;
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totalArea += fx.apply(x);
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}
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return interval * totalArea / n;
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}
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private static boolean validate(Function<Double, Double> fx, double a, double b, int n) {
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boolean isFunctionValid = fx != null;
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boolean isIntervalValid = a < b;
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boolean isSampleSizeValid = n > 0;
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return isFunctionValid && isIntervalValid && isSampleSizeValid;
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}
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}

‎src/test/java/com/thealgorithms/randomized/MonteCarloIntegrationTest.java

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package com.thealgorithms.randomized;
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import static com.thealgorithms.randomized.MonteCarloIntegration.approximate;
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import static org.junit.jupiter.api.Assertions.assertEquals;
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import static org.junit.jupiter.api.Assertions.assertNotNull;
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import static org.junit.jupiter.api.Assertions.assertThrows;
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import java.util.function.Function;
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import org.junit.jupiter.api.Test;
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class MonteCarloIntegrationTest {
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private static final double EPSILON = 0.03; // Allow 3% error margin
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@Test
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void testConstantFunction() {
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// Integral of f(x) = 2 from 0 to 1 is 2
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Function<Double, Double> constant = x -> 2.0;
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double result = approximate(constant, 0, 1, 10000);
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assertEquals(2.0, result, EPSILON);
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}
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@Test
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void testLinearFunction() {
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// Integral of f(x) = x from 0 to 1 is 0.5
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Function<Double, Double> linear = Function.identity();
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double result = approximate(linear, 0, 1, 10000);
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assertEquals(0.5, result, EPSILON);
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}
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@Test
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void testQuadraticFunction() {
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// Integral of f(x) = x^2 from 0 to 1 is 1/3
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Function<Double, Double> quadratic = x -> x * x;
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double result = approximate(quadratic, 0, 1, 10000);
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assertEquals(1.0 / 3.0, result, EPSILON);
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}
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@Test
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void testLargeSampleSize() {
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// Integral of f(x) = x^2 from 0 to 1 is 1/3
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Function<Double, Double> quadratic = x -> x * x;
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double result = approximate(quadratic, 0, 1, 50000000);
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assertEquals(1.0 / 3.0, result, EPSILON / 2); // Larger sample size, smaller error margin
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}
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@Test
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void testReproducibility() {
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Function<Double, Double> linear = Function.identity();
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double result1 = approximate(linear, 0, 1, 10000, 42L);
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double result2 = approximate(linear, 0, 1, 10000, 42L);
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assertEquals(result1, result2, 0.0); // Exactly equal
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}
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@Test
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void testNegativeInterval() {
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// Integral of f(x) = x from -1 to 1 is 0
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Function<Double, Double> linear = Function.identity();
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double result = approximate(linear, -1, 1, 10000);
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assertEquals(0.0, result, EPSILON);
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}
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@Test
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void testNullFunction() {
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Exception exception = assertThrows(IllegalArgumentException.class, () -> approximate(null, 0, 1, 1000));
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assertNotNull(exception);
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}
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@Test
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void testInvalidInterval() {
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Function<Double, Double> linear = Function.identity();
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Exception exception = assertThrows(IllegalArgumentException.class, () -> {
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approximate(linear, 2, 1, 1000); // b <= a
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});
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assertNotNull(exception);
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}
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@Test
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void testZeroSampleSize() {
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Function<Double, Double> linear = Function.identity();
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Exception exception = assertThrows(IllegalArgumentException.class, () -> approximate(linear, 0, 1, 0));
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assertNotNull(exception);
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}
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@Test
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void testNegativeSampleSize() {
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Function<Double, Double> linear = Function.identity();
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Exception exception = assertThrows(IllegalArgumentException.class, () -> approximate(linear, 0, 1, -100));
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assertNotNull(exception);
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}
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}

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