Mathematical Algorithms in TheAlgorithms/Java: Complete Guide to Number Theory, Linear Algebra, and Numerical Methods

TheAlgorithms/Java provides a comprehensive library of over 50 mathematical algorithms spanning number theory, combinatorics, linear algebra, and numerical analysis, implemented as stateless utility classes in src/main/java/com/thealgorithms/maths/.

TheAlgorithms/Java is one of the most-starred open-source algorithm repositories on GitHub, offering production-ready implementations of classic and modern mathematical algorithms. Whether you need prime number sieves, fast Fourier transforms, or numerical integration methods, this repository provides dependency-free Java code optimized for Java 17 and above. This guide catalogs the mathematical algorithms available in TheAlgorithms/Java, organized by category with practical code examples drawn directly from the source.

Architecture and Design Pattern

All mathematical algorithms reside in the com.thealgorithms.maths package under src/main/java/com/thealgorithms/maths/. The repository follows a strict stateless utility pattern: each class is declared final with a private constructor, exposing algorithms via public static methods. This design eliminates instantiation overhead and ensures thread safety. The implementations rely exclusively on the Java standard library with zero external dependencies, and comprehensive JUnit test coverage exists in src/test/java/com/thealgorithms/maths/.

Number Theory and Primality Algorithms

Prime Generation and Sieving

The repository contains multiple prime generation strategies. The SieveOfEratosthenes.java class implements the classic O(n log log n) algorithm via generatePrimes(int limit), returning a List<Integer> of all primes up to the specified bound. For advanced use cases, SieveOfAtkin.java provides an optimized alternative with superior asymptotic performance for large ranges.

Divisibility and Factorization

For divisibility operations, GCD.java implements the Euclidean algorithm through gcd(int a, int b), while LeastCommonMultiple.java calculates LCM using the mathematical relationship lcm(a, b) = (a * b) / gcd(a, b) via lcm(int a, int b). These utilities handle edge cases including negative inputs and zero values.

Special Number Sequences and Properties

The collection includes generators for fundamental sequences: FibonacciLoop.java for Fibonacci numbers, CatalanNumbers.java for combinatorial structures via catalan(int n), and BellNumbers.java for set partitions. Additionally, special number tests are available in Armstrong.java, HappyNumber.java, PerfectNumber.java, and HarshadNumber.java for validating numerical properties.

Combinatorics and Counting

The combinatorics suite provides tools for discrete mathematics calculations. BinomialCoefficient.java computes nCk using dynamic programming based on Pascal's triangle via binomialCoefficient(int n, int k), avoiding the overflow issues inherent in factorial division. CatalanNumbers.java and BellNumbers.java support advanced counting problems in graph theory and set theory applications.

Linear Algebra and Numerical Analysis

Matrix and Vector Operations

For linear algebra computations, DeterminantOfMatrix.java calculates matrix determinants recursively with O(n!) complexity, suitable for small to medium matrices. Vector mathematics are supported by VectorCrossProduct.java for three-dimensional geometric calculations.

Fast Fourier Transform

Signal processing implementations include FFT.java, which features an iterative radix-2 Cooley-Tukey Fast Fourier Transform. The method signature fft(ArrayList<Complex> signal, boolean inverse) requires input sizes that are powers of two and returns complex number results for frequency domain analysis.

Numerical Integration and Root Finding

Calculus utilities include SimpsonIntegration.java, which applies Simpson's rule for numerical integration via integrate(DoubleUnaryOperator f, double a, double b, int n). Root-finding algorithms are implemented in SquareRootWithNewtonRaphsonMethod.java and SquareRootWithBabylonianMethod.java, offering iterative approaches to approximation with configurable precision.

Arithmetic and Exponentiation

High-performance arithmetic operations are centered in BinaryPow.java and FastExponentiation.java. The BinaryPow.binPow(int a, int p) method implements binary exponentiation with O(log p) complexity, efficiently handling large powers without overflow through iterative bit manipulation. FastExponentiation.java provides additional support for long and modular arithmetic operations.

Statistical and Geometric Utilities

Statistical Tools

The statistics module in Means.java implements arithmetic, geometric, harmonic, and quadratic means, while StandardDeviation.java and variance calculations support descriptive statistics. ZScore.java provides standard score normalization for data analysis workflows.

Geometric Calculations

Geometric algorithms include Area.java for polygon and circle area calculations, PythagoreanTriple.java for generating integer triples satisfying a² + b² = c², and VectorCrossProduct.java for spatial computing applications.

Practical Implementation Examples

Binary Exponentiation

import com.thealgorithms.maths.BinaryPow;

public class DemoBinaryPow {
    public static void main(String[] args) {
        int base = 5;
        int exponent = 13;
        int result = BinaryPow.binPow(base, exponent);
        System.out.println(base + "^" + exponent + " = " + result);
        // Output: 5^13 = 1220703125
    }
}

Greatest Common Divisor

import com.thealgorithms.maths.GCD;

public class DemoGCD {
    public static void main(String[] args) {
        System.out.println("GCD(48, 180) = " + GCD.gcd(48, 180));
        // Output: GCD(48, 180) = 12
    }
}

Prime Generation with Sieve of Eratosthenes

import com.thealgorithms.maths.SieveOfEratosthenes;
import java.util.List;

public class DemoSieve {
    public static void main(String[] args) {
        List<Integer> primes = SieveOfEratosthenes.generatePrimes(50);
        System.out.println(primes);
        // Output: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
    }
}

Fast Fourier Transform

import com.thealgorithms.maths.FFT;
import com.thealgorithms.maths.FFT.Complex;
import java.util.ArrayList;
import java.util.List;

public class DemoFFT {
    public static void main(String[] args) {
        // Example signal: 4 samples (must be power of two)
        List<Complex> signal = List.of(
                new Complex(0, 0),
                new Complex(1, 0),
                new Complex(0, 0),
                new Complex(-1, 0));

        ArrayList<Complex> transformed = FFT.fft(new ArrayList<>(signal), false);
        System.out.println("FFT result:");
        transformed.forEach(c -> System.out.println(c.getReal() + " + " + c.getImaginary() + "i"));
    }
}

Numerical Integration with Simpson's Rule

import com.thealgorithms.maths.SimpsonIntegration;

public class DemoSimpson {
    public static void main(String[] args) {
        // Integrate f(x) = x^2 from 0 to 1
        double integral = SimpsonIntegration.integrate(x -> x * x, 0, 1, 100);
        System.out.println("∫₀¹ x² dx ≈ " + integral);
        // Expected value: 1/3 ≈ 0.33333
    }
}

Summary

• TheAlgorithms/Java organizes mathematical algorithms in src/main/java/com/thealgorithms/maths/ within the com.thealgorithms.maths package.
• All implementations follow a stateless utility pattern with public static methods, final classes, and private constructors to prevent instantiation.
• The repository covers eight major categories: number theory (primes, GCD, special numbers), combinatorics (binomial coefficients, Catalan numbers), linear algebra (determinants, FFT), numerical methods (Simpson's integration, root finding), exponentiation (binary pow), arithmetic, statistics, and geometry.
• Zero dependencies are required beyond the Java standard library, with full compatibility for Java 17+.
• Each algorithm includes corresponding JUnit tests in src/test/java/com/thealgorithms/maths/ to validate edge cases and correctness.

Frequently Asked Questions

How do I import and use mathematical algorithms from TheAlgorithms/Java in my project?

Import the specific utility class from the com.thealgorithms.maths package and invoke the static method directly. For example, use import com.thealgorithms.maths.GCD; followed by int result = GCD.gcd(48, 180);. No instantiation or dependency configuration is required, as all algorithms are implemented as stateless utility functions.

What Java version is required to run these mathematical algorithms?

TheAlgorithms/Java targets Java 17 and above. The mathematical algorithms in src/main/java/com/thealgorithms/maths/ utilize modern Java features such as DoubleUnaryOperator for numerical integration and enhanced List interfaces, ensuring compatibility with current Long-Term Support (LTS) versions.

Are the mathematical algorithms in TheAlgorithms/Java thread-safe?

Yes, all mathematical utility classes are inherently thread-safe. Because each class is stateless with only static methods and no instance variables, concurrent threads can safely invoke BinaryPow.binPow(), FFT.fft(), or SimpsonIntegration.integrate() without synchronization concerns or race conditions.

How can I contribute a new mathematical algorithm to the repository?

Create a new class in src/main/java/com/thealgorithms/maths/ following the existing stateless pattern: declare the class final, add a private constructor to prevent instantiation, and implement the algorithm as a public static method. Include comprehensive JUnit tests in src/test/java/com/thealgorithms/maths/ covering edge cases such as zero inputs, negative values, and maximum integer bounds.

{
  "mcpServers": {
    "instagit": {
      "command": "npx",
      "args": ["-y", "instagit@latest"]
    }
  }
}