Step by Step solution about How to resolve the algorithm Bernoulli numbers step by step in the Java programming language

The provided Java code calculates and prints the first 60 Bernoulli numbers using the Apache Commons Math library's BigFraction class. Bernoulli numbers are a sequence of rational numbers that arise in number theory and have various applications in mathematics and physics.

• BigFraction Class:

  • BigFraction is a class that represents fractions with arbitrary-precision numerators and denominators. It provides methods for performing arithmetic operations and comparing fractions.

• main Method:

  • The program starts with the main method. It iterates from n = 0 to n = 60 to compute and print the Bernoulli numbers.

• Bernoulli(int n) Method:

  • This method computes the Bernoulli number with index n. It employs a recursive approach to calculate the Bernoulli numbers.

  • The method initializes an array A of BigFraction objects with a length of n + 1. It then enters a loop to fill the array with values based on the following formula:

    A[m] = 1 / (m + 1)
    for (int j = m; j >= 1; j--)
        A[j - 1] = (A[j - 1] - A[j]) * j
    

  • After filling the array, the method returns A[0], which represents the Bernoulli number B(n).

• Output:

  • The program prints the Bernoulli numbers for indices from 0 to 60. It formats the output as B(n) = <value>, where n is the index and <value> is the calculated Bernoulli number.

import org.apache.commons.math3.fraction.BigFraction;

public class BernoulliNumbers {

    public static void main(String[] args) {
        for (int n = 0; n <= 60; n++) {
            BigFraction b = bernouilli(n);
            if (!b.equals(BigFraction.ZERO))
                System.out.printf("B(%-2d) = %-1s%n", n , b);
        }
    }

    static BigFraction bernouilli(int n) {
        BigFraction[] A = new BigFraction[n + 1];
        for (int m = 0; m <= n; m++) {
            A[m] = new BigFraction(1, (m + 1));
            for (int j = m; j >= 1; j--)
                A[j - 1] = (A[j - 1].subtract(A[j])).multiply(new BigFraction(j));
        }
        return A[0];
    }
}