Java 中的 Bernoulli 数

17 Mar 2025 | 5 分钟阅读

伯努利数是一类在数论和分析中扮演重要角色的特殊数字。这些数字主要用于展开多个三角函数、伽马函数、双曲函数等的级数。

第 n 个伯努利数用B_n 表示，可以通过以下指数生成函数定义：

伯努利数出现在三角函数级数的展开中。我们有两种不同的伯努利数定义。

根据现代用法或美国国家标准与技术研究院 (NIST) 的惯例，伯努利数写作。它也被称为“偶数下标”伯努利数。根据旧文献中的定义，伯努利数写作。

在这两种情况下，每个伯努利数都是伯努利多项式的一个特例：

B_n (x) 或 B_n^* (x)，其中 B_n=B_n (0) 且 B_n^* = B_n^* (0)

我们还分别用 B_n 和 B_n (x) 表示贝尔数和贝尔多项式。因此，我们不应将它们混淆。

我们可以使用以下轮廓积分定义伯努利数 B_n：

算法

计算“第二类伯努利数”的算法如下：

for m from 0 by 1 to n do
    A[m] → 1/(m+1)
    for j from m by -1 to 1 do
      A[j-1] → j×(A[j-1] - A[j])
  return A[0] (which is Bn)

下面我们来实现 Java 代码来获取 N 个伯努利数。在编写获取伯努利数的逻辑之前，我们先创建一个 BigRational.java 类，它执行 BigInteger 和 BigDecimal 所需的操作，并返回结果。

BigRational.java

import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.RoundingMode;
import java.util.Objects;
// create BigRational class that implements comparable interface
publicclass BigRational implements Comparable<BigRational> {
	// declare and initialization of ZERO, ONE, and TWO variables of type BigRational
publicfinalstatic BigRational ZERO = new BigRational(0);
publicfinalstatic BigRational ONE  = new BigRational(1);
publicfinalstatic BigRational TWO  = new BigRational(2);
    // declare numerator and denominator variables
    // denominator will always be a positive number
private BigInteger num;
private BigInteger den;
    // initialization of BigRational object
public BigRational(int num, int den) {
	this.num = BigInteger.valueOf(num);
	this.den = BigInteger.valueOf(den);
    }
    // create and initialize a new BigRational object
public BigRational(int numerator) {
this(numerator, 1);
    }
    // initialization of BigRational object from a string, e.g., "-343/1273"
public BigRational(String str) {
        // use split() method to break it into sub-string and store them in the array
	String[] tokens = str.split("/");
	// check whether the size of the tokens is equal to 2 or not
if (tokens.length == 2) {
	BigInteger num1 = new BigInteger(tokens[0]);
	BigInteger num2 = new BigInteger(tokens[1]);
            init(num1, num2);
        }
        // check whether the size of the tokens is equal to 1 or not
elseif (tokens.length == 1) {
	BigInteger num1 = new BigInteger(tokens[0]);
            init(num1, BigInteger.ONE);
        }
        // else throw IllegalArgumentException
else
thrownew IllegalArgumentException("For input string: \"" + str + "\"");
    }
    // creation and initialization of BigRational object from a BigInteger
public BigRational(BigInteger num, BigInteger den) {
        // use init() method for initialization object 
	init(num, den);
    }
    // create init() method for initializing object
privatevoid init(BigInteger num, BigInteger den) {
        // check whether the denominator is positive or not
if (den.equals(BigInteger.ZERO)) {
thrownew ArithmeticException("Denominator is zero");
        }
        // reduce fraction (if numerator is 0, it will always yield denominator 0)
        BigInteger g = num.gcd(den);
this.num = num.divide(g);
this.den = den.divide(g);
        // ensure whether the denominator is positive or not
if (this.den.compareTo(BigInteger.ZERO) < 0) {
this.den = den.negate();
this.num = num.negate();
        }
    }
    // return string representation of denominator and numerator
public String toString() { 
        // if numerator is one, return denominator
	if (den.equals(BigInteger.ONE)) 
	return num + "";
	// else return numerator and denominator
	else
	return num + "/" + den;
    }
    // if x < y, return -1
    // if x = y, return 0
    // if x > y, return +1
publicint compareTo(BigRational y) {
	BigRational x = this;
int res = x.num.multiply(y.den).compareTo(x.den.multiply(y.num));

return res;
    }
    // check whether the current BigRational is zero, negative or positive
publicboolean isZero() {
	boolean res = num.signum() == 0;
	return res;
    }
publicboolean isPositive() {
	boolean res = num.signum() >  0;
	return res;
    }
publicboolean isNegative() {
	boolean res = num.signum() <  0;
	return res;
    }
    // check whether the current rational object equal to y or not
publicboolean equals(Object y) {
if (y == this) 
	returntrue;
if (y == null) 
	returnfalse;  
if (y.getClass() != this.getClass()) 
	returnfalse;
        BigRational newBegRational = (BigRational) y;
return compareTo(newBegRational) == 0;
    }
    // hashCode consistent with equals() and compareTo()
publicint hashCode() {
return Objects.hash(num, den);
    }
    // return a * b
public BigRational times(BigRational b) {
        BigRational a = this;
returnnew BigRational(a.num.multiply(b.num), a.den.multiply(b.den));
    }
    // return a + b
public BigRational plus(BigRational b) {
        BigRational a = this;
        BigInteger numerator   = a.num.multiply(b.den).add(b.num.multiply(a.den));
        BigInteger denominator = a.den.multiply(b.den);
returnnew BigRational(numerator, denominator);
    }
    // return -a
public BigRational negate() {
returnnew BigRational(num.negate(), den);
    }
    // return |a|
public BigRational abs() {
if (isNegative()) return negate();
elsereturnthis;
    }
    // return a - b
public BigRational minus(BigRational b) {
        BigRational a = this;
return a.plus(b.negate());
    }
    // return 1 / a
public BigRational reciprocal() {
returnnew BigRational(den, num);
    }
    // return a / b
publicBigRational divides(BigRational b) {
        BigRational a = this;
return a.times(b.reciprocal());
    }
    // return double reprentation (within given precision)
publicdouble doubleValue() {
int SCALE = 32;        // number of digits after the decimal place
        BigDecimal numerator   = new BigDecimal(num);
        BigDecimal denominator = new BigDecimal(den);
        BigDecimal quotient    = numerator.divide(denominator, SCALE, RoundingMode.HALF_EVEN);
return quotient.doubleValue();
    }
}

BernoulliNumber.java

import java.math.BigInteger;
import java.util.Scanner;
// create BernoulliNumber class to get N Bernoulli numbers
public class BernoulliNumber {
	// main() method start
    public static void main(String[] args) {

	int N;
	
	// create scanner class object to get input from user
	Scanner sc = new Scanner(System.in);
	
	System.out.println("How many Bernoulli number you want to get.");
	// initialization N
	N = sc.nextInt();
	sc.close();
        // precompute binomial coefficients
        BigInteger[][] bin = new BigInteger[N+1][N+1];

        for (int i = 1; i <= N; i++) { 
	bin[0][i] = BigInteger.ZERO;
        }
        for (int i = 0; i <= N; i++) { 
	bin[i][0] = BigInteger.ONE;
        }
        // bottom-up dynamic programming
        for (int i = 1; i <= N; i++) {
            for (int j = 1; j <= N; j++) {
                bin[i][j] = bin[i - 1][j - 1].add(bin[i - 1][j]);
            }
        }


        // get Bernoulli numbers using BigRational class
        BigRational[] bernoulliNumbers = new BigRational[N + 1];
        bernoulliNumbers[0] = new BigRational(1, 1);
        bernoulliNumbers[1] = new BigRational(-1, 2);
        for (int i = 2; i < N; i++) {
	bernoulliNumbers[i] = new BigRational(0, 1);
            for (int j = 0; j < i; j++) {
                BigRational coef = new BigRational(bin[i + 1][i + 1 - j], BigInteger.ONE);
                bernoulliNumbers[i] = bernoulliNumbers[i].minus(coef.times(bernoulliNumbers[j]));
            }
            bernoulliNumbers[i] = bernoulliNumbers[i].divides(new BigRational(i + 1, 1));
        }

        // print out the first N Bernoulli numbers
        for (int i = 0; i < N; i++)
	System.out.println(bernoulliNumbers[i]);

    }
}

输出

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