How to resolve the algorithm Faulhaber's triangle step by step in the Ruby programming language

Published on 12 May 2024 09:40 PM

How to resolve the algorithm Faulhaber's triangle step by step in the Ruby programming language

Table of Contents

Problem Statement

Named after Johann Faulhaber, the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula:

where

B

n

{\displaystyle B_{n}}

is the nth-Bernoulli number.

The first 5 rows of Faulhaber's triangle, are:

Using the third row of the triangle, we have:

k

1

n

k

2

=

1 6

n +

1 2

n

2

1 3

n

3

{\displaystyle \sum _{k=1}^{n}k^{2}={1 \over 6}n+{1 \over 2}n^{2}+{1 \over 3}n^{3}}

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Faulhaber's triangle step by step in the Ruby programming language

The provided Ruby code implements a program to calculate and display the coefficients of Faulhaber's triangle up to a specified order. Faulhaber's triangle is a triangular array of numbers that represent the sums of powers of natural numbers. It is useful for approximating certain mathematical functions.

Here's a breakdown of the code:

  1. Frac Class:

    • The Frac class represents rational numbers as fractions. It has instance variables @num and @denom to store the numerator and denominator, respectively.
    • It provides methods for arithmetic operations like addition (+), subtraction (-), and multiplication (*).
    • It also defines a to_s method to convert the fraction to a string representation.
  2. Constants:

    • FRAC_ZERO and FRAC_ONE are constants representing fractions 0/1 and 1/1, respectively.
  3. Bernoulli Function:

    • The bernoulli function calculates the n-th Bernoulli number using the explicit formula. It first initializes an array a of size n+1, with the first element set to FRAC_ZERO.
    • It then iterates through each element m in the array and calculates a[m] based on the formula:
      a[m] = 1 / (m + 1) - Σ{j=1..m} a[j - 1] * j / 1
      
    • The result of the function is a[0] if n is not equal to 1, or -a[0] if n is equal to 1.
  4. Binomial Function:

    • The binomial function calculates the binomial coefficient C(n, k) using the formula:
      C(n, k) = n! / (k! * (n - k)!)
      
    • It does this by computing the numerator and denominator separately and performing integer division.
  5. Faulhaber Triangle Function:

    • The faulhaberTriangle function calculates the coefficients of Faulhaber's triangle up to a given order p. It initializes an array coeffs of size p+1 with the first element set to FRAC_ZERO.
    • It then iterates through each element j from 0 to p and calculates the coefficients as follows:
      coeffs[p - j] = (-1)^j * (1 / (p + 1)) * C(p + 1, j) * B(j)
      
      where B(j) is the j-th Bernoulli number computed using the bernoulli function.
  6. Main Function:

    • The main function serves as the entry point of the program. It calculates and displays the coefficients of Faulhaber's triangle for orders 0 to 9.
  7. Program Flow:

    • The program starts by invoking the main function.
    • The main function calls the faulhaberTriangle function for each order from 0 to 9.
    • The coefficients for each order are printed using the to_s method of the Frac class.

Source code in the ruby programming language

class Frac
    attr_accessor:num
    attr_accessor:denom

    def initialize(n,d)
        if d == 0 then
            raise ArgumentError.new('d cannot be zero')
        end

        nn = n
        dd = d
        if nn == 0 then
            dd = 1
        elsif dd < 0 then
            nn = -nn
            dd = -dd
        end

        g = nn.abs.gcd(dd.abs)
        if g > 1 then
            nn = nn / g
            dd = dd / g
        end

        @num = nn
        @denom = dd
    end

    def to_s
        if self.denom == 1 then
            return self.num.to_s
        else
            return "%d/%d" % [self.num, self.denom]
        end
    end

    def -@
        return Frac.new(-self.num, self.denom)
    end

    def +(rhs)
        return Frac.new(self.num * rhs.denom + self.denom * rhs.num, rhs.denom * self.denom)
    end
    def -(rhs)
        return Frac.new(self.num * rhs.denom - self.denom * rhs.num, rhs.denom * self.denom)
    end

    def *(rhs)
        return Frac.new(self.num * rhs.num, rhs.denom * self.denom)
    end
end

FRAC_ZERO = Frac.new(0, 1)
FRAC_ONE  = Frac.new(1, 1)

def bernoulli(n)
    if n < 0 then
        raise ArgumentError.new('n cannot be negative')
    end

    a = Array.new(n + 1)
    a[0] = FRAC_ZERO

    for m in 0 .. n do
        a[m] = Frac.new(1, m + 1)
        m.downto(1) do |j|
            a[j - 1] = (a[j - 1] - a[j]) * Frac.new(j, 1)
        end
    end

    if n != 1 then
        return a[0]
    end
    return -a[0]
end

def binomial(n, k)
    if n < 0 then
        raise ArgumentError.new('n cannot be negative')
    end
    if k < 0 then
        raise ArgumentError.new('k cannot be negative')
    end
    if n < k then
        raise ArgumentError.new('n cannot be less than k')
    end

    if n == 0 or k == 0 then
        return 1
    end

    num = 1
    for i in k + 1 .. n do
        num = num * i
    end

    den = 1
    for i in 2 .. n - k do
        den = den * i
    end

    return num / den
end

def faulhaberTriangle(p)
    coeffs = Array.new(p + 1)
    coeffs[0] = FRAC_ZERO
    q = Frac.new(1, p + 1)
    sign = -1
    for j in 0 .. p do
        sign = -sign
        coeffs[p - j] = q * Frac.new(sign, 1) * Frac.new(binomial(p + 1, j), 1) * bernoulli(j)
    end
    return coeffs
end

def main
    for i in 0 .. 9 do
        coeffs = faulhaberTriangle(i)
        coeffs.each do |coeff|
            print "%5s  " % [coeff]
        end
        puts
    end
end

main()


  

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