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

The Ruby code implements Faulhaber's formula, which is used to find the sum of powers of natural numbers.

The code defines three methods: binomial, bernoulli and faulhaber.

The binomial method calculates the binomial coefficient using the formula (\binom{n}{k}) = (\frac{n!}{k!(n-k)!}). It takes two parameters, n and k, and returns the binomial coefficient.

The bernoulli method calculates the Bernoulli number using the formula (B_n = \frac{1}{n+1} \sum_{k=0}^n (-1)^k \binom{n+1}{k} B_k). It takes one parameter, n, and returns the Bernoulli number.

The faulhaber method calculates the sum of powers of natural numbers using Faulhaber's formula. It takes one parameter, p, and prints the formula for the sum of p-th powers of natural numbers.

The main method calls the faulhaber method for p from 0 to 9 and prints the results.

Here is a breakdown of the code:

def binomial(n,k)
   if n < 0 or k < 0 or n < k then
       return -1
   end
   if n == 0 or k == 0 then
       return 1
   end

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

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

   return num / denom
end

The binomial method calculates the binomial coefficient (\binom{n}{k}). It handles the cases when n or k are negative or when n is less than k. It then calculates the numerator and denominator of the binomial coefficient and returns the result.

def bernoulli(n)
   if n < 0 then
       raise "n cannot be less than zero"
   end

   a = Array.new(16)
   for m in 0 .. n do
       a[m] = Rational(1, m + 1)
       for j in m.downto(1) do
           a[j-1] = (a[j-1] - a[j]) * Rational(j)
       end
   end

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

The bernoulli method calculates the Bernoulli number (B_n). It first checks if n is negative and raises an exception if it is. It then initializes an array a to store the Bernoulli numbers. It then iterates over the values of m from 0 to n and calculates the Bernoulli number for each value of m. Finally, it returns the Bernoulli number for n.

def faulhaber(p)
   print("%d : " % [p])
   q = Rational(1, p + 1)
   sign = -1
   for j in 0 .. p do
       sign = -1 * sign
       coeff = q * Rational(sign) * Rational(binomial(p+1, j)) * bernoulli(j)
       if coeff == 0 then
           next
       end
       if j == 0 then
           if coeff != 1 then
               if coeff == -1 then
                   print "-"
               else
                   print coeff
               end
           end
       else
           if coeff == 1 then
               print " + "
           elsif coeff == -1 then
               print " - "
           elsif 0 < coeff then
               print " + "
               print coeff
           else
               print " - "
               print -coeff
           end
       end
       pwr = p + 1 - j
       if pwr > 1 then
           print "n^%d" % [pwr]
       else
           print "n"
       end
   end
   print "\n"
end

The faulhaber method calculates the sum of powers of natural numbers using Faulhaber's formula. It takes one parameter, p, and prints the formula for the sum of p-th powers of natural numbers.

def main
   for i in 0 .. 9 do
       faulhaber(i)
   end
end

main()

The main method calls the faulhaber method for p from 0 to 9 and prints the results.

def binomial(n,k)
    if n < 0 or k < 0 or n < k then
        return -1
    end
    if n == 0 or k == 0 then
        return 1
    end

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

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

    return num / denom
end

def bernoulli(n)
    if n < 0 then
        raise "n cannot be less than zero"
    end

    a = Array.new(16)
    for m in 0 .. n do
        a[m] = Rational(1, m + 1)
        for j in m.downto(1) do
            a[j-1] = (a[j-1] - a[j]) * Rational(j)
        end
    end

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

def faulhaber(p)
    print("%d : " % [p])
    q = Rational(1, p + 1)
    sign = -1
    for j in 0 .. p do
        sign = -1 * sign
        coeff = q * Rational(sign) * Rational(binomial(p+1, j)) * bernoulli(j)
        if coeff == 0 then
            next
        end
        if j == 0 then
            if coeff != 1 then
                if coeff == -1 then
                    print "-"
                else
                    print coeff
                end
            end
        else
            if coeff == 1 then
                print " + "
            elsif coeff == -1 then
                print " - "
            elsif 0 < coeff then
                print " + "
                print coeff
            else
                print " - "
                print -coeff
            end
        end
        pwr = p + 1 - j
        if pwr > 1 then
            print "n^%d" % [pwr]
        else
            print "n"
        end
    end
    print "\n"
end

def main
    for i in 0 .. 9 do
        faulhaber(i)
    end
end

main()