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

This Haskell code generates and prints the Bernoulli numbers up to a specified index (60 by default). Bernoulli numbers are a sequence of rational numbers that occur naturally in many mathematical contexts.

Here's a breakdown of the code:

1. Importing Modules:

   import Data.Ratio
   import System.Environment

   These modules provide functions for working with rational numbers and for interacting with the command line.

2. main Function:

   main = getArgs >>= printM . defaultArg
    where
      defaultArg as =
        if null as
          then 60
          else read (head as)

   The main function is the entry point of the program. It uses the getArgs function to retrieve command-line arguments. If no arguments are provided, it defaults to generating the first 60 Bernoulli numbers. If arguments are provided, it reads the first argument as an integer and uses that as the upper bound for generating the numbers.

3. printM Function:

   printM m =
    mapM_ (putStrLn . printP) .
    takeWhile ((<= m) . fst) . filter (\(_, b) -> b /= 0 % 1) . zip [0 ..] $
    bernoullis

   The printM function takes a rational number m and prints the Bernoulli numbers up to that index. It applies the printP function to each Bernoulli number and uses mapM_ to print them one by one.

4. printP Function:

   printP (i, r) =
    "B(" ++ show i ++ ") = " ++ show (numerator r) ++ "/" ++ show (denominator r)

   The printP function takes a pair of a number i and a rational number r and prints it in a user-friendly format. It displays the Bernoulli number B(i) as a fraction with numerator and denominator.

5. bernoullis Function:

   bernoullis = map head . iterate (ulli 1) . map berno $ enumFrom 0
    where
      berno i = 1 % (i + 1)
      ulli _ [_] = []
      ulli i (x:y:xs) = (i % 1) * (x - y) : ulli (i + 1) (y : xs)

   The bernoullis function generates the Bernoulli numbers using a recursive approach. It starts with the initial Bernoulli number B(0) = 1 and then calculates subsequent numbers using a recurrence relation. The berno function provides the initial Bernoulli numbers, and the ulli function calculates the differences between consecutive Bernoulli numbers.

6. Improved Bernoulli Number Generation (with faulhaber):

   This version of the code provides an alternative method for generating Bernoulli numbers using the Faulhaber triangle. It defines the bernouillis function as follows:

   bernouillis :: Integer -> [Rational]
   bernouillis =
    fmap head
      . tail
      . scanl faulhaber []
      . enumFromTo 0

   The faulhaber function constructs the Faulhaber triangle and uses it to generate Bernoulli numbers. It iteratively calculates the differences between consecutive rows of the triangle.

7. faulhaber Function:

   faulhaber :: [Ratio Integer] -> Integer -> [Ratio Integer]
   faulhaber rs n =
    (:) =<< (-) 1 . sum $
      zipWith ((*) . (n %)) [2 ..] rs

   The faulhaber function takes a list of rational numbers rs and an integer n and generates the next row of the Faulhaber triangle. It multiplies each element of rs by n % and subtracts the sum from 1.

8. Testing and Formatting:

   The code includes a main function for testing and formatting the output. It generates the first 60 Bernoulli numbers, calculates their numerators' lengths, and prints the results in a table format.

   The fTable function is used for formatting the table, while showRatio and rjust are helper functions for displaying rational numbers and right-justifying strings.

Overall, this code demonstrates two different approaches for generating Bernoulli numbers in Haskell and provides a user-friendly interface for viewing the results.

import Data.Ratio
import System.Environment

main = getArgs >>= printM . defaultArg
  where
    defaultArg as =
      if null as
        then 60
        else read (head as)

printM m =
  mapM_ (putStrLn . printP) .
  takeWhile ((<= m) . fst) . filter (\(_, b) -> b /= 0 % 1) . zip [0 ..] $
  bernoullis

printP (i, r) =
  "B(" ++ show i ++ ") = " ++ show (numerator r) ++ "/" ++ show (denominator r)

bernoullis = map head . iterate (ulli 1) . map berno $ enumFrom 0
  where
    berno i = 1 % (i + 1)
    ulli _ [_] = []
    ulli i (x:y:xs) = (i % 1) * (x - y) : ulli (i + 1) (y : xs)


import Data.Bool (bool)
import Data.Ratio (Ratio, denominator, numerator, (%))

-------------------- BERNOULLI NUMBERS -------------------

bernouillis :: Integer -> [Rational]
bernouillis =
  fmap head
    . tail
    . scanl faulhaber []
    . enumFromTo 0

faulhaber :: [Ratio Integer] -> Integer -> [Ratio Integer]
faulhaber rs n =
  (:) =<< (-) 1 . sum $
    zipWith ((*) . (n %)) [2 ..] rs

--------------------------- TEST -------------------------
main :: IO ()
main = do
  let xs = bernouillis 60
      w = length $ show (numerator (last xs))
  putStrLn $
    fTable
      "Bernouillis from Faulhaber triangle:\n"
      (show . fst)
      (showRatio w . snd)
      id
      (filter ((0 /=) . snd) $ zip [0 ..] xs)

------------------------ FORMATTING ----------------------
fTable ::
  String ->
  (a -> String) ->
  (b -> String) ->
  (a -> b) ->
  [a] ->
  String
fTable s xShow fxShow f xs =
  let w = maximum (length . xShow <$> xs)
   in unlines $
        s :
        fmap
          ( ((<>) . rjust w ' ' . xShow)
              <*> ((" -> " <>) . fxShow . f)
          )
          xs

showRatio :: Int -> Rational -> String
showRatio w r =
  let d = denominator r
   in rjust w ' ' $ show (numerator r)
        <> bool [] (" / " <> show d) (1 /= d)

rjust :: Int -> a -> [a] -> [a]
rjust n c = drop . length <*> (replicate n c <>)