Cálculo de pi mediante el método de Newton

PorJosé A. Alonso 24-febrero-201725-abril-2021

El método de Newton para el cálculo de pi se basa en la relación

y en el desarrollo del arco seno

de donde se obtiene la fórmula

La primeras aproximaciones son

   a(0) = 6*(1/2)                               = 3.0
   a(1) = 6*(1/2+1/(2*3*2^3))                   = 3.125
   a(2) = 6*(1/2+1/(2*3*2^3)+(1*3)/(2*4*5*2^5)) = 3.1390625

a(0) = 6*(1/2) = 3.0

a(1) = 6*(1/2+1/(2*3*2^3)) = 3.125

a(2) = 6*(1/2+1/(2*3*2^3)+(1*3)/(2*4*5*2^5)) = 3.1390625

Definir las funciones

   aproximacionPi :: Int -> Double
   grafica        :: [Int] -> IO ()

1 2	aproximacionPi :: Int -> Double grafica :: [Int] -> IO ()

tales que

(aproximacionPi n) es la n-ésima aproximación de pi con la fórmula de Newton. Por ejemplo,

     aproximacionPi 0   ==  3.0
     aproximacionPi 1   ==  3.125
     aproximacionPi 2   ==  3.1390625
     aproximacionPi 10  ==  3.1415926468755613
     aproximacionPi 21  ==  3.141592653589793
     pi                 ==  3.141592653589793

aproximacionPi 0 == 3.0

aproximacionPi 1 == 3.125

aproximacionPi 2 == 3.1390625

aproximacionPi 10 == 3.1415926468755613

aproximacionPi 21 == 3.141592653589793

pi == 3.141592653589793

(grafica xs) dibuja la gráfica de las k-ésimas aproximaciones de pi donde k toma los valores de la lista xs. Por ejemplo, (grafica [1..30]) dibuja

Nota: Este ejercicio ha sido propuesto por Manuel Herrera.

Soluciones

import Graphics.Gnuplot.Simple

-- 1ª definición
-- =============

aproximacionPi :: Int -> Double
aproximacionPi n = 6 * arcsinX
  where arcsinX = 0.5 + sum (take n factoresN)

factoresN :: [Double]
factoresN = zipWith (*) (potenciasK 3) fraccionesPI

potenciasK :: Double -> [Double]
potenciasK k = (0.5**k)/k : potenciasK (k+2)

fraccionesPI :: [Double]
fraccionesPI =
  scanl (*) (1/2) (tail (zipWith (/) [1,3..] [2,4..]))

-- 2ª definición
-- =============

aproximacionPi2 :: Int -> Double
aproximacionPi2 n = 6 * (serie !! n)

serie :: [Double]
serie = scanl1 (+) (zipWith (/)
                            (map fromIntegral numeradores)
                            (map fromIntegral denominadores))
  where numeradores    = 1 : scanl1 (*) [1,3..]
        denominadores  = zipWith (*) denominadores1 denominadores2
        denominadores1 = 2 : scanl1 (*) [2,4..]
        denominadores2 = 1 : [n * 2^n | n <- [3,5..]]

grafica :: [Int] -> IO ()
grafica xs = 
    plotList [Key Nothing]
             [(k,aproximacionPi k) | k <- xs]

import Graphics.Gnuplot.Simple

-- 1ª definición

-- =============

aproximacionPi :: Int -> Double

aproximacionPi n = 6 * arcsinX

where arcsinX = 0.5 + sum (take n factoresN)

factoresN :: [Double]

factoresN = zipWith (*) (potenciasK 3) fraccionesPI

potenciasK :: Double -> [Double]

potenciasK k = (0.5**k)/k : potenciasK (k+2)

fraccionesPI :: [Double]

fraccionesPI =

scanl (*) (1/2) (tail (zipWith (/) [1,3..] [2,4..]))

-- 2ª definición

-- =============

aproximacionPi2 :: Int -> Double

aproximacionPi2 n = 6 * (serie !! n)

serie :: [Double]

serie = scanl1 (+) (zipWith (/)

(map fromIntegral numeradores)

(map fromIntegral denominadores))

where numeradores = 1 : scanl1 (*) [1,3..]

denominadores = zipWith (*) denominadores1 denominadores2

denominadores1 = 2 : scanl1 (*) [2,4..]

denominadores2 = 1 : [n * 2^n | n <- [3,5..]]

grafica :: [Int] -> IO ()

grafica xs =

plotList [Key Nothing]

[(k,aproximacionPi k) | k <- xs]

9 Comentarios

aproximacionPi :: Int -> Double
aproximacionPi n = 6*aux (1/2) (1/(2*3*2^3)) 0 3
 where aux  a b c d | c == n = a
                    | otherwise = aux (a+b) (b*d^2/((d+1)*(2+d)*2^2)) (c+1) (d+2)
                               
grafica        :: [Int] -> IO ()
grafica xs = plotList [] a
     where a = [aproximacionPi n | n <- xs]

aproximacionPi :: Int -> Double

aproximacionPi n = 6*aux (1/2) (1/(2*3*2^3)) 0 3

where aux a b c d | c == n = a

| otherwise = aux (a+b) (b*d^2/((d+1)*(2+d)*2^2)) (c+1) (d+2)

grafica :: [Int] -> IO ()

grafica xs = plotList [] a

where a = [aproximacionPi n | n <- xs]

Responder

aproximacionPi :: Int -> Double
aproximacionPi n = 6 * aux n (1/2)
 where aux 0 x = x
       aux a x = (x * (product $ take (a-1) xs)/(product $ take (a-1) ys) *
                  (x^(f a)/(f a))) + aux (a-1) x
       xs = [3,5..]
       ys = [4,6..]
       f 0 = 0
       f b = head $ drop b [1,3..]

grafica :: [Int] -> IO ()
grafica xs = plotLists [Key Nothing]
              [[(x,aproximacionPi x) | x <- xs]]

aproximacionPi :: Int -> Double

aproximacionPi n = 6 * aux n (1/2)

where aux 0 x = x

aux a x = (x * (product $ take (a-1) xs)/(product $ take (a-1) ys) *

(x^(f a)/(f a))) + aux (a-1) x

xs = [3,5..]

ys = [4,6..]

f 0 = 0

f b = head $ drop b [1,3..]

grafica :: [Int] -> IO ()

grafica xs = plotLists [Key Nothing]

[[(x,aproximacionPi x) | x <- xs]]

Responder

aproximacionPi :: Int -> Double
aproximacionPi n = 6 * (1/2 + aux 1 2 3 n)
                   where
                    aux _ _ _ 0 = 0
                    aux x z y n = x/(z*y*(2**y)) + aux (x*y) (z*(y+1)) (y+2) (n-1)

grafica :: [Int] -> IO ()
grafica xs = plotLists [Key Nothing]
              [[(x,aproximacionPiManu x) | x <- xs]]

aproximacionPi :: Int -> Double

aproximacionPi n = 6 * (1/2 + aux 1 2 3 n)

where

aux _ _ _ 0 = 0

aux x z y n = x/(z*y*(2**y)) + aux (x*y) (z*(y+1)) (y+2) (n-1)

grafica :: [Int] -> IO ()

grafica xs = plotLists [Key Nothing]

[[(x,aproximacionPiManu x) | x <- xs]]

Responder

import Graphics.Gnuplot.Simple (plotLists)
import Data.List (zipWith)

aproximacionPi4 :: Int -> Double
aproximacionPi4 n = 6 * (aux (take n [1,3..]) (take n [2,4..]))
  where
    aux is ps = sum $
      zipWith (*) (zipWith (/) (scanl (*) 1 is)
                               (scanl (*) 1 ps))
                  [1/(2**k*k) | k <- [1,3..]]      
      

grafica4 :: [Int] -> IO ()
grafica4 xs = do
  plotLists [] [[(x, aproximacionPi4 x) | x <- xs]]

import Graphics.Gnuplot.Simple (plotLists)

import Data.List (zipWith)

aproximacionPi4 :: Int -> Double

aproximacionPi4 n = 6 * (aux (take n [1,3..]) (take n [2,4..]))

where

aux is ps = sum $

zipWith (*) (zipWith (/) (scanl (*) 1 is)

(scanl (*) 1 ps))

[1/(2**k*k) | k <- [1,3..]]

grafica4 :: [Int] -> IO ()

grafica4 xs = do

plotLists [] [[(x, aproximacionPi4 x) | x <- xs]]

Responder

sucesion :: Double -> [Double]
sucesion x = [x**n / n | n <- [1,3..]]

sucesion2 :: Int -> [Double]
sucesion2 1 = [1]
sucesion2 x = (product (take (x-1) [1,3..])) / (product (take (x-1) [2,4..])) : sucesion2 (x-1)
  
productoPar :: (Double,Double) -> Double
productoPar (a,b) = a*b

aproximacionPi :: Int -> Double
aproximacionPi x  = 6 * sum (map (productoPar) (zip (sucesion 0.5) (reverse (sucesion2 (x+1)))))

grafica        :: [Int] -> IO ()
grafica xs = plotList [] x
            where x = [aproximacionPi k | k <- xs]

sucesion :: Double -> [Double]

sucesion x = [x**n / n | n <- [1,3..]]

sucesion2 :: Int -> [Double]

sucesion2 1 = [1]

sucesion2 x = (product (take (x-1) [1,3..])) / (product (take (x-1) [2,4..])) : sucesion2 (x-1)

productoPar :: (Double,Double) -> Double

productoPar (a,b) = a*b

aproximacionPi :: Int -> Double

aproximacionPi x = 6 * sum (map (productoPar) (zip (sucesion 0.5) (reverse (sucesion2 (x+1)))))

grafica :: [Int] -> IO ()

grafica xs = plotList [] x

where x = [aproximacionPi k | k <- xs]

Responder

aproximacionPi :: Int -> Double
aproximacionPi n = 6*(auxSuma sucesion 0)
  where auxSuma (x:xs) k | k == n = (fromRational x)
                         | otherwise = (fromRational x) + (auxSuma xs (k+1))

sucesion :: [Rational]
sucesion = [a%b | (a,b) <- zip sucesion1 sucesion2]
  where sucesion1 = 1:aux1 [1,3..] []
          where aux1 (x:xs) ts = product (x:ts): aux1 xs (x:ts)
        sucesion2 = aux2 [1,3..] (1:[2,4..]) []
          where aux2 (x:xs) (y:ys) ts = (2^x)*x*(product (y:ts)): (aux2 xs ys (y:ts))

aproximacionPi :: Int -> Double

aproximacionPi n = 6*(auxSuma sucesion 0)

where auxSuma (x:xs) k | k == n = (fromRational x)

| otherwise = (fromRational x) + (auxSuma xs (k+1))

sucesion :: [Rational]

sucesion = [a%b | (a,b) <- zip sucesion1 sucesion2]

where sucesion1 = 1:aux1 [1,3..] []

where aux1 (x:xs) ts = product (x:ts): aux1 xs (x:ts)

sucesion2 = aux2 [1,3..] (1:[2,4..]) []

where aux2 (x:xs) (y:ys) ts = (2^x)*x*(product (y:ts)): (aux2 xs ys (y:ts))

Responder

aproximacionPi :: Int -> Double
aproximacionPi n = 6*(aux (0.5) [3,5..] (factores n)) +3

factores n = take n [(product (take s [1,3..]))/(product (take s [2,4..]))| s <- [1..]]


aux n (x:xs) (y:ys) = y*(n**x)/x + (aux n xs ys)
aux n _ [] = 0

grafica        :: [Int] -> IO ()
grafica xs = plotLists [Key Nothing]
             [[(x,aproximacionPi x) | x <- xs]]

aproximacionPi :: Int -> Double

aproximacionPi n = 6*(aux (0.5) [3,5..] (factores n)) +3

factores n = take n [(product (take s [1,3..]))/(product (take s [2,4..]))| s <- [1..]]

aux n (x:xs) (y:ys) = y*(n**x)/x + (aux n xs ys)

aux n _ [] = 0

grafica :: [Int] -> IO ()

grafica xs = plotLists [Key Nothing]

[[(x,aproximacionPi x) | x <- xs]]

Responder

import Graphics.Gnuplot.Simple

aproximacionPi :: Int -> Double
aproximacionPi n = 6 * ((1/2) + aux 1 2 3)
               where fI a = fromIntegral a
                     aux x y z | n+1 == y `div` 2 = 0
                               | otherwise = product [1,3..fI x] / product [2,4..fI y]                                               * (1/2^z /fI z) + aux (x+2) (y+2) (z+2)

grafica :: [Int] -> IO ()
grafica xs = plotList [Key Nothing] (map aproximacionPi xs)

import Graphics.Gnuplot.Simple

aproximacionPi :: Int -> Double

aproximacionPi n = 6 * ((1/2) + aux 1 2 3)

where fI a = fromIntegral a

aux x y z | n+1 == y `div` 2 = 0

| otherwise = product [1,3..fI x] / product [2,4..fI y] * (1/2^z /fI z) + aux (x+2) (y+2) (z+2)

grafica :: [Int] -> IO ()

grafica xs = plotList [Key Nothing] (map aproximacionPi xs)

Responder

sucArcSen :: Double -> [Double]
sucArcSen x = [((product (take n xs)) / (product (take n ys))) * ((x**m) / (m)) |
                   (n,m) <- zip [1..] [3,5..]]
  where xs = [1,3..]
        ys = [2,4..]

arcSen :: Int -> Double -> Double
arcSen n x = x + (sum $ take n (sucArcSen x))

aproximaPiNewton :: Int -> Double
aproximaPiNewton n = 6 * arcSen n 0.5

grafica :: [Int] -> IO ()
grafica xs = plotList [Key Nothing] [(x, aproximaPiNewton x) | x <- xs]

sucArcSen :: Double -> [Double]

sucArcSen x = [((product (take n xs)) / (product (take n ys))) * ((x**m) / (m)) |

(n,m) <- zip [1..] [3,5..]]

where xs = [1,3..]

ys = [2,4..]

arcSen :: Int -> Double -> Double

arcSen n x = x + (sum $ take n (sucArcSen x))

aproximaPiNewton :: Int -> Double

aproximaPiNewton n = 6 * arcSen n 0.5

grafica :: [Int] -> IO ()

grafica xs = plotList [Key Nothing] [(x, aproximaPiNewton x) | x <- xs]

Responder

Cálculo de pi mediante el método de Newton

Soluciones

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