Cálculo de pi mediante la serie de Nilakantha

Una serie infinita para el cálculo de pi, publicada por Nilakantha en el siglo XV, es

Definir las funciones

   aproximacionPi :: Int -> Double
   tabla          :: FilePath -> [Int] -> IO ()

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

tales que

(aproximacionPi n) es la n-ésima aproximación de pi obtenido sumando los n primeros términos de la serie de Nilakantha. Por ejemplo,

     aproximacionPi 0        ==  3.0
     aproximacionPi 1        ==  3.1666666666666665
     aproximacionPi 2        ==  3.1333333333333333
     aproximacionPi 3        ==  3.145238095238095
     aproximacionPi 4        ==  3.1396825396825396
     aproximacionPi 5        ==  3.1427128427128426
     aproximacionPi 10       ==  3.1414067184965018
     aproximacionPi 100      ==  3.1415924109719824
     aproximacionPi 1000     ==  3.141592653340544
     aproximacionPi 10000    ==  3.141592653589538
     aproximacionPi 100000   ==  3.1415926535897865
     aproximacionPi 1000000  ==  3.141592653589787
     pi                      ==  3.141592653589793

aproximacionPi 0 == 3.0

aproximacionPi 1 == 3.1666666666666665

aproximacionPi 2 == 3.1333333333333333

aproximacionPi 3 == 3.145238095238095

aproximacionPi 4 == 3.1396825396825396

aproximacionPi 5 == 3.1427128427128426

aproximacionPi 10 == 3.1414067184965018

aproximacionPi 100 == 3.1415924109719824

aproximacionPi 1000 == 3.141592653340544

aproximacionPi 10000 == 3.141592653589538

aproximacionPi 100000 == 3.1415926535897865

aproximacionPi 1000000 == 3.141592653589787

pi == 3.141592653589793

(tabla f ns) escribe en el fichero f las n-ésimas aproximaciones de pi, donde n toma los valores de la lista ns, junto con sus errores. Por ejemplo, al evaluar la expresión

     tabla "AproximacionesPi.txt" [0,10..100]

1	tabla "AproximacionesPi.txt" [0,10..100]

hace que el contenido del fichero «AproximacionesPi.txt» sea

+------+----------------+----------------+
| n    | Aproximación   | Error          |
+------+----------------+----------------+
|    0 | 3.000000000000 | 0.141592653590 |
|   10 | 3.141406718497 | 0.000185935093 |
|   20 | 3.141565734659 | 0.000026918931 |
|   30 | 3.141584272675 | 0.000008380915 |
|   40 | 3.141589028941 | 0.000003624649 |
|   50 | 3.141590769850 | 0.000001883740 |
|   60 | 3.141591552546 | 0.000001101044 |
|   70 | 3.141591955265 | 0.000000698325 |
|   80 | 3.141592183260 | 0.000000470330 |
|   90 | 3.141592321886 | 0.000000331704 |
|  100 | 3.141592410972 | 0.000000242618 |
+------+----------------+----------------+

+------+----------------+----------------+

| n | Aproximación | Error |

+------+----------------+----------------+

| 0 | 3.000000000000 | 0.141592653590 |

| 10 | 3.141406718497 | 0.000185935093 |

| 20 | 3.141565734659 | 0.000026918931 |

| 30 | 3.141584272675 | 0.000008380915 |

| 40 | 3.141589028941 | 0.000003624649 |

| 50 | 3.141590769850 | 0.000001883740 |

| 60 | 3.141591552546 | 0.000001101044 |

| 70 | 3.141591955265 | 0.000000698325 |

| 80 | 3.141592183260 | 0.000000470330 |

| 90 | 3.141592321886 | 0.000000331704 |

| 100 | 3.141592410972 | 0.000000242618 |

+------+----------------+----------------+

al evaluar la expresión

     tabla "AproximacionesPi.txt" [0,500..5000]

1	tabla "AproximacionesPi.txt" [0,500..5000]

hace que el contenido del fichero «AproximacionesPi.txt» sea

+------+----------------+----------------+
| n    | Aproximación   | Error          |
+------+----------------+----------------+
|    0 | 3.000000000000 | 0.141592653590 |
|  500 | 3.141592651602 | 0.000000001988 |
| 1000 | 3.141592653341 | 0.000000000249 |
| 1500 | 3.141592653516 | 0.000000000074 |
| 2000 | 3.141592653559 | 0.000000000031 |
| 2500 | 3.141592653574 | 0.000000000016 |
| 3000 | 3.141592653581 | 0.000000000009 |
| 3500 | 3.141592653584 | 0.000000000006 |
| 4000 | 3.141592653586 | 0.000000000004 |
| 4500 | 3.141592653587 | 0.000000000003 |
| 5000 | 3.141592653588 | 0.000000000002 |
+------+----------------+----------------+

+------+----------------+----------------+

| n | Aproximación | Error |

+------+----------------+----------------+

| 0 | 3.000000000000 | 0.141592653590 |

| 500 | 3.141592651602 | 0.000000001988 |

| 1000 | 3.141592653341 | 0.000000000249 |

| 1500 | 3.141592653516 | 0.000000000074 |

| 2000 | 3.141592653559 | 0.000000000031 |

| 2500 | 3.141592653574 | 0.000000000016 |

| 3000 | 3.141592653581 | 0.000000000009 |

| 3500 | 3.141592653584 | 0.000000000006 |

| 4000 | 3.141592653586 | 0.000000000004 |

| 4500 | 3.141592653587 | 0.000000000003 |

| 5000 | 3.141592653588 | 0.000000000002 |

+------+----------------+----------------+

Soluciones

import Text.Printf

-- 1ª solución
-- ===========

aproximacionPi :: Int -> Double
aproximacionPi n = serieNilakantha !! n

serieNilakantha :: [Double]
serieNilakantha = scanl1 (+) terminosNilakantha

terminosNilakantha :: [Double]
terminosNilakantha = zipWith (/) numeradores denominadores
  where numeradores   = 3 : cycle [4,-4]
        denominadores = 1 : [n*(n+1)*(n+2) | n <- [2,4..]]

-- 2ª solución
-- ===========

aproximacionPi2 :: Int -> Double
aproximacionPi2 = aux 3 2 1
  where aux x _ _ 0 = x
        aux x y z m =
          aux (x+4/product[y..y+2]*z) (y+2) (negate z) (m-1)

-- 3ª solución
-- ===========

aproximacionPi3 :: Int -> Double
aproximacionPi3 x =
  3 + sum [(((-1)**(n+1))*4)/(2*n*(2*n+1)*(2*n+2))
          | n <- [1..fromIntegral x]]


-- Comparación de eficiencia
-- =========================

--    λ> aproximacionPi (10^6)
--    3.141592653589787
--    (1.35 secs, 729,373,160 bytes)
--    λ> aproximacionPi2 (10^6)
--    3.141592653589787
--    (2.96 secs, 2,161,766,096 bytes)
--    λ> aproximacionPi3 (10^6)
--    3.1415926535897913
--    (2.02 secs, 1,121,372,536 bytes)

-- Definicioń de tabla
-- ===================

tabla :: FilePath -> [Int] -> IO ()
tabla f ns = writeFile f (tablaAux ns)

tablaAux :: [Int] -> String
tablaAux ns =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %4d | %.12f | %.12f |\n" n a e
            | n <- ns
            , let a = aproximacionPi n
            , let e = abs (pi - a)]
  ++ linea

linea :: String
linea = "+------+----------------+----------------+\n"

cabecera :: String
cabecera = "| n    | Aproximación   | Error          |\n"

import Text.Printf

-- 1ª solución

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

aproximacionPi :: Int -> Double

aproximacionPi n = serieNilakantha !! n

serieNilakantha :: [Double]

serieNilakantha = scanl1 (+) terminosNilakantha

terminosNilakantha :: [Double]

terminosNilakantha = zipWith (/) numeradores denominadores

where numeradores = 3 : cycle [4,-4]

denominadores = 1 : [n*(n+1)*(n+2) | n <- [2,4..]]

-- 2ª solución

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

aproximacionPi2 :: Int -> Double

aproximacionPi2 = aux 3 2 1

where aux x _ _ 0 = x

aux x y z m =

aux (x+4/product[y..y+2]*z) (y+2) (negate z) (m-1)

-- 3ª solución

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

aproximacionPi3 :: Int -> Double

aproximacionPi3 x =

3 + sum [(((-1)**(n+1))*4)/(2*n*(2*n+1)*(2*n+2))

| n <- [1..fromIntegral x]]

-- Comparación de eficiencia

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

-- λ> aproximacionPi (10^6)

-- 3.141592653589787

-- (1.35 secs, 729,373,160 bytes)

-- λ> aproximacionPi2 (10^6)

-- 3.141592653589787

-- (2.96 secs, 2,161,766,096 bytes)

-- λ> aproximacionPi3 (10^6)

-- 3.1415926535897913

-- (2.02 secs, 1,121,372,536 bytes)

-- Definicioń de tabla

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

tabla :: FilePath -> [Int] -> IO ()

tabla f ns = writeFile f (tablaAux ns)

tablaAux :: [Int] -> String

tablaAux ns =

linea

++ cabecera

++ linea

++ concat [printf "| %4d | %.12f | %.12f |\n" n a e

| n <- ns

, let a = aproximacionPi n

, let e = abs (pi - a)]

++ linea

linea :: String

linea = "+------+----------------+----------------+\n"

cabecera :: String

cabecera = "| n | Aproximación | Error |\n"

Pensamiento

Bueno es saber que los vasos
nos sirven para beber;
lo malo es que no sabemos
para que sirve la sed.

Antonio Machado

5 Comentarios

import Text.Printf (printf)

aproximacionPi :: Int -> Double
aproximacionPi x =
  3 + sum [(((-1)**(n+1))*4)/(2*n*(2*n+1)*(2*n+2)) | n <- [1..fromIntegral x]]
 
tabla :: FilePath -> [Int] -> IO ()
tabla f ns = do
  writeFile f (tablaAux ns)
 
tablaAux :: [Int] -> String
tablaAux ns =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %4d | %.12f | %.12f |\n" n a e
            | n <- ns
            , let a = aproximacionPi n
            , let e = abs (pi - a)]
  ++ linea
 
linea :: String
linea = "+------+----------------+----------------+\n"
 
cabecera :: String
cabecera = "| n    | Aproximacion   | Error          |\n"

import Text.Printf (printf)

aproximacionPi :: Int -> Double

aproximacionPi x =

3 + sum [(((-1)**(n+1))*4)/(2*n*(2*n+1)*(2*n+2)) | n <- [1..fromIntegral x]]

tabla :: FilePath -> [Int] -> IO ()

tabla f ns = do

writeFile f (tablaAux ns)

tablaAux :: [Int] -> String

tablaAux ns =

linea

++ cabecera

++ linea

++ concat [printf "| %4d | %.12f | %.12f |\n" n a e

| n <- ns

, let a = aproximacionPi n

, let e = abs (pi - a)]

++ linea

linea :: String

linea = "+------+----------------+----------------+\n"

cabecera :: String

cabecera = "| n | Aproximacion | Error |\n"

Responder

import Text.Printf (printf)

aproximacionPi :: Int -> Double
aproximacionPi n = 3 + sum [(-1)**(k/2+1)*4/(k*(k+1)*(k+2))
                           | k <- [2,4..2*fromIntegral n]]
 
tabla :: FilePath -> [Int] -> IO ()
tabla f ns = do
  writeFile f (tablaAproxPiAux ns)
 
tablaAproxPiAux :: [Int] -> String
tablaAproxPiAux ns =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %2d | %.12f | %.12f |\n" n x y
            | n <- ns
            , let x = aproximacionPi n
            , let y = abs (pi - x)]
  ++ linea
 
linea, cabecera :: String
linea    = "+----+----------------+---------------+\n"
cabecera = "| n  | Aproximacion   | Error         |\n"

import Text.Printf (printf)

aproximacionPi :: Int -> Double

aproximacionPi n = 3 + sum [(-1)**(k/2+1)*4/(k*(k+1)*(k+2))

| k <- [2,4..2*fromIntegral n]]

tabla :: FilePath -> [Int] -> IO ()

tabla f ns = do

writeFile f (tablaAproxPiAux ns)

tablaAproxPiAux :: [Int] -> String

tablaAproxPiAux ns =

linea

++ cabecera

++ linea

++ concat [printf "| %2d | %.12f | %.12f |\n" n x y

| n <- ns

, let x = aproximacionPi n

, let y = abs (pi - x)]

++ linea

linea, cabecera :: String

linea = "+----+----------------+---------------+\n"

cabecera = "| n | Aproximacion | Error |\n"

Responder

import Text.Printf (printf)

aproximacionPi :: Int -> Double
aproximacionPi 0 = 3
aproximacionPi n =
  sum [((-1)^(s+1))*4/(k*(k+1)*(k+2))
      | s <- [1..n]
      , let k = 2*fromIntegral s]
  + 3
 
tabla :: FilePath -> [Int] -> IO ()
tabla f xs = do
  writeFile f (tablaAux xs)
 
tablaAux :: [Int] -> String
tablaAux xs =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %4d | %.12f | %.12f |\n" x y z
            | x <- xs
            , let y = aproximacionPi x
            , let z = abs (pi - y)]
  ++ linea
 
linea :: String
linea = "+------+----------------+----------------+\n"
 
cabecera :: String
cabecera = "| n    | Aproximacion   | Error          |\n"

import Text.Printf (printf)

aproximacionPi :: Int -> Double

aproximacionPi 0 = 3

aproximacionPi n =

sum [((-1)^(s+1))*4/(k*(k+1)*(k+2))

| s <- [1..n]

, let k = 2*fromIntegral s]

+ 3

tabla :: FilePath -> [Int] -> IO ()

tabla f xs = do

writeFile f (tablaAux xs)

tablaAux :: [Int] -> String

tablaAux xs =

linea

++ cabecera

++ linea

++ concat [printf "| %4d | %.12f | %.12f |\n" x y z

| x <- xs

, let y = aproximacionPi x

, let z = abs (pi - y)]

++ linea

linea :: String

linea = "+------+----------------+----------------+\n"

cabecera :: String

cabecera = "| n | Aproximacion | Error |\n"

Responder

import Text.Printf

aproximacionPi :: Int -> Double
aproximacionPi 0 = 3
aproximacionPi n = ((4*(-1)^(n+1))/((2*(d n))*(2*(d n)+1)*(2*(d n)+2))
                    + (aproximacionPi (n-1)))
  where d :: Int -> Double
        d x = fromIntegral x

tabla :: FilePath -> [Int] -> IO ()
tabla f ns = do
  writeFile f (tablaAproxPiAux ns)
 
tablaAproxPiAux :: [Int] -> String
tablaAproxPiAux ns =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %2d | %.12f | %.12f |n" n j k
            | n <- ns
            , let j = aproximacionPi n
            , let k = abs (pi - j)]
  ++ linea
 
linea, cabecera :: String
linea    = "+----+----------------+---------------+n"
cabecera = "| n  | Aproximacion   | Error         |n"

import Text.Printf

aproximacionPi :: Int -> Double

aproximacionPi 0 = 3

aproximacionPi n = ((4*(-1)^(n+1))/((2*(d n))*(2*(d n)+1)*(2*(d n)+2))

+ (aproximacionPi (n-1)))

where d :: Int -> Double

d x = fromIntegral x

tabla :: FilePath -> [Int] -> IO ()

tabla f ns = do

writeFile f (tablaAproxPiAux ns)

tablaAproxPiAux :: [Int] -> String

tablaAproxPiAux ns =

linea

++ cabecera

++ linea

++ concat [printf "| %2d | %.12f | %.12f |n" n j k

| n <- ns

, let j = aproximacionPi n

, let k = abs (pi - j)]

++ linea

linea, cabecera :: String

linea = "+----+----------------+---------------+n"

cabecera = "| n | Aproximacion | Error |n"

Responder

aproximacionPi :: Int -> Double
aproximacionPi n = 3 +  f (take n aux) 0
  where f [] _ = 0
        f (x:xs) 0 = x + f xs 1
        f (x:xs) 1 = -x + f xs 0
        aux = [ 4 / (x*(x+1)*(x+2)) | x <- [2,4..]] 

tabla          :: FilePath -> [Int] -> IO () 
tabla f ns = do
  writeFile f (tablaAux ns)

tablaAux :: [Int] -> String
tablaAux ns =
     linea
  ++ cabecera
  ++ linea
  ++ concat [printf "| %2d | %.12f | %.12f |\n" n x y
            | n <- ns
            , let x = (aproximacionPi n) 
            , let y = abs (pi -x)]
  ++ linea

linea :: String
linea    = "+------+----------------+----------------+\n"
cabecera :: String
cabecera = "| n    | Aproximacion   | Error          |\n"

aproximacionPi :: Int -> Double

aproximacionPi n = 3 + f (take n aux) 0

where f [] _ = 0

f (x:xs) 0 = x + f xs 1

f (x:xs) 1 = -x + f xs 0

aux = [ 4 / (x*(x+1)*(x+2)) | x <- [2,4..]]

tabla :: FilePath -> [Int] -> IO ()

tabla f ns = do

writeFile f (tablaAux ns)

tablaAux :: [Int] -> String

tablaAux ns =

linea

++ cabecera

++ linea

++ concat [printf "| %2d | %.12f | %.12f |\n" n x y

| n <- ns

, let x = (aproximacionPi n)

, let y = abs (pi -x)]

++ linea

linea :: String

linea = "+------+----------------+----------------+\n"

cabecera :: String

cabecera = "| n | Aproximacion | Error |\n"

Responder

Cálculo de pi mediante la serie de Nilakantha

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