Los polinomios se pueden representar mediante vectores usando la librer\u00eda Data.Array. En primer lugar, se define el tipo de los polinomios (con coeficientes de tipo a) mediante<\/p>\n
\n type Polinomio a = Array Int a\n<\/pre>\nComo ejemplos, definimos el polinomio<\/p>\n
\n ej_pol1 :: Array Int Int\n ej_pol1 = array (0,4) [(0,6),(1,2),(2,-5),(3,0),(4,7)]\n<\/pre>\nque representa a 6 + 2x – 5x^2 + 7x^4 y el polinomio<\/p>\n
\n ej_pol2 :: Array Int Double\n ej_pol2 = array (0,4) [(0,6.5),(1,2),(2,-5.2),(3,0),(4,7)]\n<\/pre>\nque representa a 6.5 + 2x – 5.2x^2 + 7x^4<\/p>\n
Definir la funci\u00f3n<\/p>\n
\n valor :: Num a => Polinomio a -> a -> a\n<\/pre>\ntal que (valor p b) es el valor del polinomio p en el punto b. Por ejemplo,<\/p>\n
\n valor ej_pol1 0 == 6\n valor ej_pol1 1 == 10\n valor ej_pol1 2 == 102\n valor ej_pol2 0 == 6.5\n valor ej_pol2 1 == 10.3\n valor ej_pol2 3 == 532.7\n length (show (valor (listArray (0,5*10^5) (repeat 1)) 2)) == 150516\n<\/pre>\nSoluciones<\/h4>\n\nimport Data.List (foldl')\nimport Data.Array (Array, (!), array, assocs, bounds, elems, listArray)\nimport Test.QuickCheck\n\ntype Polinomio a = Array Int a\n\nej_pol1 :: Array Int Int\nej_pol1 = array (0,4) [(1,2),(2,-5),(4,7),(0,6),(3,0)]\n\nej_pol2 :: Array Int Double\nej_pol2 = array (0,4) [(1,2),(2,-5.2),(4,7),(0,6.5),(3,0)]\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nvalor1 :: Num a => Polinomio a -> a -> a\nvalor1 p b = sum [(p!i)*b^i | i <- [0..n]]\n where (_,n) = bounds p\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nvalor2 :: Num a => Polinomio a -> a -> a\nvalor2 p b = sum [(p!i)*b^i | i <- [0..length p - 1]]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nvalor3 :: Num a => Polinomio a -> a -> a\nvalor3 p b = sum [v*b^i | (i,v) <- assocs p]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nvalor4 :: Num a => Polinomio a -> a -> a\nvalor4 = valorLista4 . elems\n\nvalorLista4 :: Num a => [a] -> a -> a\nvalorLista4 xs b =\n sum [(xs !! i) * b^i | i <- [0..length xs - 1]]\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nvalor5 :: Num a => Polinomio a -> a -> a\nvalor5 = valorLista5 . elems\n\nvalorLista5 :: Num a => [a] -> a -> a\nvalorLista5 [] _ = 0\nvalorLista5 (x:xs) b = x + b * valorLista5 xs b\n\n-- 6\u00aa soluci\u00f3n\n-- ===========\n\nvalor6 :: Num a => Polinomio a -> a -> a\nvalor6 = valorLista6 . elems\n\nvalorLista6 :: Num a => [a] -> a -> a\nvalorLista6 xs b = aux xs\n where aux [] = 0\n aux (y:ys) = y + b * aux ys\n\n-- 7\u00aa soluci\u00f3n\n-- ===========\n\nvalor7 :: Num a => Polinomio a -> a -> a\nvalor7 = valorLista7 . elems\n\nvalorLista7 :: Num a => [a] -> a -> a\nvalorLista7 xs b = foldr (\\y r -> y + b * r) 0 xs\n\n-- 8\u00aa soluci\u00f3n\n-- ===========\n\nvalor8 :: Num a => Polinomio a -> a -> a\nvalor8 = valorLista8 . elems\n\nvalorLista8 :: Num a => [a] -> a -> a\nvalorLista8 xs b = aux 0 (reverse xs)\n where aux r [] = r\n aux r (y:ys) = aux (y + r * b) ys\n\n-- 9\u00aa soluci\u00f3n\n-- ===========\n\nvalor9 :: Num a => Polinomio a -> a -> a\nvalor9 = valorLista9 . elems\n\nvalorLista9 :: Num a => [a] -> a -> a\nvalorLista9 xs b = aux 0 (reverse xs)\n where aux = foldl (\\ r y -> y + r * b)\n\n-- 10\u00aa soluci\u00f3n\n-- ============\n\nvalor10 :: Num a => Polinomio a -> a -> a\nvalor10 p b =\n foldl (\\ r y -> y + r * b) 0 (reverse (elems p))\n\n-- 11\u00aa soluci\u00f3n\n-- ============\n\nvalor11 :: Num a => Polinomio a -> a -> a\nvalor11 p b =\n foldl' (\\ r y -> y + r * b) 0 (reverse (elems p))\n\n-- 12\u00aa soluci\u00f3n\n-- ============\n\nvalor12 :: Num a => Polinomio a -> a -> a\nvalor12 p b =\n sum (zipWith (*) (elems p) (iterate (* b) 1))\n\n-- 13\u00aa soluci\u00f3n\n-- ============\n\nvalor13 :: Num a => Polinomio a -> a -> a\nvalor13 p b =\n foldl' (+) 0 (zipWith (*) (elems p) (iterate (* b) 1))\n\n-- Equivalencia de las definiciones\n-- ================================\n\n-- La propiedad es\nprop_valor :: [Integer] -> Integer -> Bool\nprop_valor xs b =\n all (== valor1 p b)\n [f p b | f <- [valor2,\n valor3,\n valor4,\n valor5,\n valor6,\n valor7,\n valor8,\n valor9,\n valor10,\n valor11,\n valor12,\n valor13]]\n where p = listArray (0, length xs - 1) xs\n\n-- La comprobaci\u00f3n es\n-- \u03bb> quickCheck prop_valor\n-- +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n-- \u03bb> length (show (valor1 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (7.62 secs, 2,953,933,864 bytes)\n-- \u03bb> length (show (valor2 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (8.26 secs, 2,953,933,264 bytes)\n-- \u03bb> length (show (valor3 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (7.49 secs, 2,954,733,184 bytes)\n-- \u03bb> length (show (valor4 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (84.80 secs, 2,956,333,712 bytes)\n-- \u03bb> length (show (valor5 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.34 secs, 1,307,347,416 bytes)\n-- \u03bb> length (show (valor6 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.26 secs, 1,308,114,752 bytes)\n-- \u03bb> length (show (valor7 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.21 secs, 1,296,843,456 bytes)\n-- \u03bb> length (show (valor8 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.28 secs, 1,309,591,744 bytes)\n-- \u03bb> length (show (valor9 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.27 secs, 1,299,191,672 bytes)\n-- \u03bb> length (show (valor10 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (1.30 secs, 1,299,191,432 bytes)\n-- \u03bb> length (show (valor11 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (0.23 secs, 1,287,654,752 bytes)\n-- \u03bb> length (show (valor12 (listArray (0,10^5) (repeat 1)) 2))\n-- 30104\n-- (0.75 secs, 1,309,506,968 bytes)\n<\/pre>\nEl c\u00f3digo se encuentra en GitHub<\/a>.<\/p>\nLa elaboraci\u00f3n de las soluciones se encuentran en el siguiente v\u00eddeo<\/p>\n