{"id":3009,"date":"2017-02-24T06:00:46","date_gmt":"2017-02-24T04:00:46","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3009"},"modified":"2021-04-25T16:14:01","modified_gmt":"2021-04-25T14:14:01","slug":"calculo-de-pi-mediante-el-metodo-de-newton-2017","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/calculo-de-pi-mediante-el-metodo-de-newton-2017\/","title":{"rendered":"C\u00e1lculo de pi mediante el m\u00e9todo de Newton"},"content":{"rendered":"<p>El m\u00e9todo de Newton para el c\u00e1lculo de pi se basa en la relaci\u00f3n<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_1.png?resize=234%2C97\" alt=\"Calculo_de_pi_mediante_el_metodo_de_Newton_1\" width=\"234\" height=\"97\" class=\"aligncenter size-full wp-image-3010\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_1.png?w=234&amp;ssl=1 234w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_1.png?resize=100%2C41&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_1.png?resize=150%2C62&amp;ssl=1 150w\" sizes=\"(max-width: 234px) 100vw, 234px\" data-recalc-dims=\"1\" \/><\/a><br \/>\ny en el desarrollo del arco seno<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png?resize=645%2C98\" alt=\"Calculo_de_pi_mediante_el_metodo_de_Newton_2\" width=\"645\" height=\"98\" class=\"aligncenter size-full wp-image-3011\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png?w=645&amp;ssl=1 645w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png?resize=300%2C45&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png?resize=100%2C15&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_2.png?resize=150%2C22&amp;ssl=1 150w\" sizes=\"(max-width: 645px) 100vw, 645px\" data-recalc-dims=\"1\" \/><\/a><br \/>\nde donde se obtiene la f\u00f3rmula<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png?resize=671%2C99\" alt=\"Calculo_de_pi_mediante_el_metodo_de_Newton_3\" width=\"671\" height=\"99\" class=\"aligncenter size-full wp-image-3012\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png?w=671&amp;ssl=1 671w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png?resize=300%2C44&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png?resize=100%2C14&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_3.png?resize=150%2C22&amp;ssl=1 150w\" sizes=\"(max-width: 671px) 100vw, 671px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>La primeras aproximaciones son<\/p>\n<pre lang=\"text\">\n   a(0) = 6*(1\/2)                               = 3.0\n   a(1) = 6*(1\/2+1\/(2*3*2^3))                   = 3.125\n   a(2) = 6*(1\/2+1\/(2*3*2^3)+(1*3)\/(2*4*5*2^5)) = 3.1390625\n<\/pre>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   aproximacionPi :: Int -> Double\n   grafica        :: [Int] -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(aproximacionPi n) es la n-\u00e9sima aproximaci\u00f3n de pi con la f\u00f3rmula de Newton. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     aproximacionPi 0   ==  3.0\n     aproximacionPi 1   ==  3.125\n     aproximacionPi 2   ==  3.1390625\n     aproximacionPi 10  ==  3.1415926468755613\n     aproximacionPi 21  ==  3.141592653589793\n     pi                 ==  3.141592653589793\n<\/pre>\n<ul>\n<li>(grafica xs) dibuja la gr\u00e1fica de las k-\u00e9simas aproximaciones de pi donde k toma los valores de la lista xs. Por ejemplo, (grafica [1..30]) dibuja<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png?resize=627%2C467\" alt=\"Calculo_de_pi_mediante_el_metodo_de_Newton_4\" width=\"627\" height=\"467\" class=\"aligncenter size-full wp-image-3013\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png?w=627&amp;ssl=1 627w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png?resize=300%2C223&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png?resize=100%2C74&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/02\/Calculo_de_pi_mediante_el_metodo_de_Newton_4.png?resize=150%2C111&amp;ssl=1 150w\" sizes=\"(max-width: 627px) 100vw, 627px\" data-recalc-dims=\"1\" \/><\/a><\/li>\n<\/ul>\n<p><strong>Nota<\/strong>: Este ejercicio ha sido propuesto por Manuel Herrera.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Graphics.Gnuplot.Simple\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\naproximacionPi :: Int -> Double\naproximacionPi n = 6 * arcsinX\n  where arcsinX = 0.5 + sum (take n factoresN)\n\nfactoresN :: [Double]\nfactoresN = zipWith (*) (potenciasK 3) fraccionesPI\n\npotenciasK :: Double -> [Double]\npotenciasK k = (0.5**k)\/k : potenciasK (k+2)\n\nfraccionesPI :: [Double]\nfraccionesPI =\n  scanl (*) (1\/2) (tail (zipWith (\/) [1,3..] [2,4..]))\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\naproximacionPi2 :: Int -> Double\naproximacionPi2 n = 6 * (serie !! n)\n\nserie :: [Double]\nserie = scanl1 (+) (zipWith (\/)\n                            (map fromIntegral numeradores)\n                            (map fromIntegral denominadores))\n  where numeradores    = 1 : scanl1 (*) [1,3..]\n        denominadores  = zipWith (*) denominadores1 denominadores2\n        denominadores1 = 2 : scanl1 (*) [2,4..]\n        denominadores2 = 1 : [n * 2^n | n <- [3,5..]]\n\ngrafica :: [Int] -> IO ()\ngrafica xs = \n    plotList [Key Nothing]\n             [(k,aproximacionPi k) | k <- xs]\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>El m\u00e9todo de Newton para el c\u00e1lculo de pi se basa en la relaci\u00f3n y en el desarrollo del arco seno de donde se obtiene la f\u00f3rmula 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 Definir las funciones aproximacionPi :: Int -> Double grafica&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[4],"tags":[8,183,376,11,309,6,78,252,40,45,47,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3009"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=3009"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3009\/revisions"}],"predecessor-version":[{"id":3050,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3009\/revisions\/3050"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}