{"id":2292,"date":"2016-04-08T05:00:00","date_gmt":"2016-04-08T03:00:00","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2292"},"modified":"2016-05-01T20:02:41","modified_gmt":"2016-05-01T18:02:41","slug":"raiz-entera","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/raiz-entera\/","title":{"rendered":"Ra\u00edz entera"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   raizEnt :: Integer -> Integer -> Integer\n<\/pre>\n<p>tal que (raizEnt x n) es la ra\u00edz entera n-\u00e9sima de x; es decir, el mayor n\u00famero entero y tal que y^n &lt;= x. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   raizEnt  8 3      ==  2\n   raizEnt  9 3      ==  2\n   raizEnt 26 3      ==  2\n   raizEnt 27 3      ==  3\n   raizEnt (10^50) 2 ==  10000000000000000000000000\n<\/pre>\n<p>Comprobar con QuickCheck que para todo n\u00famero natural n,<\/p>\n<pre lang=\"text\">\n    raizEnt (10^(2*n)) 2 == 10^n\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n\nraizEnt1 :: Integer -> Integer -> Integer\nraizEnt1 x n =\n    last (takeWhile (\\y -> y^n <= x) [0..])\n\n-- 2\u00aa definici\u00f3n         \nraizEnt2 :: Integer -> Integer -> Integer\nraizEnt2 x n =\n    floor ((fromIntegral x)**(1 \/ fromIntegral n))\n\n-- Nota. La definici\u00f3n anterior falla para n\u00fameros grandes. Por ejemplo,\n--    \u03bb> raizEnt2 (10^50) 2 == 10^25\n--    False\n          \n-- 3\u00aa definici\u00f3n          \nraizEnt3 :: Integer -> Integer -> Integer\nraizEnt3 x n = aux (1,x)\n    where aux (a,b) | d == x    = c\n                    | c == a    = c\n                    | d < x     = aux (c,b)\n                    | otherwise = aux (a,c) \n              where c = (a+b) `div` 2\n                    d = c^n\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> raizEnt1 (10^14) 2\n--    10000000\n--    (6.15 secs, 6,539,367,976 bytes)\n--    \u03bb> raizEnt2 (10^14) 2\n--    10000000\n--    (0.00 secs, 0 bytes)\n--    \u03bb> raizEnt3 (10^14) 2\n--    10000000\n--    (0.00 secs, 25,871,944 bytes)\n--    \n--    \u03bb> raizEnt2 (10^50) 2\n--    9999999999999998758486016\n--    (0.00 secs, 0 bytes)\n--    \u03bb> raizEnt3 (10^50) 2\n--    10000000000000000000000000\n--    (0.00 secs, 0 bytes)\n                        \n-- La propiedad es                        \nprop_raizEnt :: Integer -> Bool\nprop_raizEnt n =\n    raizEnt3 (10^(2*m)) 2 == 10^m\n    where m = abs n\n\n-- La comprobaci\u00f3n es              \n--    \u03bb> quickCheck prop_raizEnt\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Soluciones en Maxima<\/h4>\n<pre lang=\"text\">\nraizEnt (x,n) := inrt (x,n)$\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n raizEnt :: Integer -> Integer -> Integer tal que (raizEnt x n) es la ra\u00edz entera n-\u00e9sima de x; es decir, el mayor n\u00famero entero y tal que y^n &lt;= x. Por ejemplo, raizEnt 8 3 == 2 raizEnt 9 3 == 2 raizEnt 26 3 == 2 raizEnt 27 3 ==&#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":[282,183,134,11,6,34,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2292"}],"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=2292"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2292\/revisions"}],"predecessor-version":[{"id":2332,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2292\/revisions\/2332"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2292"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2292"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}