{"id":8134,"date":"2023-05-19T06:00:54","date_gmt":"2023-05-19T04:00:54","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8134"},"modified":"2023-05-22T16:58:22","modified_gmt":"2023-05-22T14:58:22","slug":"19-may-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/19-may-23\/","title":{"rendered":"TAD de los polinomios: Ra\u00edces enteras de un polinomio"},"content":{"rendered":"<p>Usando el <a href=\"https:\/\/bit.ly\/3KwqXYu\">tipo abstracto de los polinomios<\/a>, definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n    raicesRuffini :: Polinomio Int -> [Int]\n<\/pre>\n<p>tal que <code>raicesRuffini p<\/code> es la lista de las raices enteras de <code>p<\/code>, calculadas usando el regla de Ruffini. Por ejemplo,<\/p>\n<pre lang=\"text\">\n    \u03bb> ejPol1 = consPol 4 3 (consPol 2 (-5) (consPol 0 3 polCero))\n    \u03bb> ejPol1\n    3*x^4 + -5*x^2 + 3\n    \u03bb> raicesRuffini ejPol1\n    []\n    \u03bb> ejPol2 = consPol 5 1 (consPol 2 5 (consPol 1 4 polCero))\n    \u03bb> ejPol2\n    x^5 + 5*x^2 + 4*x\n    \u03bb> raicesRuffini ejPol2\n    [0,-1]\n    \u03bb> ejPol3 = consPol 4 6 (consPol 1 2 polCero)\n    \u03bb> ejPol3\n    6*x^4 + 2*x\n    \u03bb> raicesRuffini ejPol3\n    [0]\n    \u03bb> ejPol4 = consPol 3 1 (consPol 2 2 (consPol 1 (-1) (consPol 0 (-2) polCero)))\n    \u03bb> ejPol4\n    x^3 + 2*x^2 + -1*x + -2\n    \u03bb> raicesRuffini ejPol4\n    [1,-1,-2]\n<\/pre>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nmodule Pol_Raices_enteras_de_un_polinomio where\n\nimport TAD.Polinomio (Polinomio, consPol, polCero, esPolCero)\nimport Pol_Termino_independiente_de_un_polinomio (terminoIndep)\nimport Pol_Regla_de_Ruffini (cocienteRuffini)\nimport Pol_Reconocimiento_de_raices_por_la_regla_de_Ruffini (esRaizRuffini)\nimport Test.Hspec (Spec, hspec, it, shouldBe)\n\nraicesRuffini :: Polinomio Int -> [Int]\nraicesRuffini p\n  | esPolCero p = []\n  | otherwise   = aux (0 : divisores (terminoIndep p))\n  where aux [] = []\n        aux (r:rs)\n          | esRaizRuffini r p = r : raicesRuffini (cocienteRuffini r p)\n          | otherwise         = aux rs\n\n-- (divisores n) es la lista de todos los divisores enteros de n. Por\n-- ejemplo,\n--    divisores 4     ==  [1,-1,2,-2,4,-4]\n--    divisores (-6)  ==  [1,-1,2,-2,3,-3,6,-6]\ndivisores :: Int -> [Int]\ndivisores n = concat [[x,-x] | x <- [1..abs n], rem n x == 0]\n\n-- Verificaci\u00f3n\n-- ============\n\nverifica :: IO ()\nverifica = hspec spec\n\nspec :: Spec\nspec = do\n  it \"e1\" $\n    raicesRuffini ejPol1 `shouldBe` []\n  it \"e2\" $\n    raicesRuffini ejPol2 `shouldBe` [0,-1]\n  it \"e3\" $\n    raicesRuffini ejPol3 `shouldBe` [0]\n  it \"e4\" $\n    raicesRuffini ejPol4 `shouldBe` [1,-1,-2]\n  where\n    ejPol1 = consPol 4 3 (consPol 2 (-5) (consPol 0 3 polCero))\n    ejPol2 = consPol 5 1 (consPol 2 5 (consPol 1 4 polCero))\n    ejPol3 = consPol 4 6 (consPol 1 2 polCero)\n    ejPol4 = consPol 3 1 (consPol 2 2 (consPol 1 (-1) (consPol 0 (-2) polCero)))\n\n-- La verificaci\u00f3n es\n--    \u03bb> verifica\n--\n--    e1\n--    e2\n--    e3\n--    e4\n--\n--    Finished in 0.0013 seconds\n--    4 examples, 0 failures\n<\/pre>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom src.Pol_Reconocimiento_de_raices_por_la_regla_de_Ruffini import \\\n    esRaizRuffini\nfrom src.Pol_Regla_de_Ruffini import cocienteRuffini\nfrom src.Pol_Termino_independiente_de_un_polinomio import terminoIndep\nfrom src.TAD.Polinomio import Polinomio, consPol, esPolCero, polCero\n\n\n# (divisores n) es la lista de todos los divisores enteros de n. Por\n# ejemplo,\n#    divisores(4)  == [1, 2, 4, -1, -2, -4]\n#    divisores(-6) == [1, 2, 3, 6, -1, -2, -3, -6]\ndef divisores(n: int) -> list[int]:\n    xs = [x for x in range(1, abs(n)+1) if n % x == 0]\n    return xs + [-x for x in xs]\n\ndef raicesRuffini(p: Polinomio[int]) -> list[int]:\n    if esPolCero(p):\n        return []\n    def aux(rs: list[int]) -> list[int]:\n        if not rs:\n            return []\n        x, *xs = rs\n        if esRaizRuffini(x, p):\n            return [x] + raicesRuffini(cocienteRuffini(x, p))\n        return aux(xs)\n\n    return aux([0] + divisores(terminoIndep(p)))\n\n# Verificaci\u00f3n\n# ============\n\ndef test_raicesRuffini() -> None:\n    ejPol1 = consPol(4, 3, consPol(2, -5, consPol(0, 3, polCero())))\n    assert raicesRuffini(ejPol1) == []\n    ejPol2 = consPol(5, 1, consPol(2, 5, consPol(1, 4, polCero())))\n    assert raicesRuffini(ejPol2) == [0, -1]\n    ejPol3 = consPol(4, 6, consPol(1, 2, polCero()))\n    assert raicesRuffini(ejPol3) == [0]\n    ejPol4 = consPol(3, 1, consPol(2, 2, consPol(1, -1, consPol(0, -2, polCero()))))\n    assert raicesRuffini(ejPol4) == [1, -1, -2]\n    print(\"Verificado\")\n\n# La verificaci\u00f3n es\n#    >>> test_raicesRuffini()\n#    Verificado\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Usando el tipo abstracto de los polinomios, definir la funci\u00f3n raicesRuffini :: Polinomio Int -> [Int] tal que raicesRuffini p es la lista de las raices enteras de p, calculadas usando el regla de Ruffini. Por ejemplo, \u03bb> ejPol1 = consPol 4 3 (consPol 2 (-5) (consPol 0 3 polCero)) \u03bb> ejPol1 3*x^4 + -5*x^2&#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":[581],"tags":[265],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8134"}],"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=8134"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8134\/revisions"}],"predecessor-version":[{"id":8170,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8134\/revisions\/8170"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}