{"id":8130,"date":"2023-05-17T06:00:11","date_gmt":"2023-05-17T04:00:11","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8130"},"modified":"2023-05-09T20:15:09","modified_gmt":"2023-05-09T18:15:09","slug":"17-may-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/17-may-23\/","title":{"rendered":"TAD de los polinomios: Regla de Ruffini"},"content":{"rendered":"<p>Usando el <a href=\"https:\/\/bit.ly\/3KwqXYu\">tipo abstracto de los polinomios<\/a>, definir las funciones<\/p>\n<pre lang=\"text\">\n   cocienteRuffini :: Int -> Polinomio Int -> Polinomio Int\n   restoRuffini    :: Int -> Polinomio Int -> Int\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li><code>cocienteRuffini r p<\/code> es el cociente de dividir el polinomio <code>p<\/code> por el polinomio <code>x-r<\/code>. Por ejemplo:<\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> ejPol = consPol 3 1 (consPol 2 2 (consPol 1 (-1) (consPol 0 (-2) polCero)))\n     \u03bb> ejPol\n     x^3 + 2*x^2 + -1*x + -2\n     \u03bb> cocienteRuffini 2 ejPol\n     x^2 + 4*x + 7\n     \u03bb> cocienteRuffini (-2) ejPol\n     x^2 + -1\n     \u03bb> cocienteRuffini 3 ejPol\n     x^2 + 5*x + 14\n<\/pre>\n<ul>\n<li><code>restoRuffini r p<\/code> es el resto de dividir el polinomio <code>p<\/code> por el polinomio <code>x-r<\/code>. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> restoRuffini 2 ejPol\n     12\n     \u03bb> restoRuffini (-2) ejPol\n     0\n     \u03bb> restoRuffini 3 ejPol\n     40\n<\/pre>\n<p>Comprobar con QuickCheck que, dado un polinomio p y un n\u00famero entero r, las funciones anteriores verifican la propiedad de la divisi\u00f3n eucl\u00eddea.<\/p>\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\">\nimport TAD.Polinomio (Polinomio, consPol, polCero)\nimport Pol_Transformaciones_polinomios_densas (densaApolinomio,\n                                               polinomioAdensa)\nimport Pol_Division_de_Ruffini_con_representacion_densa (ruffiniDensa)\nimport Pol_Producto_polinomios (multPol)\nimport Pol_Suma_de_polinomios (sumaPol)\nimport Pol_Crea_termino (creaTermino)\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n de cocienteRuffini\n-- ================================\n\ncocienteRuffini :: Int -> Polinomio Int -> Polinomio Int\ncocienteRuffini r p = densaApolinomio (init (ruffiniDensa r (polinomioAdensa p)))\n\n-- 2\u00aa definici\u00f3n de cocienteRuffini\n-- ================================\n\ncocienteRuffini2 :: Int -> Polinomio Int -> Polinomio Int\ncocienteRuffini2 r = densaApolinomio . ruffiniDensa r . init . polinomioAdensa\n\n-- 1\u00aa definici\u00f3n de restoRuffini\n-- =============================\n\nrestoRuffini :: Int -> Polinomio Int -> Int\nrestoRuffini r p = last (ruffiniDensa r (polinomioAdensa p))\n\n-- 2\u00aa definici\u00f3n de restoRuffini\n-- =============================\n\nrestoRuffini2 :: Int -> Polinomio Int -> Int\nrestoRuffini2 r = last . ruffiniDensa r . polinomioAdensa\n\n-- Comprobaci\u00f3n de la propiedad\n-- ============================\n\n-- La propiedad es\nprop_diviEuclidea :: Int -> Polinomio Int -> Bool\nprop_diviEuclidea r p =\n  p == sumaPol (multPol coci divi) rest\n  where coci = cocienteRuffini r p\n        divi = densaApolinomio [1,-r]\n        rest = creaTermino 0 (restoRuffini r p)\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_diviEuclidea\n--    +++ OK, passed 100 tests.\n<\/pre>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nfrom src.Pol_Crea_termino import creaTermino\nfrom src.Pol_Division_de_Ruffini_con_representacion_densa import ruffiniDensa\nfrom src.Pol_Producto_polinomios import multPol\nfrom src.Pol_Suma_de_polinomios import sumaPol\nfrom src.Pol_Transformaciones_polinomios_densas import (densaApolinomio,\n                                                        polinomioAdensa)\nfrom src.TAD.Polinomio import (Polinomio, consPol, esPolCero, polCero,\n                               polinomioAleatorio)\n\n\ndef cocienteRuffini(r: int, p: Polinomio[int]) -> Polinomio[int]:\n    if esPolCero(p):\n        return polCero()\n    return densaApolinomio(ruffiniDensa(r, polinomioAdensa(p))[:-1])\n\ndef restoRuffini(r: int, p: Polinomio[int]) -> int:\n    if esPolCero(p):\n        return 0\n    return ruffiniDensa(r, polinomioAdensa(p))[-1]\n\n# Comprobaci\u00f3n de la propiedad\n# ============================\n\n# La propiedad es\n@given(r=st.integers(), p=polinomioAleatorio())\ndef test_diviEuclidea (r: int, p: Polinomio[int]) -> None:\n    coci = cocienteRuffini(r, p)\n    divi = densaApolinomio([1, -r])\n    rest = creaTermino(0, restoRuffini(r, p))\n    assert p == sumaPol(multPol(coci, divi), rest)\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q Pol_Regla_de_Ruffini.py\n#    1 passed in 0.32s\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Usando el tipo abstracto de los polinomios, definir las funciones cocienteRuffini :: Int -> Polinomio Int -> Polinomio Int restoRuffini :: Int -> Polinomio Int -> Int tales que cocienteRuffini r p es el cociente de dividir el polinomio p por el polinomio x-r. Por ejemplo: \u03bb> ejPol = consPol 3 1 (consPol 2 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":[265,587],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8130"}],"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=8130"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8130\/revisions"}],"predecessor-version":[{"id":8131,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8130\/revisions\/8131"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}