{"id":8315,"date":"2023-10-29T06:00:50","date_gmt":"2023-10-29T04:00:50","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8315"},"modified":"2024-05-16T19:38:18","modified_gmt":"2024-05-16T17:38:18","slug":"29-oct-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/29-oct-23\/","title":{"rendered":"M\u00e9todo de Her\u00f3n para calcular la ra\u00edz cuadrada"},"content":{"rendered":"\n<p>El m\u00e9todo de Her\u00f3n para calcular la ra\u00edz cuadrada de un n\u00famero se basa en las siguientes propiedades:<\/p>\n<ul>\n<li>Si &#92;(y&#92;) es una aproximaci\u00f3n de la ra\u00edz cuadrada de &#92;(x&#92;), entonces<br \/>\n&#92;[&#92;frac{y+&#92;frac{x}{y}}{2}&#92;] es una aproximaci\u00f3n mejor.<\/li>\n<li>El l\u00edmite de la sucesi\u00f3n definida por<br \/>\n&#92;begin{align}<br \/>\n  x_{0}   &amp;= 1 &#92;&#92;<br \/>\n  x_{n+1} &amp;= &#92;frac{x_n+&#92;frac{x}{x_n}}{2}<br \/>\n&#92;end{align}<br \/>\nes la ra\u00edz cuadrada de x.<\/li>\n<\/ul>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   raiz :: Double -> Double\n<\/pre>\n<p>tal que <code>raiz x<\/code> es la ra\u00edz cuadrada de <code>x<\/code> calculada usando la propiedad anterior con una aproximaci\u00f3n de 0.00001 y tomando como valor inicial 1. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   raiz 9  ==  3.000000001396984\n<\/pre>\n<p><!--more--><\/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\">\nmodule Metodo_de_Heron_para_calcular_la_raiz_cuadrada where\n\nimport Test.QuickCheck\nimport Test.Hspec (Spec, hspec, it, shouldBe)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nraiz :: Double -> Double\nraiz x = raizAux 1\n  where raizAux y | aceptable y = y\n                  | otherwise   = raizAux (mejora y)\n        aceptable y = abs(y*y-x) < 0.00001\n        mejora y    = 0.5*(y+x\/y)\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nraiz2 :: Double -> Double\nraiz2 x = until aceptable mejora 1\n  where aceptable y = abs(y*y-x) < 0.00001\n        mejora y    = 0.5*(y+x\/y)\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_raiz :: Positive Double -> Bool\nprop_raiz (Positive x) =\n  raiz x ~= sqrt x &&\n  raiz2 x ~= sqrt x\n  where\n    a ~= b = abs (a-b) < 0.001\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_raiz\n--    +++ OK, passed 100 tests.\n\n-- Verificaci\u00f3n\n-- ============\n\nverifica :: IO ()\nverifica = hspec spec\n\nspec :: Spec\nspec = do\n  it \"e1\" $\n    raiz 9 `shouldBe`  3.000000001396984\n  it \"e2\" $\n    raiz2 9 `shouldBe`  3.000000001396984\n\n-- La verificaci\u00f3n es\n--    \u03bb> verifica\n--\n--    e1\n--    e2\n--\n--    Finished in 0.0008 seconds\n--    2 examples, 0 failures\n<\/pre>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef raiz(x : float) -> float:\n    def aceptable(y: float) -> bool:\n        return abs(y*y-x) < 0.00001\n    def mejora(y: float) -> float:\n        return 0.5*(y+x\/y)\n    def raizAux(y: float) -> float:\n        if aceptable(y):\n            return y\n        return raizAux(mejora(y))\n    return raizAux(1)\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef raiz2(x: float) -> float:\n    def aceptable(y: float) -> bool:\n        return abs(y*y-x) < 0.00001\n    def mejora(y: float) -> float:\n        return 0.5*(y+x\/y)\n    y = 1.0\n    while not aceptable(y):\n        y = mejora(y)\n    return y\n\n# Verificaci\u00f3n\n# ============\n\ndef test_raiz() -> None:\n    assert raiz(9)  ==  3.000000001396984\n    assert raiz2(9)  ==  3.000000001396984\n    print(\"Verificado\")\n\n# La verificaci\u00f3n es\n#    >>> test_raiz()\n#    Verificado\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>El m\u00e9todo de Her\u00f3n para calcular la ra\u00edz cuadrada de un n\u00famero se basa en las siguientes propiedades: Si &#92;(y&#92;) es una aproximaci\u00f3n de la ra\u00edz cuadrada de &#92;(x&#92;), entonces &#92;[&#92;frac{y+&#92;frac{x}{y}}{2}&#92;] es una aproximaci\u00f3n mejor. El l\u00edmite de la sucesi\u00f3n definida por &#92;begin{align} x_{0} &amp;= 1 &#92;&#92; x_{n+1} &amp;= &#92;frac{x_n+&#92;frac{x}{x_n}}{2} &#92;end{align} es la ra\u00edz cuadrada&#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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8315"}],"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=8315"}],"version-history":[{"count":9,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8315\/revisions"}],"predecessor-version":[{"id":8554,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8315\/revisions\/8554"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}