{"id":6725,"date":"2022-03-08T06:00:34","date_gmt":"2022-03-08T04:00:34","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6725"},"modified":"2022-04-15T12:03:32","modified_gmt":"2022-04-15T10:03:32","slug":"matrices-de-toepliz","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/matrices-de-toepliz\/","title":{"rendered":"Matrices de Toepliz"},"content":{"rendered":"<p>Una <a href=\"https:\/\/bit.ly\/3pqjY9D\">matriz de Toeplitz<\/a> es una matriz cuadrada que es constante a lo largo de las diagonales paralelas a la diagonal principal. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   |2 5 1 6|       |2 5 1 6|\n   |4 2 5 1|       |4 2 6 1|\n   |7 4 2 5|       |7 4 2 5|\n   |9 7 4 2|       |9 7 4 2|\n<\/pre>\n<p>la primera es una matriz de Toeplitz y la segunda no lo es.<\/p>\n<p>Las anteriores matrices se pueden definir por<\/p>\n<pre lang=\"text\">\n   ej1, ej2 :: Array (Int,Int) Int\n   ej1 = listArray ((1,1),(4,4)) [2,5,1,6,4,2,5,1,7,4,2,5,9,7,4,2]\n   ej2 = listArray ((1,1),(4,4)) [2,5,1,6,4,2,6,1,7,4,2,5,9,7,4,2]\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   esToeplitz :: Eq a => Array (Int,Int) a -> Bool\n<\/pre>\n<p>tal que (esToeplitz p) se verifica si la matriz p es de Toeplitz. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   esToeplitz ej1  ==  True\n   esToeplitz ej2  ==  False\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Array (Array, (!), bounds, listArray)\n\nej1, ej2 :: Array (Int,Int) Int\nej1 = listArray ((1,1),(4,4)) [2,5,1,6,4,2,5,1,7,4,2,5,9,7,4,2]\nej2 = listArray ((1,1),(4,4)) [2,5,1,6,4,2,6,1,7,4,2,5,9,7,4,2]\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nesToeplitz1 :: Eq a => Array (Int,Int) a -> Bool\nesToeplitz1 p =\n  esCuadrada p &&\n  all todosIguales (diagonalesPrincipales p)\n\n-- (esCuadrada p) se verifica si la matriz p es cuadrada. Por ejemplo,\n--    esCuadrada (listArray ((1,1),(4,4)) [1..])  ==  True\n--    esCuadrada (listArray ((1,1),(3,4)) [1..])  ==  False\nesCuadrada :: Eq a => Array (Int,Int) a -> Bool\nesCuadrada p = m == n\n  where (_,(m,n)) = bounds p\n\n-- (diagonalesPrincipales p) es la lista de las diagonales principales\n-- de p. Por ejemplo,\n--    \u03bb> diagonalesPrincipales ej1\n--    [[2,2,2,2],[5,5,5],[1,1],[6],[2,2,2,2],[4,4,4],[7,7],[9]]\n--    \u03bb> diagonalesPrincipales ej2\n--    [[2,2,2,2],[5,6,5],[1,1],[6],[2,2,2,2],[4,4,4],[7,7],[9]]\ndiagonalesPrincipales :: Array (Int,Int) a -> [[a]]\ndiagonalesPrincipales p =\n  [[p ! i |i <- is] | is <- posicionesDiagonalesPrincipales m n]\n  where (_,(m,n)) = bounds p\n\n-- (posicionesDiagonalesPrincipales m n) es la lista de las\n-- posiciones de las diagonales principales de una matriz con m filas y\n-- n columnas. Por ejemplo,\n--   \u03bb> mapM_ print (posicionesDiagonalesPrincipales 3 4)\n--   [(3,1)]\n--   [(2,1),(3,2)]\n--   [(1,1),(2,2),(3,3)]\n--   [(1,2),(2,3),(3,4)]\n--   [(1,3),(2,4)]\n--   [(1,4)]\n--   \u03bb> mapM_ print (posicionesDiagonalesPrincipales 4 4)\n--   [(4,1)]\n--   [(3,1),(4,2)]\n--   [(2,1),(3,2),(4,3)]\n--   [(1,1),(2,2),(3,3),(4,4)]\n--   [(1,2),(2,3),(3,4)]\n--   [(1,3),(2,4)]\n--   [(1,4)]\n--   \u03bb> mapM_ print (posicionesDiagonalesPrincipales 4 3)\n--   [(4,1)]\n--   [(3,1),(4,2)]\n--   [(2,1),(3,2),(4,3)]\n--   [(1,1),(2,2),(3,3)]\n--   [(1,2),(2,3)]\n--   [(1,3)]\nposicionesDiagonalesPrincipales :: Int -> Int -> [[(Int, Int)]]\nposicionesDiagonalesPrincipales m n =\n  [zip [i..m] [1..n] | i <- [m,m-1..1]] ++\n  [zip [1..m] [j..n] | j <- [2..n]]\n\n-- (todosIguales xs) se verifica si todos los elementos de xs son\n-- iguales. Por ejemplo,\n--    todosIguales [5,5,5]  ==  True\n--    todosIguales [5,4,5]  ==  False\ntodosIguales :: Eq a => [a] -> Bool\ntodosIguales []     = True\ntodosIguales (x:xs) = all (== x) xs\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nesToeplitz2 :: Eq a => Array (Int,Int) a -> Bool\nesToeplitz2 p = m == n &&\n               and [p!(i,j) == p!(i+1,j+1) |\n                    i <- [1..n-1], j <- [1..n-1]]\n  where (_,(m,n)) = bounds p\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> esToeplitz1 (listArray ((1,1),(2*10^3,2*10^3)) (repeat 1))\n--    True\n--    (2.26 secs, 2,211,553,888 bytes)\n--    \u03bb> esToeplitz2 (listArray ((1,1),(2*10^3,2*10^3)) (repeat 1))\n--    True\n--    (4.26 secs, 3,421,651,032 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Matriz_Toeplitz.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Una matriz de Toeplitz es una matriz cuadrada que es constante a lo largo de las diagonales paralelas a la diagonal principal. Por ejemplo, |2 5 1 6| |2 5 1 6| |4 2 5 1| |4 2 6 1| |7 4 2 5| |7 4 2 5| |9 7 4 2| |9 7 4&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","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":[2],"tags":[8,507,42,11],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6725"}],"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=6725"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6725\/revisions"}],"predecessor-version":[{"id":6778,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6725\/revisions\/6778"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}