{"id":1561,"date":"2015-06-15T06:00:30","date_gmt":"2015-06-15T04:00:30","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1561"},"modified":"2015-10-28T15:18:22","modified_gmt":"2015-10-28T13:18:22","slug":"algoritmo-de-bajada-para-resolver-un-sistema-triangular-inferior","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/algoritmo-de-bajada-para-resolver-un-sistema-triangular-inferior\/","title":{"rendered":"Algoritmo de bajada para resolver un sistema triangular inferior"},"content":{"rendered":"

Un sistema de ecuaciones lineales Ax = b es triangular inferior si todos los elementos de la matriz A que est\u00e1n por encima de la diagonal principal son nulos; es decir, es de la forma<\/p>\n

\n   a(1,1)*x(1)                                               = b(1)\n   a(2,1)*x(1) + a(2,2)*x(2)                                 = b(2)\n   a(3,1)*x(1) + a(3,2)*x(2) + a(3,3)*x(3)                   = b(3)\n   ...\n   a(n,1)*x(1) + a(n,2)*x(2) + a(n,3)*x(3) +...+ a(x,x)*x(n) = b(n)\n<\/pre>\n
El sistema es compatible si, y s\u00f3lo si, el producto de los elementos de la diagonal principal es distinto de cero. En este caso, la soluci\u00f3n se puede calcular mediante el algoritmo de bajada:<\/p>\n
\n   x(1) = b(1) \/ a(1,1)\n   x(2) = (b(2) - a(2,1)*x(1)) \/ a(2,2)\n   x(3) = (b(3) - a(3,1)*x(1) - a(3,2)*x(2)) \/ a(3,3)\n   ...\n   x(n) = (b(n) - a(n,1)*x(1) - a(n,2)*x(2) -...- a(n,n-1)*x(n-1)) \/ a(n,n)\n<\/pre>\n
Definir la funci\u00f3n<\/p>\n
\n   bajada :: Matrix Double -> Matrix Double -> Matrix Double\n<\/pre>\n
tal que (bajada a b) es la soluci\u00f3n, mediante el algoritmo de bajada, del sistema compatible triangular superior ax = b. Por ejemplo,<\/p>\n
\n   ghci> let a = fromLists [[2,0,0],[3,1,0],[4,2,5.0]]\n   ghci> let b = fromLists [[3],[6.5],[10]]\n   ghci> bajada a b\n   ( 1.5 )\n   ( 2.0 )\n   ( 0.0 )\n<\/pre>\n
Es decir, la soluci\u00f3n del sistema<\/p>\n
\n   2x            = 3\n   3x + y        = 6.5\n   4x + 2y + 5 z = 10\n<\/pre>\n
es x=1.5, y=2 y z=0.<\/p>\n
Soluciones<\/h4>\n
\nimport Data.Matrix\n\nbajada :: Matrix Double -> Matrix Double -> Matrix Double\nbajada a b = fromLists [[x i] | i <- [1..m]]\n    where m = nrows a\n          x k = (b!(k,1) - sum [a!(k,j) * x j | j <- [1..k-1]]) \/ a!(k,k)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Un sistema de ecuaciones lineales Ax = b es triangular inferior si todos los elementos de la matriz A que est\u00e1n por encima de la diagonal principal son nulos; es decir, es de la forma a(1,1)*x(1) = b(1) a(2,1)*x(1) + a(2,2)*x(2) = b(2) a(3,1)*x(1) + a(3,2)*x(2) + a(3,3)*x(3) = b(3) … a(n,1)*x(1) + a(n,2)*x(2) +…<\/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":[5],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1561"}],"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=1561"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1561\/revisions"}],"predecessor-version":[{"id":1650,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1561\/revisions\/1650"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1561"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1561"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1561"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}