{"title":"迭代的乘法效率的一个界限","authors":"H. T. Kung","doi":"10.1145/800152.804902","DOIUrl":null,"url":null,"abstract":"For a convergent sequence {xi} generated by xi+1 = @@@@(xi,x1,...,xi-d+1), define the multiplication efficiency measure E to be p1/M, where p is the order of convergence, and M is the number of multiplications or divisions (except by 2) needed to compute @@@@. Then, if @@@@ is any multivariate rational function, E ≤ 2. Since E = 2 for the sequence {xi} generated by xi+1 = 1/2(xi +a/x i)with the limit @@@@a, the bound on E is sharp.","PeriodicalId":229726,"journal":{"name":"Proceedings of the fourth annual ACM symposium on Theory of computing","volume":"60 5 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1972-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"A bound on the multiplication efficiency of iteration\",\"authors\":\"H. T. Kung\",\"doi\":\"10.1145/800152.804902\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a convergent sequence {xi} generated by xi+1 = @@@@(xi,x1,...,xi-d+1), define the multiplication efficiency measure E to be p1/M, where p is the order of convergence, and M is the number of multiplications or divisions (except by 2) needed to compute @@@@. Then, if @@@@ is any multivariate rational function, E ≤ 2. Since E = 2 for the sequence {xi} generated by xi+1 = 1/2(xi +a/x i)with the limit @@@@a, the bound on E is sharp.\",\"PeriodicalId\":229726,\"journal\":{\"name\":\"Proceedings of the fourth annual ACM symposium on Theory of computing\",\"volume\":\"60 5 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1972-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the fourth annual ACM symposium on Theory of computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/800152.804902\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the fourth annual ACM symposium on Theory of computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/800152.804902","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 11
摘要
对于由xi+1 = @@@@(xi,x1，…，xi-d+1)生成的收敛序列{xi}，定义乘法效率测度E为p1/M，其中p为收敛阶数，M为计算@@@@所需的乘法或除法(除2)的次数。则，如果@@@@是任意多元有理函数，则E≤2。由于由xi+1 = 1/2(xi +a/ xi i)生成的序列{xi}的E = 2，极限为@@@@a，因此E的界很明显。
A bound on the multiplication efficiency of iteration
For a convergent sequence {xi} generated by xi+1 = @@@@(xi,x1,...,xi-d+1), define the multiplication efficiency measure E to be p1/M, where p is the order of convergence, and M is the number of multiplications or divisions (except by 2) needed to compute @@@@. Then, if @@@@ is any multivariate rational function, E ≤ 2. Since E = 2 for the sequence {xi} generated by xi+1 = 1/2(xi +a/x i)with the limit @@@@a, the bound on E is sharp.
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