{"title":"Toeplitz系统的最优循环预调节器","authors":"T. Chan","doi":"10.1137/0909051","DOIUrl":null,"url":null,"abstract":"Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\\|C - A\\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1988-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"557","resultStr":"{\"title\":\"An Optimal Circulant Preconditioner for Toeplitz Systems\",\"authors\":\"T. Chan\",\"doi\":\"10.1137/0909051\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\\\\|C - A\\\\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.\",\"PeriodicalId\":200176,\"journal\":{\"name\":\"Siam Journal on Scientific and Statistical Computing\",\"volume\":\"2 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1988-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"557\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Siam Journal on Scientific and Statistical Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/0909051\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siam Journal on Scientific and Statistical Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/0909051","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 557
摘要
给定一个Toeplitz矩阵a，在最小化${\|C - a \|}_F $的意义上，我们得到了一个最优循环预条件C。总的来说，它不同于Strang早先提出的[A proposal for Toeplitz matrix computation]， Stud。达成。数学。， 74(1986)，第171-176页]，但A本身是循环的情况除外。新的预条件易于计算，在初步的数值实验中，在减少$C^{- 1} A$的条件数方面优于Strang预条件，在围绕单位的谱聚类方面也优于Strang预条件。
An Optimal Circulant Preconditioner for Toeplitz Systems
Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\|C - A\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.
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