{"title":"k 对角圆周矩阵和环带矩阵的高效计算","authors":"Chen Wang, Chao Wang","doi":"arxiv-2403.05048","DOIUrl":null,"url":null,"abstract":"Calculating the inverse of $k$-diagonal circulant matrices and cyclic banded\nmatrices is a more challenging problem than calculating their determinants.\nAlgorithms that directly involve or specify linear or quadratic complexity for\nthe inverses of these two types of matrices are rare. This paper presents two\nfast algorithms that can compute the complexity of a $k$-diagonal circulant\nmatrix within complexity $O(k^3 \\log n+k^4)+kn$, and for $k$-diagonal cyclic\nbanded matrices it is $O(k^3 n+k^5)+kn^2$. Since $k$ is generally much smaller\nthan $n$, the cost of these two algorithms can be approximated as $kn$ and\n$kn^2$.","PeriodicalId":501256,"journal":{"name":"arXiv - CS - Mathematical Software","volume":"44 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Efficient Calculations for k-diagonal Circulant Matrices and Cyclic Banded Matrices\",\"authors\":\"Chen Wang, Chao Wang\",\"doi\":\"arxiv-2403.05048\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Calculating the inverse of $k$-diagonal circulant matrices and cyclic banded\\nmatrices is a more challenging problem than calculating their determinants.\\nAlgorithms that directly involve or specify linear or quadratic complexity for\\nthe inverses of these two types of matrices are rare. This paper presents two\\nfast algorithms that can compute the complexity of a $k$-diagonal circulant\\nmatrix within complexity $O(k^3 \\\\log n+k^4)+kn$, and for $k$-diagonal cyclic\\nbanded matrices it is $O(k^3 n+k^5)+kn^2$. Since $k$ is generally much smaller\\nthan $n$, the cost of these two algorithms can be approximated as $kn$ and\\n$kn^2$.\",\"PeriodicalId\":501256,\"journal\":{\"name\":\"arXiv - CS - Mathematical Software\",\"volume\":\"44 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Mathematical Software\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2403.05048\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Mathematical Software","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2403.05048","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Efficient Calculations for k-diagonal Circulant Matrices and Cyclic Banded Matrices
Calculating the inverse of $k$-diagonal circulant matrices and cyclic banded
matrices is a more challenging problem than calculating their determinants.
Algorithms that directly involve or specify linear or quadratic complexity for
the inverses of these two types of matrices are rare. This paper presents two
fast algorithms that can compute the complexity of a $k$-diagonal circulant
matrix within complexity $O(k^3 \log n+k^4)+kn$, and for $k$-diagonal cyclic
banded matrices it is $O(k^3 n+k^5)+kn^2$. Since $k$ is generally much smaller
than $n$, the cost of these two algorithms can be approximated as $kn$ and
$kn^2$.