{"title":"两个矩阵的几何平均值的计算方面：综述","authors":"Dario A. Bini, Bruno Iannazzo","doi":"10.1007/s44146-024-00155-5","DOIUrl":null,"url":null,"abstract":"<div><p>Algorithms for the computation of the (weighted) geometric mean <i>G</i> of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product <span>\\(y=Gb\\)</span>, and of solving the linear system <span>\\(Gx=b\\)</span>, without forming <i>G</i>, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 3-4","pages":"349 - 389"},"PeriodicalIF":0.5000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s44146-024-00155-5.pdf","citationCount":"0","resultStr":"{\"title\":\"Computational aspects of the geometric mean of two matrices: a survey\",\"authors\":\"Dario A. Bini, Bruno Iannazzo\",\"doi\":\"10.1007/s44146-024-00155-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Algorithms for the computation of the (weighted) geometric mean <i>G</i> of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product <span>\\\\(y=Gb\\\\)</span>, and of solving the linear system <span>\\\\(Gx=b\\\\)</span>, without forming <i>G</i>, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.</p></div>\",\"PeriodicalId\":46939,\"journal\":{\"name\":\"ACTA SCIENTIARUM MATHEMATICARUM\",\"volume\":\"90 3-4\",\"pages\":\"349 - 389\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s44146-024-00155-5.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACTA SCIENTIARUM MATHEMATICARUM\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s44146-024-00155-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACTA SCIENTIARUM MATHEMATICARUM","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s44146-024-00155-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Computational aspects of the geometric mean of two matrices: a survey
Algorithms for the computation of the (weighted) geometric mean G of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product \(y=Gb\), and of solving the linear system \(Gx=b\), without forming G, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.
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