{"title":"A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations","authors":"P. Concus, D. O’Leary, G. Golub","doi":"10.1016/B978-0-12-141050-6.50023-4","DOIUrl":"https://doi.org/10.1016/B978-0-12-141050-6.50023-4","url":null,"abstract":"","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114682937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Calculating the singular values and pseudo-inverse of a matrix","authors":"G. Golub, W. Kahan","doi":"10.1137/0702016","DOIUrl":"https://doi.org/10.1137/0702016","url":null,"abstract":"A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form $Sigma = U^ * AV$, thus exhibiting A’s singular values on $Sigma $’s diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse $A^I = VSigma ^I U^* $ to solve least squares problems in a way which dampens spurious oscillation and cancellation.","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126733591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The block Lanczos method for computing eigenvalues","authors":"G. Golub, Richard R. Underwood","doi":"10.1016/B978-0-12-587260-7.50018-2","DOIUrl":"https://doi.org/10.1016/B978-0-12-587260-7.50018-2","url":null,"abstract":"","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125511502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Commentary","authors":"G. W. Stewart","doi":"10.2307/j.ctv1228h00.9","DOIUrl":"https://doi.org/10.2307/j.ctv1228h00.9","url":null,"abstract":"","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116714289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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