{"title":"On fast multiplication of a matrix by its transpose","authors":"J. Dumas, Clément Pernet, A. Sedoglavic","doi":"10.1145/3373207.3404021","DOIUrl":null,"url":null,"abstract":"We present a non-commutative algorithm for the multiplication of a 2 × 2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any field of prime characteristic. We use geometric considerations on the space of bilinear forms describing 2 × 2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field. To conclude, we show how to use our result in L · D · LT factorization.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"23 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3373207.3404021","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
Abstract
We present a non-commutative algorithm for the multiplication of a 2 × 2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any field of prime characteristic. We use geometric considerations on the space of bilinear forms describing 2 × 2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field. To conclude, we show how to use our result in L · D · LT factorization.