A Mixed Precision Jacobi SVD Algorithm
We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess and then performs one-sided Jacobi rotations in the working precis...
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| Published in | ACM transactions on mathematical software Vol. 51; no. 1; pp. 1 - 33 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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| ISSN | 0098-3500 1557-7295 1557-7295 |
| DOI | 10.1145/3721124 |
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| Abstract | We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess and then performs one-sided Jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about \(2\times\) speedup on the x86-64 architecture compared to the usual one-sided Jacobi SVD algorithm in LAPACK, without sacrificing the accuracy. |
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| AbstractList | We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess and then performs one-sided Jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about \(2\times\) speedup on the x86-64 architecture compared to the usual one-sided Jacobi SVD algorithm in LAPACK, without sacrificing the accuracy. |
| ArticleNumber | 5 |
| Author | Ma, Yuxin Gao, Weiguo Shao, Meiyue |
| Author_xml | – sequence: 1 givenname: Weiguo orcidid: 0009-0003-6675-8754 surname: Gao fullname: Gao, Weiguo email: wggao@fudan.edu.cn organization: School of Mathematical Sciences and School of Data Science, Fudan University, Shanghai, China – sequence: 2 givenname: Yuxin orcidid: 0000-0002-2860-0134 surname: Ma fullname: Ma, Yuxin email: yuxin.ma@matfyz.cuni.cz organization: Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic – sequence: 3 givenname: Meiyue orcidid: 0000-0002-4914-7666 surname: Shao fullname: Shao, Meiyue email: myshao@fudan.edu.cn organization: School of Data Science and MOE Key Laboratory for Computational Physical Sciences, Fudan University, Shanghai, China |
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| Keywords | mixed precision algorithm high relative accuracy Jacobi algorithm Singular value decomposition |
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| References | (Bib0017) 2008; 29 (Bib0002) 2016; 25 (Bib0021) 2002 (Bib0031) 1966; 9 (Bib0008) 2023; 45 (Bib0032) 1968; 1 (Bib0012) 1989; 10 (Bib0025) 2022; 43 (Bib0028) 2018; 35 (Bib0001) 2021; 35 (Bib0014) 1997 (Bib0006) 2018; 40 (Bib0005) 1965; 7 (Bib0007) 2020; 42 (Bib0016) 2017; 44 (Bib0024) 1977; 17 (Bib0029) 2019; 36 (Bib0011) 2000; 40 (Bib0009) 1983; 91 (Bib0015) 2008; 35 (Bib0018) 2008; 29 (Bib0020) 2008 (Bib0026) 1967; 14 (Bib0033) 2015; 37 (Bib0003) 2020; 12043 (Bib0004) 2023; 92 (Bib0010) 2001; 88 (Bib0019) 1958; 6 (Bib0027) 2022 (Bib0022) 2022; 31 (Bib0013) 1992; 13 (Bib0030) 2020; 369 (Bib0023) 1846; 30 e_1_3_3_17_2 e_1_3_3_16_2 e_1_3_3_19_2 e_1_3_3_18_2 e_1_3_3_13_2 e_1_3_3_12_2 e_1_3_3_15_2 e_1_3_3_34_2 e_1_3_3_14_2 e_1_3_3_32_2 e_1_3_3_33_2 e_1_3_3_11_2 e_1_3_3_30_2 e_1_3_3_10_2 e_1_3_3_31_2 Jacobi C. G. J. (e_1_3_3_24_2) 1846; 30 e_1_3_3_6_2 e_1_3_3_5_2 e_1_3_3_8_2 e_1_3_3_7_2 e_1_3_3_28_2 e_1_3_3_9_2 e_1_3_3_27_2 e_1_3_3_29_2 e_1_3_3_23_2 Hida Yozo (e_1_3_3_21_2) 2008 e_1_3_3_26_2 e_1_3_3_25_2 e_1_3_3_2_2 e_1_3_3_20_2 e_1_3_3_4_2 e_1_3_3_22_2 e_1_3_3_3_2 |
| References_xml | – volume: 9 start-page: 19 year: 1966 end-page: 22 ident: Bib0031 article-title: On the quadratic convergence of the special cyclic Jacobi method publication-title: Numer. Math. doi: 10.1007/BF02165225 – volume: 45 start-page: C131 issue: 3 year: 2023 end-page: C153 ident: Bib0008 article-title: Mixed precision iterative refinement with sparse approximate inverse preconditioning publication-title: SIAM J. Sci. Comput. doi: 10.1137/22M1487709 – volume: 17 start-page: 303 year: 1977 end-page: 311 ident: Bib0024 article-title: Iterative refinement implies numerical stability publication-title: BIT doi: 10.1007/BF01932150 – volume: 35 start-page: 1 issue: 2 year: 2008 end-page: 28 ident: Bib0015 article-title: On the failure of rank-revealing QR factorization software–A case study publication-title: ACM Trans. Math. Softw. doi: 10.1145/1377612.1377616 – volume: 29 start-page: 1343 issue: 4 year: 2008 end-page: 1362 ident: Bib0018 article-title: New fast and accurate Jacobi SVD algorithm II publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/05063920X – volume: 35 start-page: 344 issue: 4 year: 2021 end-page: 369 ident: Bib0001 article-title: A survey of numerical linear algebra methods utilizing mixed-precision arithmetic publication-title: Int. J. High Perform. Comput. Appl. doi: 10.1177/10943420211003313 – volume: 369 start-page: 112512 year: 2020 ident: Bib0030 article-title: Iterative refinement for singular value decomposition based on matrix multiplication publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2019.112512 – volume: 30 start-page: 51 year: 1846 end-page: 94 ident: Bib0023 article-title: Über ein leichtes Verfahren die in der Theorie der Säcularströrungen vorkommenden Gleichungen numerisch aufzulösen publication-title: J. Reine Angew. Math. – year: 2008 ident: Bib0020 publication-title: Library for Double–Double and Quad–Double Arithmetic – year: 2022 ident: Bib0027 article-title: MPLAPACK version 2.0.1 user manual – volume: 10 start-page: 359 issue: 2 year: 1989 end-page: 371 ident: Bib0012 article-title: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer publication-title: SIAM J. Sci. Stat. Comput. doi: 10.1137/0910023 – volume: 35 start-page: 1007 issue: 3 year: 2018 end-page: 1035 ident: Bib0028 article-title: Iterative refinement for symmetric eigenvalue decomposition publication-title: Jpn. J. Indust. Appl. Math. doi: 10.1007/s13160-018-0310-3 – volume: 1 start-page: 1 issue: 1 year: 1968 end-page: 12 ident: Bib0032 article-title: Almost diagonal matrices with multiple or close eigenvalues publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(68)90042-6 – volume: 12043 start-page: 555 year: 2020 end-page: 566 ident: Bib0003 article-title: Preconditioned Jacobi SVD algorithm outperforms PDGESVD publication-title: Parallel Processing and Applied Mathematics (PPAM ’19) Lecture Notes in Computer Science doi: 10.1007/978-3-030-43229-4_47 – volume: 42 start-page: A4063 issue: 6 year: 2020 end-page: A4083 ident: Bib0007 article-title: Three-precision GMRES-based iterative refinement for least squares problems publication-title: SIAM J. Sci. Comput. doi: 10.1137/20M1316822 – volume: 92 start-page: 247 issue: 1 year: 2023 end-page: 267 ident: Bib0004 article-title: Iterative refinement of Schur decompositions publication-title: Numer. Algorithms doi: 10.1007/s11075-022-01327-6 – volume: 44 start-page: 1 issue: 1 year: 2017 end-page: 30 ident: Bib0016 article-title: Algorithm 977: A QR-preconditioned QR SVD method for computing the SVD with high accuracy publication-title: ACM Trans. Math. Software doi: 10.1145/3061709 – volume: 36 start-page: 435 issue: 2 year: 2019 end-page: 459 ident: Bib0029 article-title: Iterative refinement for symmetric eigenvalue decomposition II: clustered eigenvalues publication-title: Jpn. J. Indust. Appl. Math. doi: 10.1007/s13160-019-00348-4 – volume: 40 start-page: A817 issue: 2 year: 2018 end-page: A847 ident: Bib0006 article-title: Accelerating the solution of linear systems by iterative refinement in three precisions publication-title: SIAM J. Sci. Comput. doi: 10.1137/17M1140819 – volume: 88 start-page: 319 year: 2001 end-page: 345 ident: Bib0010 article-title: Componentwise perturbation analyses for the QR factorization publication-title: Numer. Math. doi: 10.1007/PL00005447 – volume: 13 start-page: 1204 issue: 4 year: 1992 end-page: 1245 ident: Bib0013 article-title: Jacobi’s method is more accurate than QR publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/0613074 – year: 1997 ident: Bib0014 publication-title: Applied Numerical Linear Algebra – volume: 6 start-page: 144 issue: 2 year: 1958 end-page: 162 ident: Bib0019 article-title: On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing the eigenvalues of Hermitian matrices publication-title: SIAM J. Appl. Math. doi: 10.1137/0106008 – volume: 91 start-page: 562 year: 1983 end-page: 566 ident: Bib0009 article-title: Diagonal elements and eigenvalues of a real symmetric matrix publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(83)90171-3 – volume: 14 start-page: 316 issue: 2 year: 1967 end-page: 321 ident: Bib0026 article-title: Iterative refinement in floating point publication-title: J. ACM doi: 10.1145/321386.321394 – volume: 29 start-page: 1322 issue: 4 year: 2008 end-page: 1342 ident: Bib0017 article-title: New fast and accurate Jacobi SVD algorithm I publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/050639193 – volume: 40 start-page: 640 year: 2000 end-page: 651 ident: Bib0011 article-title: Numerically stable generation of correlation matrices and their factors publication-title: BIT doi: 10.1023/A:1022384216930 – volume: 31 start-page: 347 year: 2022 end-page: 414 ident: Bib0022 article-title: Mixed precision algorithms in numerical linear algebra publication-title: Acta Numer. doi: 10.1017/S0962492922000022 – volume: 7 start-page: 269 issue: 3 year: 1965 end-page: 276 ident: Bib0005 article-title: Linear least squares solutions by Householder transformations publication-title: Numer. Math. doi: 10.1007/BF01436084 – volume: 37 start-page: C307 issue: 3 year: 2015 end-page: C330 ident: Bib0033 article-title: Mixed-precision Cholesky QR factorization and its case studies on multicore CPU with multiple GPUs publication-title: SIAM J. Sci. Comput. doi: 10.1137/14M0973773 – year: 2002 ident: Bib0021 publication-title: Accuracy and Stability of Numerical Algorithms – volume: 25 start-page: 1 issue: 1 year: 2016 end-page: 160 ident: Bib0002 article-title: Linear algebra software for large-scale accelerated multicore computing publication-title: Acta Numer. doi: 10.1017/S0962492916000015 – volume: 43 start-page: 1573 issue: 4 year: 2022 end-page: 1598 ident: Bib0025 article-title: A fast iterative algorithm for near-diagonal eigenvalue problems publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/21M1401474 – ident: e_1_3_3_8_2 doi: 10.1137/20M1316822 – ident: e_1_3_3_20_2 doi: 10.1137/0106008 – ident: e_1_3_3_18_2 doi: 10.1137/050639193 – ident: e_1_3_3_10_2 doi: 10.1016/0022-247X(83)90171-3 – volume: 30 start-page: 51 year: 1846 ident: e_1_3_3_24_2 article-title: Über ein leichtes Verfahren die in der Theorie der Säcularströrungen vorkommenden Gleichungen numerisch aufzulösen publication-title: J. Reine Angew. 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| Snippet | We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the... |
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| Title | A Mixed Precision Jacobi SVD Algorithm |
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