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Low rank approximation of a sparse matrix based on LU factorization with column and row tournament pivoting. (English) Zbl 1453.65090


MSC:

65F55 Numerical methods for low-rank matrix approximation; matrix compression
65F50 Computational methods for sparse matrices
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[1] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. W. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, SIAM, Philadelphia, 1999.
[2] G. Ballard, J. Demmel, O. Holtz, and O. Schwartz, Minimizing communication in numerical linear algebra, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 866–901. · Zbl 1246.68128
[3] C. H. Bischof, A parallel QR factorization algorithm with controlled local pivoting, SIAM J. Sci. and Stat. Comput., 12 (1991), pp. 36–57. · Zbl 0718.65017
[4] P. A. Businger and G. H. Golub, Linear least squares solutions by Householder transformations, Numer. Math., 7 (1965), pp. 269–276. · Zbl 0142.11503
[5] S. Chandrasekaran and I. C. F. Ipsen, On rank-revealing factorisations, SIAM. J. Matrix Anal. Appl., 15 (1994), pp. 592–622. · Zbl 0796.65030
[6] K. L. Clarkson and D. P. Woodruff, Low rank approximation and regression in input sparsity time, in Proceedings of the 45th Annual ACM Symposium on Theory of Computing, 2013, pp. 81–90. · Zbl 1293.65069
[7] J. K. Cullum and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory, SIAM, Philadelphia, 2002. · Zbl 1013.65033
[8] T. Davis, Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization, ACM Trans. Math. Software, 38 (2011), pp. 8:1–8:22. · Zbl 1365.65122
[9] T. A. Davis, J. R. Gilbert, S. I. Larimore, and E. G. Ng, A column approximate minimum degree ordering algorithm, ACM Trans. Math. Software, 30 (2004), pp. 353–376. · Zbl 1073.65039
[10] T. A. Davis and Y. Hu, The University of Florida Sparse Matrix Collection, ACM Trans. Math. Software, 38 (2011), pp. 1–25. . · Zbl 1365.65123
[11] J. Demmel, L. Grigori, M. Gu, and H. Xiang, Communication-avoiding rank-revealing QR decomposition, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 55–89. · Zbl 1327.65078
[12] J. Demmel, L. Grigori, M. Gu, and H. Xiang, LU with Tournament Pivoting for Nearly Singular Matrices, Technical report, Inria and UC Berkeley, 2018, in preparation.
[13] J. W. Demmel, L. Grigori, M. Hoemmen, and J. Langou, Communication-optimal parallel and sequential QR and LU factorizations, SIAM J. Sci. Comput., 34 (2012), pp. 206–239. · Zbl 1241.65028
[14] J. W. Demmel, N. J. Higham, and R. Schreiber, Block LU factorization, Numer. Linear Algebra Appl., 2 (1995), pp. 173–190.
[15] L. Foster, San Jose State University Singular Matrix Database, .
[16] A. George, Nested Dissection of a Regular Finite Element Mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345–363. · Zbl 0259.65087
[17] A. George and J. W.-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981. · Zbl 0516.65010
[18] A. George, J. W.-H. Liu, and E. G.-Y. Ng, A data structure for sparse QR and LU factors, SIAM J. Sci. and Stat. Comput., 9 (1988), pp. 100–121. · Zbl 0648.65019
[19] J. R. Gilbert, E. G. Ng, and B. W. Peyton, Separators and structure prediction in sparse orthogonal factorization, Linear Algebra Appl., 262 (1997). · Zbl 0881.15015
[20] G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), pp. 206–216. · Zbl 0142.11502
[21] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., The Johns Hopkins University Press, Baltimore, MD, 2013. · Zbl 1268.65037
[22] Z. N. Goreinov SA, Tyrtyshnikov EE, A theory of pseudoskeleton approximations, Linear Algebra Appl., 261 (1997), pp. 1–21. · Zbl 0877.65021
[23] L. Grigori, S. Cayrols, and J. Demmel, Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting, Technical report 8910, Inria, 2016. · Zbl 1453.65090
[24] L. Grigori, J. W. Demmel, and H. Xiang, CALU: A communication optimal LU factorization algorithm, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1317–1350. · Zbl 1242.65089
[25] M. Gu and S. C. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848–869. · Zbl 0858.65044
[26] N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217–288. · Zbl 1269.65043
[27] P. C. Hansen, Regularization tools version 4.1 for MATLAB 7.3.
[28] Y. P. Hong and C.-T. Pan, Rank-revealing QR factorizations and the singular value decomposition, Math. Comp., 58 (1992), pp. 213–232. · Zbl 0743.65037
[29] G. Karypis and V. Kumar, A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes and Computing Fill-Reducing Orderings of Sparse Matrices—Version 4.0, , 1998.
[30] A. Khabou, J. W. Demmel, L. Grigori, and M. Gu, Communication avoiding LU factorization with panel rank revealing pivoting, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1401–1429. · Zbl 1279.65034
[31] R. L. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM J. Appl. Math., 36 (1979), pp. 177–189. · Zbl 0432.05022
[32] M. W. Mahoney, Randomized algorithms for matrices and data, Found. Trends Mach. Learn., 3 (2011), pp. 123–224. · Zbl 1232.68173
[33] L. Miranian and M. Gu, Strong rank revealing LU factorizations, Linear Algebra Appl., (2003), pp. 1–16. · Zbl 1020.65016
[34] C.-T. Pan, On the existence and computation of rank-revealing LU factorizations, Linear Algebra Appl., 316 (2000), pp. 199–222. · Zbl 0962.65023
[35] Y. Saad, Numerical Methods for Large Eigenvalue Problems, 2nd ed., SIAM, Philadelphia, 2011. · Zbl 1242.65068
[36] G. Stewart, Four algorithms for the efficient computation of truncated QR approximations to a sparse matrix, Numer. Math., 83 (1999), pp. 313–323. · Zbl 0957.65031
[37] G. W. Stewart, The QLP approximation to the singular value decomposition, SIAM J. Sci. Comput., 20 (1999), pp. 1336–1348. · Zbl 0939.65062
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