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Multiplicative perturbation analysis for QR factorizations
1. | School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2A7, Canada |
2. | Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, United States |
References:
[1] |
R. Bhatia, Matrix factorizations and their perturbations, Linear Algebra Appl., 197/198 (1994), 245-276.
doi: doi:10.1016/0024-3795(94)90490-1. |
[2] |
R. Bhatia, "Matrix Analysis," Graduate Texts in Mathematics, Vol. 169. Springer, New York, 1997. |
[3] |
R. Bhatia and K. Mukherjea, Variation of the unitary part of a matrix, SIAM J. Matrix Anal. Appl., 15 (1994), 1007-1014.
doi: doi:10.1137/S0895479892243237. |
[4] |
Åke Björck, "Numerical Methods for Least Squares Problems," SIAM, Philadelphia, 1996. |
[5] |
X. W. Chang and C. C. Paige, Componentwise perturbation analyses for the QR factorization, Numer. Math., 88 (2001), 319-345.
doi: doi:10.1007/PL00005447. |
[6] |
X. W. Chang, C. C. Paige and G. W. Stewart, Perturbation analyses for the QR factorization, SIAM J. Matrix Anal. Appl., 18 (1997), 775-791.
doi: doi:10.1137/S0895479896297720. |
[7] |
X. W. Chang and D. Stehlé, Rigorous perturbation bounds of some matrix factorizations, SIAM J. Matrix Anal. Appl., 31 (2010), 2841-2859.
doi: doi:10.1137/090778535. |
[8] |
X. W. Chang, D. Stehlé and G. Villard, Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction, 25 pages, submitted. |
[9] |
G. H. Golub and C. F. Van Loan, "Matrix Computations," Johns Hopkins University Press, Baltimore, Maryland, 3rd edition, 1996. |
[10] |
N. J. Higham, "Accuracy and Stability of Numerical Algorithms," SIAM, Philadephia, 2nd edition, 2002. |
[11] |
R. C. Li, Relative perturbation bounds for the unitary polar factor, BIT, 37 (1997), 67-75.
doi: doi:10.1007/BF02510173. |
[12] |
R. C. Li, Relative perturbation theory: I. eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19 (1998), 956-982.
doi: doi:10.1137/S089547989629849X. |
[13] |
R. C. Li, Relative perturbation theory: II. eigenspace and singular subspace variations, SIAM J. Matrix Anal. Appl., 20 (1999), 471-492.
doi: doi:10.1137/S0895479896298506. |
[14] |
R. C. Li, Relative perturbation bounds for positive polar factors of graded matrices, SIAM J. Matrix Anal. Appl., 25 (2005), 424-433.
doi: doi:10.1137/S0895479803437153. |
[15] |
R. C. Li and G. W. Stewart, A new relative perturbation theorem for singular subspaces, Linear Algebra Appl., 313 (2000), 41-51.
doi: doi:10.1016/S0024-3795(00)00074-4. |
[16] |
G. W. Stewart, Perturbation bounds for the QR factorization of a matrix, SIAM J. Numer. Anal., 14 (1977), 509-518.
doi: doi:10.1137/0714030. |
[17] |
G. W. Stewart, On the perturbation of LU, Cholesky and QR factorizations, SIAM J. Matrix Anal. Appl., 14 (1993), 1141-1146.
doi: doi:10.1137/0614078. |
[18] |
G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory," Academic Press, Boston, 1990. |
[19] |
J. G. Sun, Perturbation bounds for the Cholesky and QR factorizations, BIT, 31 (1991), 341-352.
doi: doi:10.1007/BF01931293. |
[20] |
J. G. Sun, Componentwise perturbation bounds for some matrix decompositions, BIT, 32 (1992), 702-714.
doi: doi:10.1007/BF01994852. |
[21] |
J. G. Sun, On perturbation bounds for the QR factorization, Linear Algebra Appl., 215 (1995), 95-112.
doi: doi:10.1016/0024-3795(93)00077-D. |
[22] |
H. Zha, A componentwise perturbation analysis of the QR decomposition, SIAM J. Matrix Anal. Appl., 14 (1993), 1124-1131.
doi: doi:10.1137/0614076. |
show all references
References:
[1] |
R. Bhatia, Matrix factorizations and their perturbations, Linear Algebra Appl., 197/198 (1994), 245-276.
doi: doi:10.1016/0024-3795(94)90490-1. |
[2] |
R. Bhatia, "Matrix Analysis," Graduate Texts in Mathematics, Vol. 169. Springer, New York, 1997. |
[3] |
R. Bhatia and K. Mukherjea, Variation of the unitary part of a matrix, SIAM J. Matrix Anal. Appl., 15 (1994), 1007-1014.
doi: doi:10.1137/S0895479892243237. |
[4] |
Åke Björck, "Numerical Methods for Least Squares Problems," SIAM, Philadelphia, 1996. |
[5] |
X. W. Chang and C. C. Paige, Componentwise perturbation analyses for the QR factorization, Numer. Math., 88 (2001), 319-345.
doi: doi:10.1007/PL00005447. |
[6] |
X. W. Chang, C. C. Paige and G. W. Stewart, Perturbation analyses for the QR factorization, SIAM J. Matrix Anal. Appl., 18 (1997), 775-791.
doi: doi:10.1137/S0895479896297720. |
[7] |
X. W. Chang and D. Stehlé, Rigorous perturbation bounds of some matrix factorizations, SIAM J. Matrix Anal. Appl., 31 (2010), 2841-2859.
doi: doi:10.1137/090778535. |
[8] |
X. W. Chang, D. Stehlé and G. Villard, Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction, 25 pages, submitted. |
[9] |
G. H. Golub and C. F. Van Loan, "Matrix Computations," Johns Hopkins University Press, Baltimore, Maryland, 3rd edition, 1996. |
[10] |
N. J. Higham, "Accuracy and Stability of Numerical Algorithms," SIAM, Philadephia, 2nd edition, 2002. |
[11] |
R. C. Li, Relative perturbation bounds for the unitary polar factor, BIT, 37 (1997), 67-75.
doi: doi:10.1007/BF02510173. |
[12] |
R. C. Li, Relative perturbation theory: I. eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19 (1998), 956-982.
doi: doi:10.1137/S089547989629849X. |
[13] |
R. C. Li, Relative perturbation theory: II. eigenspace and singular subspace variations, SIAM J. Matrix Anal. Appl., 20 (1999), 471-492.
doi: doi:10.1137/S0895479896298506. |
[14] |
R. C. Li, Relative perturbation bounds for positive polar factors of graded matrices, SIAM J. Matrix Anal. Appl., 25 (2005), 424-433.
doi: doi:10.1137/S0895479803437153. |
[15] |
R. C. Li and G. W. Stewart, A new relative perturbation theorem for singular subspaces, Linear Algebra Appl., 313 (2000), 41-51.
doi: doi:10.1016/S0024-3795(00)00074-4. |
[16] |
G. W. Stewart, Perturbation bounds for the QR factorization of a matrix, SIAM J. Numer. Anal., 14 (1977), 509-518.
doi: doi:10.1137/0714030. |
[17] |
G. W. Stewart, On the perturbation of LU, Cholesky and QR factorizations, SIAM J. Matrix Anal. Appl., 14 (1993), 1141-1146.
doi: doi:10.1137/0614078. |
[18] |
G. W. Stewart and J. G. Sun, "Matrix Perturbation Theory," Academic Press, Boston, 1990. |
[19] |
J. G. Sun, Perturbation bounds for the Cholesky and QR factorizations, BIT, 31 (1991), 341-352.
doi: doi:10.1007/BF01931293. |
[20] |
J. G. Sun, Componentwise perturbation bounds for some matrix decompositions, BIT, 32 (1992), 702-714.
doi: doi:10.1007/BF01994852. |
[21] |
J. G. Sun, On perturbation bounds for the QR factorization, Linear Algebra Appl., 215 (1995), 95-112.
doi: doi:10.1016/0024-3795(93)00077-D. |
[22] |
H. Zha, A componentwise perturbation analysis of the QR decomposition, SIAM J. Matrix Anal. Appl., 14 (1993), 1124-1131.
doi: doi:10.1137/0614076. |
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