In this paper, we investigate the normwise condition numbers of the indefinite least squares problem with multiple right-hand sides with respect to the weighted Frobenius norm and $ 2 $-norm. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the indefinite least squares problem with single right-hand side. Numerical experiments are performed to illustrate the tightness of the upper bounds.
Citation: Limin Li. Normwise condition numbers of the indefinite least squares problem with multiple right-hand sides[J]. AIMS Mathematics, 2022, 7(3): 3692-3700. doi: 10.3934/math.2022204
In this paper, we investigate the normwise condition numbers of the indefinite least squares problem with multiple right-hand sides with respect to the weighted Frobenius norm and $ 2 $-norm. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the indefinite least squares problem with single right-hand side. Numerical experiments are performed to illustrate the tightness of the upper bounds.
| [1] |
M. Arioli, M. Baboulin, S. Gratton, A partial condition number for linear least squares problems, SIAM J. Matrix Anal. Appl., 29 (2007), 413–433. doi: 10.1137/050643088. doi: 10.1137/050643088
|
| [2] |
E. H. Bergou, S. Gratton, J. Tshimanga, The exact condition number of the truncated singular value solution of a linear ill-posed problem, SIAM J. Matrix Anal. Appl., 35 (2014), 1073–1085. doi: 10.1137/120869286. doi: 10.1137/120869286
|
| [3] |
A. Bojanczyk, N. J. Higham, H. Patel, Solving the indefinite least squares problem by hyperbolic QR factorization, SIAM J. Matrix Anal. Appl., 24 (2003), 914–931. doi: 10.1137/S0895479802401497. doi: 10.1137/S0895479802401497
|
| [4] |
S. Chandrasekaran, M. Gu, A. H. Sayed, A stable and efficient algorithm for the indefinite linear least-squares problem, SIAM J. Matrix Anal. Appl., 20 (1998), 354–362. doi: 10.1137/S0895479896302229. doi: 10.1137/S0895479896302229
|
| [5] |
H. Diao, T. Zhou, Backward error and condition number analysis for the indefinite linear least squares problem, Int. J. Comput. Math., 96 (2019), 1603–1622. doi: 10.1080/00207160.2018.1467007. doi: 10.1080/00207160.2018.1467007
|
| [6] |
J. I. Giribet, A. Maestripieri, F. M. Peria, A geometrical approach to indefinite least squares problem, Acta Appl. Math., 111 (2011), 65–81. doi: 10.1007/s10440-009-9532-3. doi: 10.1007/s10440-009-9532-3
|
| [7] | A. Graham, Kronecker Products and Matrix Calculus with Application, Wiley, New York, 1981. |
| [8] |
S. Gratton, On the condition number of linear least squares problems in a weighted Frobenius norm, BIT, 36 (1996), 523–530. doi: 10.1007/BF01731931. doi: 10.1007/BF01731931
|
| [9] |
B. Hassibi, A. H. Sayed, T. Kailath, Linear estimation in Krein spaces. Part I: Theory, IEEE Trans. Automat. Contr., 41 (1996), 18–33. doi: 10.1109/9.481605. doi: 10.1109/9.481605
|
| [10] | N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2Eds., SIAM, Philadelphia, 2002. |
| [11] |
H. Li, S. Wang, H. Yang, On mixed and componentwise condition numbers for indefinite least squares problem, Linear Algebra Appl., 448 (2014), 104–129. doi: 10.1016/j.laa.2014.01.030. doi: 10.1016/j.laa.2014.01.030
|
| [12] |
H. Li, S. Wang, On the partial condition numbers for the indefinite least squares problem, Appl. Numer. Math., 123 (2018), 200–220. doi: 10.1016/j.apnum.2017.09.006. doi: 10.1016/j.apnum.2017.09.006
|
| [13] |
Q. Liu, X. Li, Preconditioned conjugate gradient methods for the solution of indefinite least squares problems, Calcolo, 48 (2011), 261–271. doi: 10.1007/s10092-011-0039-8. doi: 10.1007/s10092-011-0039-8
|
| [14] |
Q. Liu, A. Liu, Block SOR methods for the solution of indefinite least squares problems, Calcolo, 51 (2014), 367–379. doi: 10.1007/s10092-013-0090-8. doi: 10.1007/s10092-013-0090-8
|
| [15] | Y. Ou, Z. Peng, The solvability conditions for one kind of indefinite problems, Proceedings of the ninth International Conference on Matrix Theory and its Applications, World Academic Union, UK, 2010,142–145. |
| [16] |
A. H. Sayed, B. Hassibi, T. Kailath, Inertia properties of indefinite quadratic forms, IEEE. Signal Process. Lett., 3 (1996), 57–59. doi: 10.1109/97.484217. doi: 10.1109/97.484217
|
| [17] | J. Song, USSOR method for solving the indefinite least squares problem, Int. J. Comput. Math., doi: 10.1080/00207160.2019.1658869. |
| [18] | S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, 1991. doi: 10.2307/2153088. |
| [19] | L. Yang, H. Li, Condition numbers for indefinite least squares problem with multiple right-hand sides, J. Math. Res. Appl., 37 (2017), 725–742. |
Limin Li. Normwise condition numbers of the indefinite least squares problem with multiple right-hand sides[J]. AIMS Mathematics, 2022, 7(3): 3692-3700. doi: 10.3934/math.2022204
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