JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (10): 40-45.doi: 10.6040/j.issn.1671-9352.0.2023.046

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Structured backward error for a class of generalized saddle point problems

Yuling LIU()   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2023-02-14 Online:2024-10-20 Published:2024-10-10

Abstract:

In this paper, an explicit expression of the structured backward error of a generalized saddle point system of a structure is derived, and the structured and unstructured backward errors are compared, and it is found that they can be arbitrarily far apart in some cases.

Key words: structured backward error, generalized saddle point system, strong stability

CLC Number: 

  • O241.1
1 BENZI M , GOLUB G H , LIESEN J . Numerical solution of saddle point problems[J]. Acta Numer, 1999, 14 (1): 1- 137.
2 XIANG H , WEI Y M , DIAO H A . Perturbation analysis of generalized saddle point systems[J]. Linear Algebra Appl, 2006, 419 (1): 8- 23.
doi: 10.1016/j.laa.2006.03.041
3 XU W W , LI W . New perturbation analysis for generalized saddle point systems[J]. Calcolo, 2009, 46 (1): 25- 36.
doi: 10.1007/s10092-009-0157-8
4 XU W W , LIU M M , ZHU L , et al. New perturbation bounds analysis of a kind of generalized saddle point systems[J]. East Asian J Appl Math, 2017, 7 (1): 116- 124.
doi: 10.4208/eajam.100616.031216a
5 SUN J G . Structured backward errors for KKT systems[J]. Linear Algebra Appl, 1999, 288 (1): 75- 88.
6 YANG X D , DAI H , HE Q Q . Condition numbers and backward perturbation bound for linear matrix equations[J]. Numer Linear Algebra Appl, 2011, 18 (1): 155- 165.
doi: 10.1002/nla.725
7 RIGAL J L , GACHES J . On the compatibility of a given solution with the data of a linear system[J]. J Assoc Comput Mach, 1967, 14 (3): 543- 548.
doi: 10.1145/321406.321416
8 WILKINSON J . The algebraic eigenvalue problem[M]. Oxford: Oxford University Press, 1965.
9 XIANG H , WEI Y M . On normwise structured backward errors for saddle point systems[J]. SIAM J Matrix Anal Appl, 2008, 29 (3): 838- 849.
10 CHEN X S , LI W , CHEN X J , et al. Structured backward errors for generalized saddle point systems[J]. Linear Algebra Appl, 2012, 436 (9): 3109- 3119.
doi: 10.1016/j.laa.2011.10.012
11 EISENSTAT S C , GRATTON S , TITLEY-PELOQUIN D . On the symmetric componentwise relative backward error for linear systems of equations[J]. SIAM J Matrix Anal Appl, 2017, 38 (4): 1100- 1115.
doi: 10.1137/140986566
12 HIGHAM D J , HIGHAM N J . Backward error and condition of structured linear systems[J]. SIAM J Matrix Anal Appl, 1992, 13 (1): 162- 175.
doi: 10.1137/0613014
13 HIGHAM N J. Accuracy and stability of numerical algorithms[M/OL]. SIAM, 2002. https://doi.org/10.1137/1.9780898718027.fm.
14 RUMP S M . The componentwise structured and unstructured backward errors can be arbitrarily far apart[J]. SIAM J Matrix Anal Appl, 2015, 36 (2): 385- 392.
doi: 10.1137/140985500
15 STEWART G W , SUN J G . Matrix perturbation theory[M]. Boston: Academic Press, 1990.
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