Schur Complement-Based Infinity Norm Bounds for the Inverse of SDD Matrices
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Schur Complement-Based Infinity Norm Bounds for the Inverse of SDD Matrices Chaoqian Li1 Received: 8 July 2019 / Revised: 5 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Based on the Schur complement, two upper bounds for the infinity norm of the inverse of strictly diagonally dominant matrices are presented. We apply these new bounds to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of B-matrices. A lower bound for the smallest singular value is also given. Keywords Infinity norm · SDD matrices · Linear complementary problems · Singular values Mathematics Subject Classification 15A18 · 15A42 · 15A69
1 Introduction The class of strictly diagonally dominant (SDD) matrices was presented by Lévy in 1881 [37]. But it was restricted to real matrices and then extended to the complex matrices case by Minkowski [48] and Desplanques [16], respectively. This general result was later independently obtained by Hadamard [25] in his book. Definition 1 A matrix A = [ai j ] ∈ Cn×n is called an SDD matrix if |aii | > ri (A) for each i ∈ N := {1, 2, . . . , n}, where ri (A) :=
j∈N , |ai j |. j=i
Communicated by Fuad Kittaneh.
B 1
Chaoqian Li [email protected] School of Mathematics and Statistics, Yunnan University, Kunming 650091, Yunnan, People’s Republic of China
123
C. Li
SDD matrices appear now and then in various fields, for details, see [1,7,8,26,33,53]. In order to solve problems arising in different fields, it needs various properties for SDD matrices, which leads to various problems related to SDD matrices, such as Schur complement problem for SDD matrices [43,44], estimating the infinity norm for the inverse of SDD matrices [55], error bound for linear complementarity problems [19,38], etc. In this paper, we focus on infinity norm bounds for the inverse of SDD matrices, as it can be used for the convergence analysis of matrix splitting and matrix multisplitting iterative methods for solving large sparse systems of linear equations [56]. The first work may be due to Varah [55], who in 1975 presented an upper bound for the infinity norm of the inverse for SDD matrices. Theorem 1 [55, Theorem 1] If A = [ai j ] ∈ Cn×n is S D D, then ||A−1 ||∞ ≤ max i∈N
1 . |aii | − ri (A)
(1)
Although Varah’s bound (1) is simple and beautiful, it could only provide a rough bound for the special case that |aii | → ri (A) for some i ∈ N . To improve it, other new upper bounds for a wider class of matrices including the SDD case are given, such as bounds for doubly strictly diagonally dominant (DSDD) matrices [32], SSDD (CKV) matrices [42,49,50], weakly chained diagonally dominant matrices [54], Nekrasov matrices [10,35,36,39,41], S-Nekrasov matrices [9], Dashnic–Zusmanovich type matrices [40], H -matrices [34] and eventually SDD matrices [11]. On the other hand, by considering strictly diagonally dominant M-matrices as a special case of SDD matrices, some improved upper bounds are also obtained, see [5,42,57]. By using the Sc
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