1、第44卷第10 期2023年10 月宁夏师范学院学报Journal of Ningxia Normal UniversityVol.44 No.10Oct.2023NDSDD矩阵逆的无穷大范数上界敖地珍,刘兰兰,刘艳(贵州民族大学数据科学与信息工程学院,贵州贵阳550 0 0 0)摘要:给出NDSDD矩阵逆的无穷范数上界.此外,当A是NDSDD矩阵,C是一般矩阵时,给出IA-1Cll。的上界通过数值算例,说明该方法的可行性.关键词:无穷范数;NDSDD矩阵;H-矩阵中图分类号:0 151.2 1收稿日期:2 0 2 3-0 6-0 2基金项目:贵州省高等学校大数据分析与智能计算重点实验室(黔教
2、技 2 0 2 30 12 号)。作者简介:敖地珍(1998 一),女,贵州遵义人,硕士研究生,研究方向:数值代数。文献标识码:A文章编号:16 7 4-1331(2 0 2 3)10-0 0 13-0 6H-矩阵在数学研究中有着非常重要的作用,特别是非奇异H-矩阵逆的无穷范数的上界可以用于矩阵分裂的收敛性分析,以及求解大稀疏性线性方程的矩阵多分裂迭代方程.近年来,H-矩阵逆的无穷大范数上界估计得到广泛的关注和研究.NDSDD矩阵是在文献 1中定义的一类新的H-矩阵,目前对这个矩阵的研究还比较少.本文主要讨论了它的无穷大范数问题的上界,当A是NDSDD矩阵,C是一般矩阵时,给出A-C。上界.特
3、别是当C为单位矩阵时,得出A-1I。的上界.1预备知识在整个文章中,将使用以下符号,N:=1,2,n,M:=1,2,m 表示指标集;S表示N的任意非空子集,S:=N/S表示S的补集,表示所有个复(实)矩阵的集合;r;(A):=,l a x|,r(A):=a x|,显然,对任意的S和任意的keN.k+i(A)iE N,r(A)=r(A)+ri(A);R(A):=.Iaal,其中AECx,laal+0.KESI(i)首先介绍严格对角占优(SDD)和(DSDD)矩阵等常用定义.定义1C2设A=a EC x ,对于任意的iEN,都有则称A是SDD严格对角占优矩阵.定义2 2 设A=a j EC x ,
4、对于任意的iEN,则称A是DSDD严格双对角占优矩阵.kEs.k+iI ai1r;(A),Iau|l ai|r(A)r;(A),(1)(2)14文献 3证明SDD矩阵和DSDD矩阵是H-矩阵的子类,文献 1中提出一个新的H-矩阵的子类.然而,文献 1中并未对该矩阵命名,本文称它为NDSDD矩阵,定义如下。定义 3 1设矩阵A=a,EC ,n 2 对于任意的iEN,r;(A)r,(A)RN(A)RN(A),则称A是NDSDD矩阵.显然,SDD矩阵包含NDSDD矩阵,即(SDD)C(ND SD D).以下的例子说明反包含不成立,即(NDSDD)C(SDD)不成立.例1通过计算可得对任意的iE N,
5、ri(A)rN(A)RN(A)=kEM(iYR(A)=9.8505,RY(A)=13.4514,R(A)=4.2250,所以有 r:(A)r,(A)RN(A)RY(A),所以A为NDSDD矩阵,但A不是SDD矩阵.定义44设A=a,EC x ,若A的比较矩阵(A)=,是M-矩阵,(A)-1 O,则A是一个H-矩阵.文献4表明若 A是一个H-矩阵,则(A)-1A I-i,这里AB是指 RN(A)RN(A)=r;(B)r;(B),因此矩阵B是DSDD矩阵.由定理2 可知I+r;(A)eNr:(A)r,(A)-RN(A)R,(A)max敖地珍,等:NDSDD矩阵逆的无穷大范数上界r;(A)+R;(A
6、)I bi I+r;(B)ai15(6)(7)令AX=B=b,Eak因为 AX=B,所以IA-I 。=x(X-A-)I l=X B-l I x l/B-I lri(A)maxmaxi.jENaiENr;(A)r,(A)-RN(A)RN(A)计计定理4设A=a i EC 是NDSDD矩阵,C=c j EC X 则r;(A)Zck|+RY(A)21cklr(A)IA-Cll.maxmaxiENaii.jENr;(A)r;(A)-RN(A)RN(A)证明因为A=a,E C 是一个NDSDD矩阵,所以A是H-矩阵,因此L(A)-1A|-1,设W=|A-Cl e=(wi,w2,w,)T,y=(A)-1|
7、C l e=(y1,y2,y,)T,并且e=(1,1,1)T 是m维列向量,则=(A)-1|Cle|A-I/Cle|A-Cle=w.设X是定理3中的X,则有 y=XX-(A)-Cle,因此(A)-XX-y=|C e.令 X-y=z=(zi,z2,z),故aIak12.故有iENiEN.itp(A)ZIc/=T,(A)z,-1alzr(A)-appkEMkEN,kpr,(A)+RN(A)kEMmx.所以2 1ca/-r(A)-2kEMkEN.k+par(A)apkappkEN,k卡pkEM(8)T(A)(9)16:Z1c/=r(A)2-kEM(9)Xr(A)得(A)(10)XkeN.ktpapp
8、kEN.kpappkEN.kpapp(11)+(12)得r,(A)abk/Z1cg/r(A)r,(A)zpkEMkEN.kpap(A)apkakEN,kp力力于是之;r;(A)Ji所以可得;(A)maxaii,jENr:(A)r;(A)-不妨设yi=maxyi,则iENIA-Cll=maxyi=yiiEN推论1设A=a;EC 是NDSDD矩阵,取C为n 阶的单位矩阵,得到的结果如下:宁夏师范学院学报(A)a1zrg(A)zg-kEN.ktqara(A)ZIc/r.(A)r,(A)z,-r,(A)kEMapk1得(A)r.(A)21cx1+2kEMkEN.k丰pappr(A)r,(A)r.(A)
9、-apkappkeN.k卡pkEN.k子ikEMmaxijENr;(A)r;(A)-r;(A)ZIcI+kEMkEN.ktiair:(A)aikeN,ktiair;(A)ZIcI+(A)kEMmaxalli.jENr:(A)r;(A)-r,(A)ZIcuI+r:(A)kEMmaxmaxiENaii,jENr;(A)r;(A)-r;(A)ZIca I+RY(A)ZIcxlr:(A)kEMmaxmaxiENaii.jEN2023年10 月(A)(10)kEN.k+qap(A)kEN,kpapp(A)a1rg(A)za-kEMkEN.ktppp(A)(A)akEN,k丰akEMr,(A)akEN,k
10、卡qr(A)aikkEN.k丰iair;(A)akkEMr(A)kEN,k+jr;(A)aikEN.kir:(A)akEN.kiaikEN.kiair;(A)aikeN,k+iaikEMr;(A)r,;(A)-RN(A)R)(A)(11)(12)(A)apkkEN,k+qr;(A)akaiikEMr(A)kEN.k牛jajiIckIckkEMr(A)kEN,kjajr;(A)akkEMr;(A)kEN.ktjaikEMr,(A)aa第10 期3数值算例下面给出数值算例,对上述的理论进行说明。例2 考虑以下的NDSDD矩阵及一般矩阵F38124130234251411 3153116312209
11、337131A=,C0320494144113612102312241291L523152447通过计算可得出ri(A)=13,r2(A)=19,rs(A)=19,r4(A)=23,rs(A)=14,re(A)=17,rz(A)=14,rs(A)=22,RN(A)=6.1094,R(A)=8.0987,R(A)=6.8557,RN(A)=10.2572,R(A)=5.5000,R(A)=7.1751,R(A)=5.8499,R(A)=8.7916.所以可以得出矩阵A是一个NDSDD矩阵,故IA-icll.maxmaxiENaii.jENr;(A)r;(A)-RN(A)R,(A)实际上,A-1C
12、ll=0.39381.5095.例3考虑以下的NDSDD矩阵6.81470.1576 0.65570.90588.97060.03570.1270.957211.84910.27690.91340.48540.9340.63240.80030.6787A0.09750.14190.75770.27850.42180.74310.54690.91570.39220.95750.79220.6555L0.96490.95950.1712通过计算可得出ri(A)=4.2536,r2(A)=3.4288,rs(A)=5.2181,r(A)=5.5621,rs(A)=5.9989,r(A)=4.0354
13、,r(A)=5.0661,rg(A)=4.5203,rg(A)=5.4273,rio(A)=4.4071,RN(A)=1.8776,R(A)=1.6752,R(A)=2.2158,RN(A)=2.5941,R(A)=2.5254,R(A)=1.8592,R(A)=2.1434,R(A)=2.0905,R(A)=2.0 90 5,Ri(A)=2.2 0 2 4.所以可以得出矩阵A是一个NDSDD矩阵,故敖地珍,等:NDSDD矩阵逆的无穷大范数上界r;(A)r,(A)+RN(A)I/A-Ill.maxmaxiENaii.jEN.i+ir;(A)r,(A)-RN(A)R,(A)41745226701
14、32135 3412 5344232L1715r;(A)2IcaI+RY(A)ZIcklr:(A)kEM0.7060.43870.2760.03180.38160.67970.76550.655112.04620.79520.16260.09719.18690.1190.82350.489815.49840.95930.350.69480.44560.95970.31710.64630.34040.95020.70940.58530.03440.75470.22381723kEM=1.5095.0.75130.84070.25510.25430.5060.81430.69910.24350.8
15、9090.929312.54720.19660.138611.25110.75370.14930.6160.25750.47330.35170.83080.0540.58530.53080.54970.77920.91720.9340.28580.12990.75720.56880.469410.38040.01190.567817.33710.075918实际上,A-1。=0.159 3 0.532 6.参考文献:1 LJILJANA C,VLADIMIR K.New criteria for identi-fying H-matrices J.Journal of Computationa
16、l andApplied Mathematics,2005,180(2):265-278.2BERMAN A,PLEMMONS R.Nonnegative matricesin the mathematical sciencesMJ.New York:Societyfor Industrial and Applied Mathematics,1994.3CVETKOVIC L.H-matrix theory vs.eigenvalue lo-calization J.Numerical Algorithms,2006,42:229-245.4LI Y,WANG Y.Schur compleme
17、nt-based infinitynorm bounds for the inverse of GDSDD matricesJI.Mathematics,2022,10(2):186.5HUANG T.Estimation of IIA-Il and the smallestsingular valueJ.Computers&Mathematics withApplications,2008,55(6):1075-1080.6JOHNSON C.A Gersgorin-type lower bound for thesmallest singular valueJ.Linear Algebra
18、 and itsApplications,1989,112:1-7.7KOSTIC V,CVETKOVI L,C VET K O VI D.Pseudospectra localizations and their applicationsJ.Numerical Linear Algebra with Applications,2016,23(2):356-372.8LI C,CVETKOVIC L,WEI Y,et al.An infinity宁夏师范学院学报r:(A)r;(A)+RN(A)I/A-1IlmaxmaxiENa.jeNr;(A)r;(A)-RN(A)R,(A)计2023年10
19、月0.5326.norm bound for the inverse of Dashnic-Zusmanovichtype matrices with applications J.Linear Algebraand its Applications,2019,565:99-122.9LIU J,ZHANG J,LIU Y.The Schur complement ofstrictly doubly diagonally dominant matrices and itsapplicationJJ.Linear Algebra and its Applications,2012,437(1):
20、168-183.10VARAH J M.A lower bound for the smallest sin-gular value of a matrixJ.Linear Algebra and itsApplications,1975,11(1):3-5.11LI C.Schur complement-based infinity norm boundsfor the inverse of SDD matricesJ.Bulletin of theMalaysian Mathematical Sciences Society,2020,43:3829-3845.12赵仁庆,何建锋IA-BI
21、。的无穷大范数的上界估计 J.数学的实践与认识,2 0 2 1,51(18):2 8 8-2 92.13SANG C.Schur complement-based infinity normbounds for the inverse of DSDD matricesJ.Bulle-tin of the Iranian Mathematical Society,2021,47:1379-1398.14胡家赣B-A I。的估计及其应用J计算数学,1982,(3):272-282.Infinity norm bounds for the inverse of NDSDD matricesAO Di
22、zhen,LIU Lanlan,LIU Yan(College of Data Science and Information Engineering,Guizhou Minzu University,GuiyangGuizhou550000)Abstract In this paper,the upper bound of the infinity norm of inverse for NDSDD matrices is given.Moreover,whenA is an NDSDD matrix and Cis a general matrix,the upper bound of IlA-Cll is presented.Numerical examples are givento illustrate our methods.Key words Infinity norm;NDSDD matrix;H-matrix责任编辑董白英
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