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1、广义不定最小二乘问题的扰动分析第4期2009年7月华东师范大学(自然科学版)No.4JournalofEastChinaNormalUniversity(NaturalScience)July2009ArticleID:1000-5641(2009)04.-0047-07PerturbationanalysisforgeneralizedindefiniteleastsquaresroblemsWANGOian(DepartmentD,Mathematics,EastChinaNormalUniversity,Shang
2、hai200062,Chi竹01Abstract:AnewkindofweightedgeneralizedinverseWasdefinedtostudyindefiniteleastsquares(ILS)problemandequalityconstrainedindefiniteleastsquares(ILSE)problem.Byapplyinghyperbolic-QRfactorization,thesolutionstothetwoproblemswereobtalned.A1s0theperturba
3、tionboundsofthesolutionswerederived.Keywords:perturbationanalysis;indefiniteleastsquaresproblem;equalitycon—strainedindefiniteleastsquaresproblem;weightedgeneralizedinverseCLCnumber:O157.21;O241.1Documentcode:A广义不定最小二乘问题的扰动分析王茜(华东师范大学数学系,上海2000621摘要:通过定义一种新的加权广义逆
4、,研究不定最小二乘问题和等式约束不定最小二乘问题.应用矩阵的双曲QR分解,得到这两个问题的解的表达形式,并且推出了关于这两个问题的解的扰动界.关键词:扰动分析;不定最小二乘问题;等式约束不定最小二乘问题;加权广义逆0IntroductionThroughoutthispaper,weusethefollowingnotation.andC×aretheset8of仇×礼matricesofrealandcomplexentries,respectively.ForamatrixA,rank(A)istherankofA,
5、nul1(A)isthenullspaceofA,ATandAHare,respectively,thetransposeandtheconjugatetransposeofA.1I.Il=ll_1l2istheEuclidianvectornormorthespectralmatrixnorm.TheindefiniteleastsquaresproblemhastheformILS:n(b—Ax)TJ(b—Ax),whereA∈",m≥n,andb∈aregivenandJisthesignaturematrixIp
6、0f0一ql'P+qm收稿日期:2008.11基金项目:国家自然科学基金(10771073);上海市重点学科建设项目(S30405)作者简介:王茜,女,博士研究生,研究方向为数值代数.E-mail:wqn27@163.com华东师范大学(自然科学版)2009笠TheproblemwasfirstintroducedbyChandrasekaran,GuandSayed[,andfurtherstudiedbyBojanczyk.HighamandPatel[.The'extensionoftheILSprobleminc
7、ludingtheconsistentequalityconstraintsBx=dwasstudiedin【3】,whereBisassumedtohavefullrowrank.LiuandWang[studiedtheILSproblemwithleastsquaresconstraints.whichhastheformILSEmin(b—Ax)TJ(b—Ax),subjecttoX∈,ZwhereisthesolutionsettotheleastsquaresproblemB—dIJm∈inIIBy—dIIj
8、(0.2)(0.3)whereB∈z.d∈Rz.HereBmaynothavefullranksandtheconstraintsBx=dmaybeinconsistent.Itiswellknownthat(see[5】)theILSproblemhasauniquesolutionifandonlyifATJAi
