非线性矩阵问题的若干结果
|
摘要
本文研究如下几类非线性矩阵问题和特殊结构矩阵问题．
1．无阻尼陀螺系统
给出无阻尼陀螺系统G(λ)=Mλ~2+Cλ+K的谱分解定理，其中M为质量矩阵，K为刚性矩阵，C为陀螺矩阵，并利用该定理解决了特征值反问题和无干扰特征值嵌入问题．谱分解定理表明矩阵M，C，K的含反对称参数矩阵S的谱分解式与实标准对(X，T)是一一对应的．若由系统的谱信息构造的矩阵T是分块对角矩阵，则S具有类上三角Hankel块结构．特别地，当系统只有半单的特征值时，存在一个标准对使得S只有2n个非零元素，并且非零元素为1或-1．另外，谱分解定理还给出无阻尼陀螺系统的二次特征值反问题有解的充要条件和通解表达式．当无阻尼陀螺系统只有单特征值时，无干扰特征值嵌入问题一定有解，并且可以推导出只包含已知的特征信息的通解表达式．
2．非线性矩阵方程的可解性和代数扰动分析
系统地研究矩阵方程X~s±A~*X~tA=Q(s＞0，t＞0)的可解性、迭代算法和代数扰动分析．首先利用矩阵分解方法推导X~s+A~*X~tA=Q存在Hermtian正定解的充要条件和通解表达式(如果解存在)．然后利用Brouwer不动点原理和特征值方法寻找Hermtian正定解的存在区间和存在唯一解的充分条件．将X~s-A~*X~tA=Q分为三种情况进行讨论：当s＜t时给出类最大解和类最小解的定义及存在的充分条件，并且推导出最大解存在的充分条件和能够收敛到最大解的迭代公式；当s＞t时设计求唯一解的一个新的迭代算法；当s=t时给出唯一解的精确表达式．另外建立代数扰动理论，分别给出唯一解、类最大解和类最小解的扰动上界．
3．实多项式矩阵方程的敏感性分析
考虑实系数多项式矩阵方程X~s±A′X~tA=Q的对称正定解的敏感性．首先给出扰动方程有唯一的对称正定解的充分条件，然后给出能反应对称正定解的敏感性的测度—条件数的定义及其精确表达式．数值实验表明该条件数能够反映对称正定解的敏感性．
4．子矩阵约束对称广义特征值反问题和最佳逼近问题
给出对称的广义特征值模型的部分修正问题和最佳逼近问题的数学描述，并给出问题有解的充要条件及通解表达式．
5．广义对称矩阵
给出k次(R，S)-对称矩阵和(R，ω_η)-对称矩阵的定义，并刻画其特殊结构．解决了相应的特征问题、奇异值分解、最小二乘问题、特征值反问题、极小范数问题和最佳逼近问题．
In this thesis some nonlinear matrix problems and special structure matrix problemsare investigated.
1. The undamped gyroscopic system
The spectral decomposition theorem of undamped gyroscopic system G(λ)=Mλ~2+Cλ+K is given, where M is mass matrix, K is stiffness matrix, C is gyroscopic matrix,and the inverse eigenvalue problem as well as the spill-over eigenvalue embeddingproblem is solved by the spectral decomposition theorem. The spectral decompositiontheorem shows that once the real standard pair (X, T) is given, the spectral decompositionsof M, C, K, including a skew-symmetric parameter matrix S,are decided. If T is ablock diagonal matrix, S becomes an upper tridiagonal Hankel-like matrix. Especially,if G(λ) only has semi-simple eigenvalues, then there exists a real standard pair, such thatS only has 2n nonzero elements 1 or-1. On the other hand, the spectral decompositiontheorem presents a necessary and sufficient condition for the solvability of the quadraticinverse eigenvalue problem, and an expression of the general solution(if exists). Whenall of the eigenvalues of G(λ) are simple, the spill-over eigenvalue embedding problemis always solvable and an expression of the general solution, only including the knowneigen information, is given.
2. Solvability and algebraic perturbation analysis of a class of nonlinear matrixequation
The nonlinear matrix equation X~s±A~*X~tA=Q(s＞0, t＞0) is systematicallystudied, including solvability, iterative algorithms and algebraic perturbation analysis. Atfirst, an necessary and sufficient condition and an expression of the general solution (ifexists) of X~s+A~*X~tA=Q are derived by a matrix decomposition method. Secondly,an interval in which an Hermitian positive solution exists and a sufficient condition forthe existence and uniqueness of the Hermitian positive solution are introduced by theBrouwer fixed point theorem and the eigenvalue method. For X~s-A~*X~tA=Q, we studythree cases: when s＜t, the maximal-like solution and the minimal-like solution aredefined, and sufficient conditions for their existences are given, furthermore, a sufficientcondition for the existence of the maximal solution and an iterative formula are given;when s＞t, a new iterative algorithm for the unique solution is given; when s=t, anexpression of the unique solution is obtained. Besides, the algebraic perturbation analysis theory for the unique solution, the maximal-like solution and the minimal-like solutionare established.
3. Sensitivity Analysis for a class of real polynomial matrix equation
The sensitivity of the symmetric positive solution of real polynomial matrix equationX~s±A′X~tA=Q is considered. A sufficient condition of the existence of the uniquesymmetric positive solution of the perturbed equation is established. As an estimator ofthe sensitivity of the unique symmetric positive solution, a condition number is definedand its expression is derived. Numerical experiments show that the defined conditionnumber works well.
4. Submatrix-constraint symmetric generalized inverse eigenvalue problem andoptimal solution problem
The descriptions in math language of the partial embedding problem and optimalsolution problem for the symmetric generalized eigenvalue model are given. A necessaryand sufficient condition for solvability and an expression of the general solution arederived.
5. Generalized symmetric matrix
Degree k (R, S )-symmetric matrix and (R,ω_η)-symmetric matrix are defined and theirstructures are characterized. The corresponding eigenvalue problem, singular value decomposition,least squares problem, inverse eigenvalue problem, minimal norm problemand optimal solution problem are studied.
1．无阻尼陀螺系统
给出无阻尼陀螺系统G(λ)=Mλ~2+Cλ+K的谱分解定理，其中M为质量矩阵，K为刚性矩阵，C为陀螺矩阵，并利用该定理解决了特征值反问题和无干扰特征值嵌入问题．谱分解定理表明矩阵M，C，K的含反对称参数矩阵S的谱分解式与实标准对(X，T)是一一对应的．若由系统的谱信息构造的矩阵T是分块对角矩阵，则S具有类上三角Hankel块结构．特别地，当系统只有半单的特征值时，存在一个标准对使得S只有2n个非零元素，并且非零元素为1或-1．另外，谱分解定理还给出无阻尼陀螺系统的二次特征值反问题有解的充要条件和通解表达式．当无阻尼陀螺系统只有单特征值时，无干扰特征值嵌入问题一定有解，并且可以推导出只包含已知的特征信息的通解表达式．
2．非线性矩阵方程的可解性和代数扰动分析
系统地研究矩阵方程X~s±A~*X~tA=Q(s＞0，t＞0)的可解性、迭代算法和代数扰动分析．首先利用矩阵分解方法推导X~s+A~*X~tA=Q存在Hermtian正定解的充要条件和通解表达式(如果解存在)．然后利用Brouwer不动点原理和特征值方法寻找Hermtian正定解的存在区间和存在唯一解的充分条件．将X~s-A~*X~tA=Q分为三种情况进行讨论：当s＜t时给出类最大解和类最小解的定义及存在的充分条件，并且推导出最大解存在的充分条件和能够收敛到最大解的迭代公式；当s＞t时设计求唯一解的一个新的迭代算法；当s=t时给出唯一解的精确表达式．另外建立代数扰动理论，分别给出唯一解、类最大解和类最小解的扰动上界．
3．实多项式矩阵方程的敏感性分析
考虑实系数多项式矩阵方程X~s±A′X~tA=Q的对称正定解的敏感性．首先给出扰动方程有唯一的对称正定解的充分条件，然后给出能反应对称正定解的敏感性的测度—条件数的定义及其精确表达式．数值实验表明该条件数能够反映对称正定解的敏感性．
4．子矩阵约束对称广义特征值反问题和最佳逼近问题
给出对称的广义特征值模型的部分修正问题和最佳逼近问题的数学描述，并给出问题有解的充要条件及通解表达式．
5．广义对称矩阵
给出k次(R，S)-对称矩阵和(R，ω_η)-对称矩阵的定义，并刻画其特殊结构．解决了相应的特征问题、奇异值分解、最小二乘问题、特征值反问题、极小范数问题和最佳逼近问题．
In this thesis some nonlinear matrix problems and special structure matrix problemsare investigated.
1. The undamped gyroscopic system
The spectral decomposition theorem of undamped gyroscopic system G(λ)=Mλ~2+Cλ+K is given, where M is mass matrix, K is stiffness matrix, C is gyroscopic matrix,and the inverse eigenvalue problem as well as the spill-over eigenvalue embeddingproblem is solved by the spectral decomposition theorem. The spectral decompositiontheorem shows that once the real standard pair (X, T) is given, the spectral decompositionsof M, C, K, including a skew-symmetric parameter matrix S,are decided. If T is ablock diagonal matrix, S becomes an upper tridiagonal Hankel-like matrix. Especially,if G(λ) only has semi-simple eigenvalues, then there exists a real standard pair, such thatS only has 2n nonzero elements 1 or-1. On the other hand, the spectral decompositiontheorem presents a necessary and sufficient condition for the solvability of the quadraticinverse eigenvalue problem, and an expression of the general solution(if exists). Whenall of the eigenvalues of G(λ) are simple, the spill-over eigenvalue embedding problemis always solvable and an expression of the general solution, only including the knowneigen information, is given.
2. Solvability and algebraic perturbation analysis of a class of nonlinear matrixequation
The nonlinear matrix equation X~s±A~*X~tA=Q(s＞0, t＞0) is systematicallystudied, including solvability, iterative algorithms and algebraic perturbation analysis. Atfirst, an necessary and sufficient condition and an expression of the general solution (ifexists) of X~s+A~*X~tA=Q are derived by a matrix decomposition method. Secondly,an interval in which an Hermitian positive solution exists and a sufficient condition forthe existence and uniqueness of the Hermitian positive solution are introduced by theBrouwer fixed point theorem and the eigenvalue method. For X~s-A~*X~tA=Q, we studythree cases: when s＜t, the maximal-like solution and the minimal-like solution aredefined, and sufficient conditions for their existences are given, furthermore, a sufficientcondition for the existence of the maximal solution and an iterative formula are given;when s＞t, a new iterative algorithm for the unique solution is given; when s=t, anexpression of the unique solution is obtained. Besides, the algebraic perturbation analysis theory for the unique solution, the maximal-like solution and the minimal-like solutionare established.
3. Sensitivity Analysis for a class of real polynomial matrix equation
The sensitivity of the symmetric positive solution of real polynomial matrix equationX~s±A′X~tA=Q is considered. A sufficient condition of the existence of the uniquesymmetric positive solution of the perturbed equation is established. As an estimator ofthe sensitivity of the unique symmetric positive solution, a condition number is definedand its expression is derived. Numerical experiments show that the defined conditionnumber works well.
4. Submatrix-constraint symmetric generalized inverse eigenvalue problem andoptimal solution problem
The descriptions in math language of the partial embedding problem and optimalsolution problem for the symmetric generalized eigenvalue model are given. A necessaryand sufficient condition for solvability and an expression of the general solution arederived.
5. Generalized symmetric matrix
Degree k (R, S )-symmetric matrix and (R,ω_η)-symmetric matrix are defined and theirstructures are characterized. The corresponding eigenvalue problem, singular value decomposition,least squares problem, inverse eigenvalue problem, minimal norm problemand optimal solution problem are studied.
引文
[1]H.Abou-Kandil et al.,Matrix Riccati Equations in control and systems theory.Birkhauser Verlag,2003.
[2]W.N.Anderson et al.,Consistent easrimates of the parameters of linear system.Ann.Math.Statist.1969,40:2064-2075.
[3]A.L.Andrew,Centrosymmetric matrices.SIAM,Rev.1998,40(3):697-698.
[4]J.P.Aubin,Applied Functional analysis.John Wiley & Sons,Inc,1979.
[5]Z.Bai,The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation.SIAM.J.Matrix Anal.Appl.2005,26(4):1100-1114.
[6]Z.Bai and Y.Su,SOAR:A second-order Arnoldi method for the solution of the quatratic eigenvalue problem.SIAM J.matrix.anal.appl.2005,26(3):640-659.
[7]M.J.Balas,Trends in large space structure control theory:fondest hopes,wildest dreams.IEEE Trans.Automat.Contr.1982,27:522-535.
[8]L.Barkwell,P.Lancaster and A.S.Markus,Gyroscopically Stabilized Systems:A Class of Quadratic Eigenvalue Problems with Real Spectrum.Canadian J.of Math.1992,44(1):42-53.
[9]M.Baruch,Optimization procedure to correct stiffness and flexibility matrices using cibration tests.AIAAJ.1978,16:1208-1210.
[10]M.Baruch,Optimal correction of mass and stiffness matrices using measured modes.AIAA J.1982,20:1623-1626.
[11]A.Berman,W.G.Flannely,Theory of incomplete models of dynamic structures.AIAA J.1971,9:1481-1487.
[12]A.Berman,E.J.Nagy,Improvement of a large analytical model using test data.AIAA J.1983,21:1168-1173.
[13]P.Benner,V.Mehrmann,H.Xu,A numerically stable,structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils.Numer.Math.1998,78(3):329-358.
[14]P.Benner,A.J.Laub and V.Mehrmann,Benchmarks for the Numerical Solution of Algebraic Riccati Equations.IEEE Control Systems Magazine 1997,7:18-28
[15]A.Ben-Israel,T.N.E.Greville.Generalized Inverses:Theory and Applications (2nd edition).Springer Verlag,New York,2003.
[16]A.Bhaya,C.A.Desoer,On the design of large flexible space structures.IEEE Trans.Auto.Contr.1985,30:1118-1120.
[17]R.Bhatia,Matrix Analysis,Graduate Texts in Mathematics,vol.169,Springer Verlag,1997.
[18]R.Bhatia,K.B.Sinha,Variation of real powers of positive operators.Indiana Univ.Math.J.1994,43:913-925.
[19]S.Bittanti,A.Laub and J.C.Willems,The Riccati Equation,Berlin:Springer-Verlag,1991
[20]R.Byers and S.Nash,On the singular “vectors” of the Lyapunov operator.SIAM J.Alg.Disc.Meth.1987,8:59-66.
[21]Y.Cai et al.,Solutions to a quadratic inverse eigenvalue problem,Linear Algebra Appl.2008,doi:10.1016/j.laa.2008.04.015
[22]C.Chen,Perturbation analysis for solutions of algebraic Riccati equations,J.Comput.Math.1988,6:336-347.
[23]H.C.Chen.Generalized reflexive matrices:Special properties and applicationes.SIAM.J.Matrix.Anal.Appl.1998,19(1):140-153.
[24]M.Cheng,S.Xu,Perturbation analysis of the Hermitian positive definite solution of the matrix equation X-A~*X~(-2)A=I.Linear Algebra Appl.2005,394:39-51.
[25]D.Chu,X.Liu,V.Mehrmann,A numerical method form computing the Hamiltonian Schur form.Numer.Math.2007,105:375-412.
[26]E.K.W.Chu et al.,Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations.Internat.J.Control 2004,77:767-788.
[27]E.K.W.Chu,H.Y.Fan,W.W.Lin,A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations.Linear Algebra Appl.2005,396:55-80.
[28]M.T.Chu,W.-W.Lin,S.-F.Xu,Updating quadratic models with no spill-over effect on unmeasured spectral data.Inverse Prob.2007,23:243-256.
[29]M.T.Chu et al.,Spill-over phenomenon in quadratic model updating.AIAA J.2008,46 (2):420-428.
[30]H.H.F.Chu,Mudal Testing and Modal Refinement.American Society of Mechanical Engineers,New York,1983.
[31]R.E.Cline,Representations for the generalized inverse of a partitioned matrix.SIAM J.Appl.Math.1964,12:588-600.
[32]R.G.Cobb,B.Liebst,Structural damage identification using assigned partial eigenstructure,AIAA J.1997,35:152-158.
[33]戴华，线性约束下的矩阵束最佳逼近及其应用．计算数学1989，11：29-37．
[34]戴华，谱约束下实对称矩阵束的最隹逼近．高等学校计算数学学报1990，12：177-187．
[35]戴华，用试验数据修正刚度矩阵．航空学报1994，15：1091-1094．
[36]H.Dai,About an inverse eigenvalue problem arising in vibration analysis.RAIRO Math.Model.Numer.Anal.1995,29:421-434.
[37]H.Dai,An algorithm for symmetric generalizedinverse eigenvalue problems,Linear algebra appl.1999,296:79-98.
[38]S.M.El-Sayed,M.G.Petkov,Iterative methods for nonlinear equations X + A~*X~(-α)A=I.Linear Algebra Appl.2005,403:45-52.
[39]S.M.El-Sayed,A.C.M.Ran,On an iteration method for solving a class of nonlinear matrix equations.SIAM J.Matrix Anal.Appl.2001,23:632-645.
[40]J.C.Engwerda,A.C.M.Ran,A.L.Rijkeboer,Necessary and sufficient conditions for the existence of a positive solution of the matrix equation X + A~*X~(-1)A=I.Linear Algebra Appl.1993,184:255-275.
[41]M.I.Friswell,J.E.Mottershead,Finite Element Model Updating in Structural Dynamics.Kluwer Academic Publishers,Dordrecht,1995.
[42]T.Furuta,Operator inequalities associated with Holder-McCarthy and Kantorovich inequalities,J.Inequal.Appl.1998,6:137-148.
[43]A.Ferrante,B.C.Levy,Hermitian solutions of the equation X=Q + NX~(-1)N~*,Linear Algebra Appl.1996,247:359-373.
[44]F.R.Gantmacher,The Theory of Matrices,Vol.1,Translated from the Russian by K.A.Hirsch.AMS Chelsea Publishing,Providence,RI,1998.MR1657129 (99f:15001)
[45]高东杰，张玉海，矩阵方程x-A~*X~qA=I(0＜q＜1)Hermite正定解的扰动分析．计算数学2007，29(4)：403-412
[46]I.Gohberg,P.Lancaster,L.Rodman,Spectral analysis of selfadjoint matrix polynomials.Ann.Math.1980,112:33-71.
[47]I.Gohberg,P.Lancaster,L.Rodman,Matrix Polynomials.Academic Press,New York,1982.
[48]C.Guo,Numerical solution of a quadratic eigenvalue problem.Linear Algebra Appl.2004,385:391-406.
[49]G.H.Golub,C.F.Van Loan,Matrix Computations,3rd Edition.The Johns Hopkins University Press,Baltimore,1996.
[50]C.R.H.Hanusa.Pseudo-centrosymmetric matrices,with applications to counting perfect matchings.Linear Algebra Appl.2007,427:206-217.
[51]J.He,Structural modification.Phil.Trans.R.Soc.Lond.2001,A359:187-204.
[52]N.J.Higham,Perturbation theory and backward error for AX-XB=C.BIT 1993,33:124-136.
[53]R.A.Horn,C.R.Johnson,Topics in Matrix Analysis.Cambridge University Press,1991.
[54]R.A.Horn,C.R.Johnson,Matrix Analysis.Cambridge University Press,1990.
[55]V.I.Hasanov,Positive definite solutions of the matrix equations X±A~*X~(-q)A=Q.Linear Algebra Appl.2005,404:166-182.
[56]I.G.Ivanov,S.M.Sayed,Properties of positive definite solutions of the equations X + A~*X~(-2)A=I.LinearAlgebraAppl.1998,279:303-316.
[57]I.G.Ivanov,V.I.Hasavov,B.V.Minchev,On matrix equations X±A~*X~(-2)A=I.Linear Algebra Appl.2001,326:27-44.
[58]I.G.Ivanov,On positive definite solutions of the family of matrix equations X + A~*X~(-n)A=Q.J.Comput.Appl.Math.2006,193:277-301.
[59]Y.Kuo,W.Lin,S.Xu,Solutions of the partially described inverse quadratic eigenvalue problem.SIAMJ.Matrix Anal.Appl.2006,29(1):33-53.
[60]M.Kimura,Doubling algorithm for continuous-time algebraic Riccati equation.Internat.J.Systems Sci.1989,20:191-202.
[61]M.Konstantinov,P.Petkov,Note on Perturbation theory for algebraic Riccati equations.SIAMJ.MatrixAnal.Appl.1999,21:327-327.
[62]P.Lancaster,Lambda-Matrices and Vibrating Systems.Pergamon Press,1966.
[63]P.Lancaster,M.Tismenetsky,The Theory of Matrices Second Edition with Applications.ACADEMIC PRESS,1985.
[64]P.Lancaster,L.Rodman,The Algebraic Riccati Equation.Oxford University Press,Oxford,1995
[65]A.J.Laub,A Schur method for solving algebraic Riccati equations,IEEE Trans.Automat.Contr.1979,24:913-921.
[66]A.J.Laub,“Invariant Subspace Methods for the Numerical Solution of Riccati Equations” in S.Bittanti,A.J.Laub and J.C.Willems editors The Riccari Equation.Berlin,Springer-Verlag,1991,163-196.
[67] H. Lee, Y. Lim, Invariant metrics, contractions and nonlinear matrix equations. Non-linearity 2008, 21: 857-878.
[68] L.W.Li, A new method for structural model updating and stiffness identification.Mech. Syst. Signal Process 2002, 16: 155-167.
[69] M.Link, Identification and correction of errors in analytical models using test data-theoretical and practical bounda. Proc. of the 8th International Modal Analysis Conference 1990:570-578.
[70] X. Liu, H. Gao, On the positive definite solutions of the matrix equations X~s±A~TX~(-t)A =I_n. Linear Algebra Appl. 2003, 368: 83-97
[71] W.Luber, A.Lotze, Application of sensitivity methods for error localization in finite element systems. Proc. of the 8th International Modal Analysis Conference 1990:598-604.
[72] G. Martin-Herr(?)n, Symplectic Methods for the Solution to Riccati Matrix Equations Related to Macroeconomic Models. Computational Economics 1999, 13: 61-91.
[73] V. Mehrmann, A step towards a unified treatment of continuous and discrete time control problems. Linear Algebra Appl. 1996,241-243:749-779.
[74] C.Minas, D.J. Inman, Matching finite element models to modal data. J. Vibration and Acoustics 1990, 112: 84-92.
[75] J.E.Mottershead, M.I.Friswell, Model uptating in structural dynamics: a survey. J.Sound and vibration1993, 167: 347-375.
[76] J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equation in several variables. Academic Press, New York, 1970.
[77] D.V. Ouellette, Schur complements and statistics. Linear Algebra Appl. 1981, 36:187-295.
[78] J.C.O'Callahan,C.-M. Chou, Localzation of model errors in optimized mass and stiffness matrices using modal test data. Internat.J. Analytical and Experimental Analysis 1989,4: 8-14.
[79]Zhuohua Peng,Xiyan Hu,Lei Zhang,An efficient algorithm for the least-squares reflexive solution of the matrix equation A_1XB_1=C_1,A_2XB_2=C_2.Appl.Math.Comput.2006,181(2):988-999.
[80]Zhenyun Peng,The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation.Appl.Math.Comput.2005,162(3):1377-1389.
[81]J.Qian,W.Lin,A numerical method for quadratic eigenvalue problems of gyroscopic systems.J.Sound Vib.2007,306:284-296.
[82]A.C.M.Ran,M.C.B.Reurings,On the nonlinear matrix equation X+A~*(?)(X)A=Q:solutions and perturbation theory.Linear Algebra Appl.2002,346:15-26.
[83]A.C.M.Ran,M.C.B.Reurings,L.Rodman,A perturbation analysis for nonlinear selfadjoint operator equations,SIAM.J.Matrix.Anal.Appl.2006,28:89-104.
[84]A.C.M.Ran,M.C.B.Reurings,The Symmetric Linear Matrix Equation.Electron.Linear Algebra 2002,9:93-107.
[85]A.C.M.Ran,M.C.B.Reurings,A fixed point theorem in partially ordered sets and some applications to matrix equations,Proceedings of the AMS 2004,132:1435-1443.
[86]A.C.M.Ran,M.C.B.Reurings,A nonlinear matrix equation connected to interpolation theory.Linear Algebra Appl.2004,379:289-302.
[87]M.C.B.Reurings,Contractive maps on normed linear spaces and their applications to nonlinear matrix equations.Linear Algebra Appl.2006,418:292-311.
[88]M.C.B.Reurings,Symmetric Matrix Equations.PhD thesis,Vrije Universiteit Amsterdam 2003(http://www.math.vu.nl/proefschriften/Reurings).
[89]A.Ruhe,Algorithms for the nonlinear eigenvalue problem.SIAM J.Numer.Anal.1973,10:674-689
[90]J.R.Rice,A theory of condition.SIAMJ.Numer.Anal.1966,3:287-310.
[91]J.Sun,Residual bounds of approximate solutions of the algebraic Riccati equation.Numer.Math.1997,76:249-263.
[92]J.Sun,Backward error for the discrete-time algebraic Riccati equation.Linear Algebra Appl.1997,259:183-208
[93]J.Sun,Backward perturbation analysis of the periodic discrete-time algebraic Riccati equation.SIAMJ.Matrix Anal.Appl.2004,26:1-19.
[94]J.Sun,Sensitivity analysis of the discrete-time algebraic Riccati equation.Linear AlgebraAppl.1998,275-276:595-615.
[95]J.Sun,Eigenvalues and eigenvectors of a matrix dependent on several parameters,J.Comput.Math.1985,3:351-364.
[96]J.Sun,Perturbation theory for algebraic Riccati equations.SIAM J.Matrix Anal.Appl.1998,19(1):39-65.
[97]F.Tisseur,K.Meerbergen,The quadratic eigenvalue problem,SIAM Review 2001,43(2):235-286.
[98]F.Tissseur,A chart of backward errors for singly and doubly structured eigenvalue problems,SIAM J.Matrix Anal 2003,24:877-897.
[99]G.E.Trapp,The Ricatti equation and the geometric mean.Contemp.Math.1985,47:437-445.
[100]W.F.Trench,Characterization and properties of matrices with generalized symmetry or skew symmetry.Linear Algebra Appl.2004,377:207-218.
[101]W.F.Trench,Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices.Linear Algebra Appl.2004,389(15):23-31.
[102]W.F.Trench,Characterization and properties of (R,S)-symmetric,(R,S)-skew symmetric,and (R,S)-conjugate matrices,SIAM J.Matrix Anal.Appl.2005,26:748-757.
[103]W.F.Trench,Hermitian,Hermitian R-symmetric and Hermitian R-skew symmetric Procrustes Problems.Linear Algebra Appl.2004,38:783-98.
[104]W.F.Trench,Multilevel matrices with involutory symmetries and skew symmetries.Linear Algebra Appl.2005,403:53-74.
[105]W.F.Trench,Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry,Linear Algebra Appl.2004,380:199-211.
[106]V.A.Tsachouridis et al,Scaling of general quadeativ matrix equations.Internat.J.Control 2006,7:207-223.
[107]M.Wang,M.Wei,On the maximal-like solution of matrix equation X +A~*X~(-2)A=I.Numer.Math.J.Chinese Univ.(English Ser.)2006,15:67-73.
[108]魏木生，广义最小二乘问题：理论与计算科学出版社，北京，2006．
[109]M.Wei,Q.Wang,On rank-constrained Hermitian nonnegative-definite least square solutions to the matrix equation AXA~H=B.Int.J.Comp.Math.2007,00:1-8.
[110]F.S.Wei,D.W.Zhang,Mass matrix modification using element correction method.AIAA J.1989,27:119-121.
[111]F.S.Wei,Structural dynamic model improvement using vibration test data.AIAA J.1990,28:175-177.
[112]F.S.Wei,Mass and stiffness interaction effects in analytical model modification.AIAA J.1990,28:1686-1688.
[113]Y.M.Wei,Solving singular linear system and generalized inverse,Ph.D.Thesis,Institute of Mathematics,Fudan University,Shanghai,March 1997.
[114]Y.M.Wei,G.R.Wang,Perturbation theory for the Drazin inverse and its applications.Linear Algebra Appl.1997,258:179-186.
[115]H.Xu,Transformations between discrete-time and continuous-time algebraic Riccati equations.Linear Algebra Appl.2007,425:77-101.
[116]S.Xu,Perturbation analysis of the maximal solution of the matrix equation X + A~*X~(-1)A=P.Linear Algebra Appl.2001,336:61-70.
[117]S.Xu,Sensitivity analysis of the algebraic Riccati equations,Numer.Math.1996,75:121-134.
[118]H.Q.Xie,Sensitivity analysis of eigenvalue problem,Ph.D.Dissertation,Nanjin University of Aeronautics and Astronautica,2003.
[119]尹小艳，刘三阳，房亮，矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析．计算数学，2008，30(1)：37-48．
[120]Y.Yang,The iterative mathod for solving nolinear matrix equation X~s + A~*X~(-t)A=Q.Appl.Math.Comp.2006,dot:10.1016/j.amc.2006.09.085.
[121]Y.Yuan,H.Dai,Inverse problems for symmetric matrices with a submatrix constraint,Applied Numerical Mathematics 57(2007) 646-656.
[122]Y.Yuan,H.Dai,A genetalized inverse eigenvalue problem in structural dynamic model updating.J.Comp.Appl.Math.2009,226(1):42-49.
[123]J.Zabezyk,Non-uniform semi-infinite grounded grids.SIAM J.Appl.Math.1982,13:770-788.
[124]X.Zhan,Matrix inequalities.Springer-Verlag,Berlin,2002.
[125]X.Zhan,Computing the extremal positive definite solutions of a matrix equation.SIAM J.Sci.Comput.1996,17:1167-1174.
[126]X.Zhan,J.Xie,On the matrix equation X + A~TX~(-1)A=I.Linear Algebra Appl.1996,247:337-345.
[127]Fuzhao Zhou,Lei Zhang,Xiyan Hu,Least-square solutions for inverse problems of centrosymmetric matrices.Comp.Math.Appl 2003,45(10-11):1581-1589.
[128]O.Zhang,A.Zerva,D.W.Zhang,Stiffness matrix adjustment using incomplete measured modes.AIAA J.1997,35:917-919.
[2]W.N.Anderson et al.,Consistent easrimates of the parameters of linear system.Ann.Math.Statist.1969,40:2064-2075.
[3]A.L.Andrew,Centrosymmetric matrices.SIAM,Rev.1998,40(3):697-698.
[4]J.P.Aubin,Applied Functional analysis.John Wiley & Sons,Inc,1979.
[5]Z.Bai,The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation.SIAM.J.Matrix Anal.Appl.2005,26(4):1100-1114.
[6]Z.Bai and Y.Su,SOAR:A second-order Arnoldi method for the solution of the quatratic eigenvalue problem.SIAM J.matrix.anal.appl.2005,26(3):640-659.
[7]M.J.Balas,Trends in large space structure control theory:fondest hopes,wildest dreams.IEEE Trans.Automat.Contr.1982,27:522-535.
[8]L.Barkwell,P.Lancaster and A.S.Markus,Gyroscopically Stabilized Systems:A Class of Quadratic Eigenvalue Problems with Real Spectrum.Canadian J.of Math.1992,44(1):42-53.
[9]M.Baruch,Optimization procedure to correct stiffness and flexibility matrices using cibration tests.AIAAJ.1978,16:1208-1210.
[10]M.Baruch,Optimal correction of mass and stiffness matrices using measured modes.AIAA J.1982,20:1623-1626.
[11]A.Berman,W.G.Flannely,Theory of incomplete models of dynamic structures.AIAA J.1971,9:1481-1487.
[12]A.Berman,E.J.Nagy,Improvement of a large analytical model using test data.AIAA J.1983,21:1168-1173.
[13]P.Benner,V.Mehrmann,H.Xu,A numerically stable,structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils.Numer.Math.1998,78(3):329-358.
[14]P.Benner,A.J.Laub and V.Mehrmann,Benchmarks for the Numerical Solution of Algebraic Riccati Equations.IEEE Control Systems Magazine 1997,7:18-28
[15]A.Ben-Israel,T.N.E.Greville.Generalized Inverses:Theory and Applications (2nd edition).Springer Verlag,New York,2003.
[16]A.Bhaya,C.A.Desoer,On the design of large flexible space structures.IEEE Trans.Auto.Contr.1985,30:1118-1120.
[17]R.Bhatia,Matrix Analysis,Graduate Texts in Mathematics,vol.169,Springer Verlag,1997.
[18]R.Bhatia,K.B.Sinha,Variation of real powers of positive operators.Indiana Univ.Math.J.1994,43:913-925.
[19]S.Bittanti,A.Laub and J.C.Willems,The Riccati Equation,Berlin:Springer-Verlag,1991
[20]R.Byers and S.Nash,On the singular “vectors” of the Lyapunov operator.SIAM J.Alg.Disc.Meth.1987,8:59-66.
[21]Y.Cai et al.,Solutions to a quadratic inverse eigenvalue problem,Linear Algebra Appl.2008,doi:10.1016/j.laa.2008.04.015
[22]C.Chen,Perturbation analysis for solutions of algebraic Riccati equations,J.Comput.Math.1988,6:336-347.
[23]H.C.Chen.Generalized reflexive matrices:Special properties and applicationes.SIAM.J.Matrix.Anal.Appl.1998,19(1):140-153.
[24]M.Cheng,S.Xu,Perturbation analysis of the Hermitian positive definite solution of the matrix equation X-A~*X~(-2)A=I.Linear Algebra Appl.2005,394:39-51.
[25]D.Chu,X.Liu,V.Mehrmann,A numerical method form computing the Hamiltonian Schur form.Numer.Math.2007,105:375-412.
[26]E.K.W.Chu et al.,Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations.Internat.J.Control 2004,77:767-788.
[27]E.K.W.Chu,H.Y.Fan,W.W.Lin,A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations.Linear Algebra Appl.2005,396:55-80.
[28]M.T.Chu,W.-W.Lin,S.-F.Xu,Updating quadratic models with no spill-over effect on unmeasured spectral data.Inverse Prob.2007,23:243-256.
[29]M.T.Chu et al.,Spill-over phenomenon in quadratic model updating.AIAA J.2008,46 (2):420-428.
[30]H.H.F.Chu,Mudal Testing and Modal Refinement.American Society of Mechanical Engineers,New York,1983.
[31]R.E.Cline,Representations for the generalized inverse of a partitioned matrix.SIAM J.Appl.Math.1964,12:588-600.
[32]R.G.Cobb,B.Liebst,Structural damage identification using assigned partial eigenstructure,AIAA J.1997,35:152-158.
[33]戴华，线性约束下的矩阵束最佳逼近及其应用．计算数学1989，11：29-37．
[34]戴华，谱约束下实对称矩阵束的最隹逼近．高等学校计算数学学报1990，12：177-187．
[35]戴华，用试验数据修正刚度矩阵．航空学报1994，15：1091-1094．
[36]H.Dai,About an inverse eigenvalue problem arising in vibration analysis.RAIRO Math.Model.Numer.Anal.1995,29:421-434.
[37]H.Dai,An algorithm for symmetric generalizedinverse eigenvalue problems,Linear algebra appl.1999,296:79-98.
[38]S.M.El-Sayed,M.G.Petkov,Iterative methods for nonlinear equations X + A~*X~(-α)A=I.Linear Algebra Appl.2005,403:45-52.
[39]S.M.El-Sayed,A.C.M.Ran,On an iteration method for solving a class of nonlinear matrix equations.SIAM J.Matrix Anal.Appl.2001,23:632-645.
[40]J.C.Engwerda,A.C.M.Ran,A.L.Rijkeboer,Necessary and sufficient conditions for the existence of a positive solution of the matrix equation X + A~*X~(-1)A=I.Linear Algebra Appl.1993,184:255-275.
[41]M.I.Friswell,J.E.Mottershead,Finite Element Model Updating in Structural Dynamics.Kluwer Academic Publishers,Dordrecht,1995.
[42]T.Furuta,Operator inequalities associated with Holder-McCarthy and Kantorovich inequalities,J.Inequal.Appl.1998,6:137-148.
[43]A.Ferrante,B.C.Levy,Hermitian solutions of the equation X=Q + NX~(-1)N~*,Linear Algebra Appl.1996,247:359-373.
[44]F.R.Gantmacher,The Theory of Matrices,Vol.1,Translated from the Russian by K.A.Hirsch.AMS Chelsea Publishing,Providence,RI,1998.MR1657129 (99f:15001)
[45]高东杰，张玉海，矩阵方程x-A~*X~qA=I(0＜q＜1)Hermite正定解的扰动分析．计算数学2007，29(4)：403-412
[46]I.Gohberg,P.Lancaster,L.Rodman,Spectral analysis of selfadjoint matrix polynomials.Ann.Math.1980,112:33-71.
[47]I.Gohberg,P.Lancaster,L.Rodman,Matrix Polynomials.Academic Press,New York,1982.
[48]C.Guo,Numerical solution of a quadratic eigenvalue problem.Linear Algebra Appl.2004,385:391-406.
[49]G.H.Golub,C.F.Van Loan,Matrix Computations,3rd Edition.The Johns Hopkins University Press,Baltimore,1996.
[50]C.R.H.Hanusa.Pseudo-centrosymmetric matrices,with applications to counting perfect matchings.Linear Algebra Appl.2007,427:206-217.
[51]J.He,Structural modification.Phil.Trans.R.Soc.Lond.2001,A359:187-204.
[52]N.J.Higham,Perturbation theory and backward error for AX-XB=C.BIT 1993,33:124-136.
[53]R.A.Horn,C.R.Johnson,Topics in Matrix Analysis.Cambridge University Press,1991.
[54]R.A.Horn,C.R.Johnson,Matrix Analysis.Cambridge University Press,1990.
[55]V.I.Hasanov,Positive definite solutions of the matrix equations X±A~*X~(-q)A=Q.Linear Algebra Appl.2005,404:166-182.
[56]I.G.Ivanov,S.M.Sayed,Properties of positive definite solutions of the equations X + A~*X~(-2)A=I.LinearAlgebraAppl.1998,279:303-316.
[57]I.G.Ivanov,V.I.Hasavov,B.V.Minchev,On matrix equations X±A~*X~(-2)A=I.Linear Algebra Appl.2001,326:27-44.
[58]I.G.Ivanov,On positive definite solutions of the family of matrix equations X + A~*X~(-n)A=Q.J.Comput.Appl.Math.2006,193:277-301.
[59]Y.Kuo,W.Lin,S.Xu,Solutions of the partially described inverse quadratic eigenvalue problem.SIAMJ.Matrix Anal.Appl.2006,29(1):33-53.
[60]M.Kimura,Doubling algorithm for continuous-time algebraic Riccati equation.Internat.J.Systems Sci.1989,20:191-202.
[61]M.Konstantinov,P.Petkov,Note on Perturbation theory for algebraic Riccati equations.SIAMJ.MatrixAnal.Appl.1999,21:327-327.
[62]P.Lancaster,Lambda-Matrices and Vibrating Systems.Pergamon Press,1966.
[63]P.Lancaster,M.Tismenetsky,The Theory of Matrices Second Edition with Applications.ACADEMIC PRESS,1985.
[64]P.Lancaster,L.Rodman,The Algebraic Riccati Equation.Oxford University Press,Oxford,1995
[65]A.J.Laub,A Schur method for solving algebraic Riccati equations,IEEE Trans.Automat.Contr.1979,24:913-921.
[66]A.J.Laub,“Invariant Subspace Methods for the Numerical Solution of Riccati Equations” in S.Bittanti,A.J.Laub and J.C.Willems editors The Riccari Equation.Berlin,Springer-Verlag,1991,163-196.
[67] H. Lee, Y. Lim, Invariant metrics, contractions and nonlinear matrix equations. Non-linearity 2008, 21: 857-878.
[68] L.W.Li, A new method for structural model updating and stiffness identification.Mech. Syst. Signal Process 2002, 16: 155-167.
[69] M.Link, Identification and correction of errors in analytical models using test data-theoretical and practical bounda. Proc. of the 8th International Modal Analysis Conference 1990:570-578.
[70] X. Liu, H. Gao, On the positive definite solutions of the matrix equations X~s±A~TX~(-t)A =I_n. Linear Algebra Appl. 2003, 368: 83-97
[71] W.Luber, A.Lotze, Application of sensitivity methods for error localization in finite element systems. Proc. of the 8th International Modal Analysis Conference 1990:598-604.
[72] G. Martin-Herr(?)n, Symplectic Methods for the Solution to Riccati Matrix Equations Related to Macroeconomic Models. Computational Economics 1999, 13: 61-91.
[73] V. Mehrmann, A step towards a unified treatment of continuous and discrete time control problems. Linear Algebra Appl. 1996,241-243:749-779.
[74] C.Minas, D.J. Inman, Matching finite element models to modal data. J. Vibration and Acoustics 1990, 112: 84-92.
[75] J.E.Mottershead, M.I.Friswell, Model uptating in structural dynamics: a survey. J.Sound and vibration1993, 167: 347-375.
[76] J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equation in several variables. Academic Press, New York, 1970.
[77] D.V. Ouellette, Schur complements and statistics. Linear Algebra Appl. 1981, 36:187-295.
[78] J.C.O'Callahan,C.-M. Chou, Localzation of model errors in optimized mass and stiffness matrices using modal test data. Internat.J. Analytical and Experimental Analysis 1989,4: 8-14.
[79]Zhuohua Peng,Xiyan Hu,Lei Zhang,An efficient algorithm for the least-squares reflexive solution of the matrix equation A_1XB_1=C_1,A_2XB_2=C_2.Appl.Math.Comput.2006,181(2):988-999.
[80]Zhenyun Peng,The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation.Appl.Math.Comput.2005,162(3):1377-1389.
[81]J.Qian,W.Lin,A numerical method for quadratic eigenvalue problems of gyroscopic systems.J.Sound Vib.2007,306:284-296.
[82]A.C.M.Ran,M.C.B.Reurings,On the nonlinear matrix equation X+A~*(?)(X)A=Q:solutions and perturbation theory.Linear Algebra Appl.2002,346:15-26.
[83]A.C.M.Ran,M.C.B.Reurings,L.Rodman,A perturbation analysis for nonlinear selfadjoint operator equations,SIAM.J.Matrix.Anal.Appl.2006,28:89-104.
[84]A.C.M.Ran,M.C.B.Reurings,The Symmetric Linear Matrix Equation.Electron.Linear Algebra 2002,9:93-107.
[85]A.C.M.Ran,M.C.B.Reurings,A fixed point theorem in partially ordered sets and some applications to matrix equations,Proceedings of the AMS 2004,132:1435-1443.
[86]A.C.M.Ran,M.C.B.Reurings,A nonlinear matrix equation connected to interpolation theory.Linear Algebra Appl.2004,379:289-302.
[87]M.C.B.Reurings,Contractive maps on normed linear spaces and their applications to nonlinear matrix equations.Linear Algebra Appl.2006,418:292-311.
[88]M.C.B.Reurings,Symmetric Matrix Equations.PhD thesis,Vrije Universiteit Amsterdam 2003(http://www.math.vu.nl/proefschriften/Reurings).
[89]A.Ruhe,Algorithms for the nonlinear eigenvalue problem.SIAM J.Numer.Anal.1973,10:674-689
[90]J.R.Rice,A theory of condition.SIAMJ.Numer.Anal.1966,3:287-310.
[91]J.Sun,Residual bounds of approximate solutions of the algebraic Riccati equation.Numer.Math.1997,76:249-263.
[92]J.Sun,Backward error for the discrete-time algebraic Riccati equation.Linear Algebra Appl.1997,259:183-208
[93]J.Sun,Backward perturbation analysis of the periodic discrete-time algebraic Riccati equation.SIAMJ.Matrix Anal.Appl.2004,26:1-19.
[94]J.Sun,Sensitivity analysis of the discrete-time algebraic Riccati equation.Linear AlgebraAppl.1998,275-276:595-615.
[95]J.Sun,Eigenvalues and eigenvectors of a matrix dependent on several parameters,J.Comput.Math.1985,3:351-364.
[96]J.Sun,Perturbation theory for algebraic Riccati equations.SIAM J.Matrix Anal.Appl.1998,19(1):39-65.
[97]F.Tisseur,K.Meerbergen,The quadratic eigenvalue problem,SIAM Review 2001,43(2):235-286.
[98]F.Tissseur,A chart of backward errors for singly and doubly structured eigenvalue problems,SIAM J.Matrix Anal 2003,24:877-897.
[99]G.E.Trapp,The Ricatti equation and the geometric mean.Contemp.Math.1985,47:437-445.
[100]W.F.Trench,Characterization and properties of matrices with generalized symmetry or skew symmetry.Linear Algebra Appl.2004,377:207-218.
[101]W.F.Trench,Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices.Linear Algebra Appl.2004,389(15):23-31.
[102]W.F.Trench,Characterization and properties of (R,S)-symmetric,(R,S)-skew symmetric,and (R,S)-conjugate matrices,SIAM J.Matrix Anal.Appl.2005,26:748-757.
[103]W.F.Trench,Hermitian,Hermitian R-symmetric and Hermitian R-skew symmetric Procrustes Problems.Linear Algebra Appl.2004,38:783-98.
[104]W.F.Trench,Multilevel matrices with involutory symmetries and skew symmetries.Linear Algebra Appl.2005,403:53-74.
[105]W.F.Trench,Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry,Linear Algebra Appl.2004,380:199-211.
[106]V.A.Tsachouridis et al,Scaling of general quadeativ matrix equations.Internat.J.Control 2006,7:207-223.
[107]M.Wang,M.Wei,On the maximal-like solution of matrix equation X +A~*X~(-2)A=I.Numer.Math.J.Chinese Univ.(English Ser.)2006,15:67-73.
[108]魏木生，广义最小二乘问题：理论与计算科学出版社，北京，2006．
[109]M.Wei,Q.Wang,On rank-constrained Hermitian nonnegative-definite least square solutions to the matrix equation AXA~H=B.Int.J.Comp.Math.2007,00:1-8.
[110]F.S.Wei,D.W.Zhang,Mass matrix modification using element correction method.AIAA J.1989,27:119-121.
[111]F.S.Wei,Structural dynamic model improvement using vibration test data.AIAA J.1990,28:175-177.
[112]F.S.Wei,Mass and stiffness interaction effects in analytical model modification.AIAA J.1990,28:1686-1688.
[113]Y.M.Wei,Solving singular linear system and generalized inverse,Ph.D.Thesis,Institute of Mathematics,Fudan University,Shanghai,March 1997.
[114]Y.M.Wei,G.R.Wang,Perturbation theory for the Drazin inverse and its applications.Linear Algebra Appl.1997,258:179-186.
[115]H.Xu,Transformations between discrete-time and continuous-time algebraic Riccati equations.Linear Algebra Appl.2007,425:77-101.
[116]S.Xu,Perturbation analysis of the maximal solution of the matrix equation X + A~*X~(-1)A=P.Linear Algebra Appl.2001,336:61-70.
[117]S.Xu,Sensitivity analysis of the algebraic Riccati equations,Numer.Math.1996,75:121-134.
[118]H.Q.Xie,Sensitivity analysis of eigenvalue problem,Ph.D.Dissertation,Nanjin University of Aeronautics and Astronautica,2003.
[119]尹小艳，刘三阳，房亮，矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析．计算数学，2008，30(1)：37-48．
[120]Y.Yang,The iterative mathod for solving nolinear matrix equation X~s + A~*X~(-t)A=Q.Appl.Math.Comp.2006,dot:10.1016/j.amc.2006.09.085.
[121]Y.Yuan,H.Dai,Inverse problems for symmetric matrices with a submatrix constraint,Applied Numerical Mathematics 57(2007) 646-656.
[122]Y.Yuan,H.Dai,A genetalized inverse eigenvalue problem in structural dynamic model updating.J.Comp.Appl.Math.2009,226(1):42-49.
[123]J.Zabezyk,Non-uniform semi-infinite grounded grids.SIAM J.Appl.Math.1982,13:770-788.
[124]X.Zhan,Matrix inequalities.Springer-Verlag,Berlin,2002.
[125]X.Zhan,Computing the extremal positive definite solutions of a matrix equation.SIAM J.Sci.Comput.1996,17:1167-1174.
[126]X.Zhan,J.Xie,On the matrix equation X + A~TX~(-1)A=I.Linear Algebra Appl.1996,247:337-345.
[127]Fuzhao Zhou,Lei Zhang,Xiyan Hu,Least-square solutions for inverse problems of centrosymmetric matrices.Comp.Math.Appl 2003,45(10-11):1581-1589.
[128]O.Zhang,A.Zerva,D.W.Zhang,Stiffness matrix adjustment using incomplete measured modes.AIAA J.1997,35:917-919.