矩阵扰动若干问题研究
|
摘要
本文主要研究有关矩阵问题中各种解的扰动分析。利用代数和分析的方法对各类问题给出一些新的扰动界，包括广义极分解，矩阵特征空间与奇异空间，奇异值Rayleigh商，广义特征值问题，非线性矩阵方程以及结构线性方程组等。
第一章，介绍了矩阵扰动分析中一些重要概念。同时对本文的主要内容作了总的叙述。
第二章，当原矩阵与其扰动矩阵的秩不相等时，对广义极分解中的次酉极因子和半正定极因子给出了一些扰动界。对次酉极因子我们改进了以往的一些结论。对半正定极因子，获得任意酉不变范数下的扰动界。特别改进了正定极因子在Frobenius范数下的一个经典的结果。
第三章，研究Hermite矩阵特征空间在Frobenius范数的加法和乘法扰动。使用特征值的双分离度，给出了一些绝对与相对扰动界，从某种意义上来说改进了以往相应的结论。
第四章，研究矩阵奇异空间在Frobenius范数下的绝对与相对扰动界。使用奇异值的双分离度，获得左、右奇异空间各自的绝对扰动界和左右奇异空间联合的绝对扰动界；由此导出奇异子空间的条件数的上界。同时也导出Wedin's sin(?)型定理的相对扰动界和左右奇异空间各自的相对扰动界。
第五章，研究奇异值的Rayleigh商问题。一方面将Hermite矩阵的Rayleigh商问题的一些结论推广到奇异值的Rayleigh商上。另一方面给出近似奇异值在任意酉不变范数下的一些界。
第六章，研究可对角化矩阵束的广义特征值的扰动界。将孙[66]的三个结论从正规矩阵对推广到可对角化矩阵对上。
第七章，研究非线性矩阵方程X±A~*X~(-1)A=P的Hermite正定解的存在性及其扰动分析。首先，给出矩阵方程X+A~X~(-1)A=P的Hermite正定解的分布情况和利用微分的方法给出最大解的一阶扰动界，部分地改进了文献[75，79]的结论。其次，使用微积分的技术对矩阵方程X±A~*X~(-1)A=I的Hermite正定解给出了一些新的扰动界，并用数值例子加以说明。
第八章，研究结构线性方程组的结构向后扰动误差分析。给出了两类结构线性方程组解的结构向后误差的计算公式。
In this thesis we investigate perturbation analysis of some matrix computational prob-lems. Applying algebra and elementary calculus techniques, some new perturbation boundsare presented for a variety of matrix computational problems, including the generlized polardecomposition, eigenspace and singular space, Rayleigh quotient for singular values, generalizedeigenvalue problems, nonlinear matrix equation and structured linear systems and so on.
In Chapter 1 we introduce some important conceptions in the matrix perturbation analysis,and give overview of our results.
In Chapter 2 some perturbation bounds of the generalized polar decomposition for thematrices with different ranks are given. For the subunitary polar factor some former results areimproved. For the Hermitian positive semidefinite polar factor, some perturbation bounds areobtained under any unitarily invariant norm. In particular, the classical result of Hermitianpositive definite polar factors is improved for the Frobenius norm.
In Chapter 3 we condiser the additive and multiplicative perturbations of eigenspaces.Some absolute and relative bounds are presented by using double gaps of eigenvalues, whichare better than the existing bounds in some sense.
In Chapter 4 the absolute and relative perturbations of singular spaces are investigated forthe Frobenius norm. Using double gaps of singular values, some separate and joint perturbationbounds for left and right singular subspaces are obtained. Moreover, the upper bounds of theircondition numbers are derived. At the same time a relative bound for the Wedin's sin(?) typebound and both right and left singular subspaces relative bounds are given, respectively.
In Chapter 5 we consider the Rayleigh quotient of singular value problems. Some results ofthe Rayleigh quotient of Hermitian matrices are extended to those for arbitrary matrix. On theothe hand, some unitarily invariant norm bounds for singular values are presented for Rayleighquotient matrices. Our results improve the existing bounds.
In Chapter 6 the generalized eigenvalue perturbation bounds for diagonalizable pairs areinvestigated. Three results of Sun on normal pencils are extened to diagonalizable pencils.
In Chapter 7 we consider the exsitence and perturbation analysis of Hermitian positivedefinite solution for X+A~*X~(-1)A=P. Firstly, the existence of the Hermitian positive definitesolutions is proved and the first order perturbation bound of the maximal solution is presentedby using differetial method for the matrix equation X+A~*X~(-1)A=P, which improves thecorresponding partial results in; Secondly(some new perturbation bounds for Hermitianpositive definite solutions of the matrix equations X±A~*X~(-1)A=I are derived by usingelementary calculus techniques and the new results are illustrated by numerical examples.
In Chapter 8 the structured backward errors for solving structured linear systems arediscussed, we present a computable formulae of the structured backward error for two kinds ofstrutured linear systems.
第一章，介绍了矩阵扰动分析中一些重要概念。同时对本文的主要内容作了总的叙述。
第二章，当原矩阵与其扰动矩阵的秩不相等时，对广义极分解中的次酉极因子和半正定极因子给出了一些扰动界。对次酉极因子我们改进了以往的一些结论。对半正定极因子，获得任意酉不变范数下的扰动界。特别改进了正定极因子在Frobenius范数下的一个经典的结果。
第三章，研究Hermite矩阵特征空间在Frobenius范数的加法和乘法扰动。使用特征值的双分离度，给出了一些绝对与相对扰动界，从某种意义上来说改进了以往相应的结论。
第四章，研究矩阵奇异空间在Frobenius范数下的绝对与相对扰动界。使用奇异值的双分离度，获得左、右奇异空间各自的绝对扰动界和左右奇异空间联合的绝对扰动界；由此导出奇异子空间的条件数的上界。同时也导出Wedin's sin(?)型定理的相对扰动界和左右奇异空间各自的相对扰动界。
第五章，研究奇异值的Rayleigh商问题。一方面将Hermite矩阵的Rayleigh商问题的一些结论推广到奇异值的Rayleigh商上。另一方面给出近似奇异值在任意酉不变范数下的一些界。
第六章，研究可对角化矩阵束的广义特征值的扰动界。将孙[66]的三个结论从正规矩阵对推广到可对角化矩阵对上。
第七章，研究非线性矩阵方程X±A~*X~(-1)A=P的Hermite正定解的存在性及其扰动分析。首先，给出矩阵方程X+A~X~(-1)A=P的Hermite正定解的分布情况和利用微分的方法给出最大解的一阶扰动界，部分地改进了文献[75，79]的结论。其次，使用微积分的技术对矩阵方程X±A~*X~(-1)A=I的Hermite正定解给出了一些新的扰动界，并用数值例子加以说明。
第八章，研究结构线性方程组的结构向后扰动误差分析。给出了两类结构线性方程组解的结构向后误差的计算公式。
In this thesis we investigate perturbation analysis of some matrix computational prob-lems. Applying algebra and elementary calculus techniques, some new perturbation boundsare presented for a variety of matrix computational problems, including the generlized polardecomposition, eigenspace and singular space, Rayleigh quotient for singular values, generalizedeigenvalue problems, nonlinear matrix equation and structured linear systems and so on.
In Chapter 1 we introduce some important conceptions in the matrix perturbation analysis,and give overview of our results.
In Chapter 2 some perturbation bounds of the generalized polar decomposition for thematrices with different ranks are given. For the subunitary polar factor some former results areimproved. For the Hermitian positive semidefinite polar factor, some perturbation bounds areobtained under any unitarily invariant norm. In particular, the classical result of Hermitianpositive definite polar factors is improved for the Frobenius norm.
In Chapter 3 we condiser the additive and multiplicative perturbations of eigenspaces.Some absolute and relative bounds are presented by using double gaps of eigenvalues, whichare better than the existing bounds in some sense.
In Chapter 4 the absolute and relative perturbations of singular spaces are investigated forthe Frobenius norm. Using double gaps of singular values, some separate and joint perturbationbounds for left and right singular subspaces are obtained. Moreover, the upper bounds of theircondition numbers are derived. At the same time a relative bound for the Wedin's sin(?) typebound and both right and left singular subspaces relative bounds are given, respectively.
In Chapter 5 we consider the Rayleigh quotient of singular value problems. Some results ofthe Rayleigh quotient of Hermitian matrices are extended to those for arbitrary matrix. On theothe hand, some unitarily invariant norm bounds for singular values are presented for Rayleighquotient matrices. Our results improve the existing bounds.
In Chapter 6 the generalized eigenvalue perturbation bounds for diagonalizable pairs areinvestigated. Three results of Sun on normal pencils are extened to diagonalizable pencils.
In Chapter 7 we consider the exsitence and perturbation analysis of Hermitian positivedefinite solution for X+A~*X~(-1)A=P. Firstly, the existence of the Hermitian positive definitesolutions is proved and the first order perturbation bound of the maximal solution is presentedby using differetial method for the matrix equation X+A~*X~(-1)A=P, which improves thecorresponding partial results in; Secondly(some new perturbation bounds for Hermitianpositive definite solutions of the matrix equations X±A~*X~(-1)A=I are derived by usingelementary calculus techniques and the new results are illustrated by numerical examples.
In Chapter 8 the structured backward errors for solving structured linear systems arediscussed, we present a computable formulae of the structured backward error for two kinds ofstrutured linear systems.
引文
[1] ARAKI H AND YAMAGAMI S. An inequality for the Hilbert-Schmidt norm, Comm. Math. Phys., 81(1981), 89-98.
[2] ANDERSON W N, MORLEY T D AND TRAPP G E. Positive solutions to X=A-BX~(-1)B~*, Linear Algebra Appl., 134(1990) 53-62.
[3] BOGGS P T AND TOLLE J W. Sequential quadratic programming, Acta Numerica, Cambridge University Press, Cambridge, UK, 1995, pp.1-51.
[4] BARLOW J L AND SLAPNICAR I. Optimal perturbation bounds for the Hermitian eigenvalue problem, Linear Algebra Appl., 309(2000), 19-43.
[5] BARRLUND A. Perturbation bounds on the polar decomposition, BIT, 31(1990), 101-113.
[6] BARRLUND A. Perturbation bounds for the LDL~H and LU decompositions, BIT, 31(1991),.358-363.
[7] BENZI M AND GOLUB G H, LIESEN J. Numerical solution of saddle point problems. Acta Numerica., 14(2005), 1-137.
[8] BHATIA R. Perturbation bounds for matrix eigenvalues, Longman, Indian, 1987.
[9] BHATIA R. Matrix Analysis, Springer Press, New York, 1997.
[10] CHEN X-S , LI W AND SUN W. Some new perturbation bounds for the generalized polar decomposition, BIT, 44(2004), 237-244.
[11] 陈小山，黎稳．酉不变范数极分解的扰动界，计算数学，2(2005)，121-128．
[12] CHEN C H, AND SUN J G. Perturbation bounds for the polar factors, J. Comput. Math., 7(1989), 397-401.
[13] CHATELIN F AND GRATTON S . On the condition numbers associated with the polar factorization of a matrix, Numer. Linear Algebra Appl., 7(2000), 337-354.
[14] DAWS C AND KAHAN W . The rotation of eigenvectors by a perturbation. Ⅲ, SIAM J. Numer. Anal., 7(1970), 1-46.
[15] DOPICO F M, MORO J AND MOLERA J M. Weyl-type relative perturbation bounds for eigensysterns of Hermitian matrices, Linear Algebra Appl., 309(2000), 3-18.
[16] DOPICO F M AND MORO J. Perturbation theory for simultaneous bases of singular subspaces, BIT, 42(2002), 84-109.
[17] DOLLAR H S AND WATHEN A J. Incomplete factorization constraint preconditioners for saddle point matrices. Technical report, NA-04/01, Oxford University Computing Laboratory, Numerical Analysis Group, 2004.
[18] ELMAN H C AND WATHEN D J A J. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90(2002), 665-688.
[19] ELSNER L AND FRIEDLAND S. Singular value, doubly stochastic matrices, and applications, Linear Algebar Appl., 220(1995), 161-169.
[20] ENGWERDA J C. On the existence of a positive definite solution of the matrix equation X+ A~TX~(-1)A=I, Linear Algebra Appl., 194(1993) 91-108.
[21] ENGWERDA J C, RAN A C M AND RIJKEBOER A L. Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A~*X~(-1)A=I, Linear Algebra Appl., 186(1993) 255-275.
[22] 戈卢布 G H，范洛恩 C F．矩阵计算，科学出版社，北京，2001．(袁亚湘等译)
[23] GULLIKSSON M, JINX Q AND WEI Y M. Perturbation bounds for constrained and weighted least squares problems, Linear Algebra Appl., 349(2002), 221-232.
[24] HANSON R J AND NORRIS M J. Analysis of measurements based on the singular value decomposition, SIAM J. Sci. Statist. Comput., 2(1981), 363-373.
[25] HASANOV V I, IVANOV I G AND UHLIG F. Improved perturbation estimates for the matrix equations X±A~*X~(-1)A=Q, Linear Algebra Appl., 379(2004), 113-135.
[26] HASANOV V I AND IVANOV I G. On two perturbation estimates of the extreme solutions to the equations X±A~*X~(-1) A=Q, Linear Algebra Appl., 413(2006), 81-92.
[27] HIGHAM N J. Computing the polar decomposition-with applications, SIAM J. Sci. Statist. Comput., 7(1986), 1160-1174.
[28] HORN R A AND JOHNSON C R. Matrix Analysis, Cambrdge University Press, New York, 1985.
[29] HORN R A AND JOHNSON C R. Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
[30] IPSEN I C F. An overview of relative sin(?) theorems for invariant subspaces of complex matrices, J. Comput. Appl. Math., 123(2000), 131-153.
[31] IPSEN I C F. Absolute and relative perturbation bounds for invariant subspaces of matrices, Linear Algebra Appl., 309(2000), 45-56.
[32] IVANOV I G, HASANOV V I AND MINCHEV B V. On matrix equations X±A~*X~(-2)A=I, Linear Algebra Appl., 326(2001), 27-44.
[33] IVANOV I G . Perturbation analysis for solutions of X±A~*X~(-n)A=Q, Linear Algebra Appl., 395(2005), 313-331.
[34] 贾仲孝，张萍．计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法，计算数学，3(2003)，293-304．
[35] 贾仲孝，牛大田．计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法 计算数学，2(2004)，13-24．
[36] KAHAN W. Numerical Linear Algebra, Canadian Mathematical Bulletin, 9(1966), 757-801.
[37] KATO T. A Sshort Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1982.
[38] KELLER C, GOULD N I M AND WATHEN A J. Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21(2000), 1300-1317.
[39] KENNEY C AND LAUB A J. Polar decomposition and matrix sign function condition estimates, SIAM J. Sci. Statist. Comput., 12(1991), 488-504.
[40] LI W AND SUN W-W. Perturbation bounds for unitary and subunitary polar factors, SIAM J. Matrix Anal. Appl., 23(2002), 1183-1193.
[41] LI W AND SUN W-W. New perturbation bounds for unitary polar factors, SIAM J. Matrix Anal. Appl., 25(2003), 362-372.
[42] LI W. Some new perturbation bounds for subunitary polar factors, Acta Math. Sinica (Engl. Sir.), 21(2005), 1515-1520.
[43] LI W AND SUN W-W, Some remarks on the perturbation of the polar decomposition for rectangular matrices, Numer. Lin. Alg. Appl., 13(2006), 327-338
[44] LI R-C. On eigenvalues of a Rayleigh quotient matrix, Linear Algebra Appl., 169(1992), 249-255.
[45] LI R-C. A perturbation bound for the generalized polar decomposition, BIT, 33(1993), 304-308.
[46] LI R C. New perturbation bounds for the unitary polar factor, SIAM J Matrix Anal. Appl., 16(1995), 327-332.
[47] LI R-C. Relative perturbation bounds for the unitary polar factor, BIT, 37(1997), 67-75.
[48] LI R-C. Relative perturbation theory:I.Eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19(1998):956-982.
[49] LI R-C. Relative perturbation theory: (Ⅱ) Eigenspace and singular subspace variations, SIAM J. Matrix Anal. Appl., 20(1999), 471-492.
[50] LI R-C. On perturbations of matrix pencils with real spectra, a revist, Math. Comput., 72(2002), 715-728.
[51] LI R-C. Accuracy of computed eigenvectors via optimizing a Rayleigh quotient, BIT, 44(2004), 585-593.
[52] LIU X-G. On Rayleigh quotient theory for the eigenproblem and the singular value problem, J. Comput. Math., Supplementary issue (1992), 216-224.
[53] 刘新国，许雅各．关于Rayleign商矩阵，计算数学，2(1990)，208-213．
[54] 刘新国．不变子空间上特征值的摄动，青岛海洋大学学报．19(1989)，91-95．
[55] LIU X-G AND GAO H. On the positive definite solutions of the matrix equations X~s±A~TX~(-t)A=I_n, Linear Algebra Appl., 368(2003) 83-97.
[56] 刘新国，王卫国．关于结构KKT方程组的扰动分析，计算数学，2(2004)，179-188．
[57] 刘新国，王学峰．一类KKT系统的结构敏度分析，计算数学，4(2004)，427-436．
[58] LONDRE T AND RHEE N H. A note on relative perturbation bounds, SIAM J. Matrix Anal. Appl., 21(1999), 357-361.
[59] MATHIAS R. Perturbation bounds for the polar decomposition, SIAM J Matrix Anal. Appl., 14(1993), 588-597.
[60] Mathias R AND Veselic K. A relative perturbation bound for positive definite matrices, Linear Algebra Appl., 270(1998), 315-321.
[61] RAN A C M AND REURINGS M C B. On the nonlinear matrix equation X+A~*F(X)A=Q: solutions and perturbation theory, Linear Algebra Appl., 346(2002) 15-26.
[62] SCHONEMANN P H. A generalized solution of the orthogonal Procrustes problem, Psychometrika, 31(1966), 1-10.
[63] Stewart G. W., On the continuity of the generalized inverse, SIAM J. Appl. Math., 17(1969), 33-45.
[64] STEWART G W AND Sun J G. Matrix Perturbation Theory, Academic Press, Boston, 1990.
[65] STEWART G W. Two simple residual bounds for the eigenvalues of a Hermitian matrix, SIAM J. Matrix Anal. Appl., 12(1991), 205-208.
[66] 孙继广．一类矩阵对的广义特征值的扰动界限，计算数学，4(1982)，23-29．
[67] 孙继广，陈春晖．广义极分解，计算数学，11(1989)，262-273．
[68] SUN J-G. Eigenvalues of Rayleigh quotient matrices, Numer. Math., 59(1991), 603-614.
[69] SUN J-G. Perturbation bounds for the cholesky and QR factorizations, BIT, 31(1991), 341-352.
[70] SUN J-G. Backward perturbation analysis of certain characteristic subspaces, Numer. Math., 65(1993), 357-382.
[71] SUN J-G. Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math., 73(1996), 235-263.
[72] SUN J-G. Optimal backward perturbation bounds for linear systems and linear least squares problems, UMINF 96.15, ISSN-0348-0542, Deparment of Computing Science, Umea University, 1996.
[73] SUN J-G. Structured backward errors for KKT systems, Linear Algebra Appl., 288(1999), 75-88.
[74] 孙继广．矩阵扰动分析，科学出版社，北京，2001．
[75] SUN J-G AND XU S F. Perturbation analysis of the maximal solution of the matrix equation X+A~*X~(-1)A=P.(?), Linear Algebra Appl., 362(2003) 211-228.
[76] SUN J-G. Perturbation analysis of the matrix equation X=Q+A~H((?)-C)~(-1)A, Linear Algebra Appl., 362(2003), 211-228.
[77] SUN J-G. A note on backward errors for structured linear systems, Numer. Linear Algebra Appl., 12(2005), 585-603.
[78] STURLER E D AND EIESEN J. Block-diagonal and constraint preconditioners for nonsysmmetric indefinite linear systems, Part Ⅰ: theory, SIAM J. Sci. Comput., 26(2005), 1598-1619.
[79] Xu S-F. Perturbation analysis of the maximal solution of the matrix equation X+A~*X~(-1)A=P, Linear Algebra Appl., 336(2001), 61-70.
[80] ZHAN X. On the matrix equation X+A~*X~(-1)A=I, Linear Algebra Appl., 247(1996), 337-345.
[81] 威尔金森J H．代数特征值问题，科学出版社，北京，2001．(石钟慈，邓健新译)
[82] WEDIN P A. Perturbation bounds in connection with singular value decomposition, BIT, 12(1972), 99-111.
[83] ZHANG Y-H. On Hermitian positive definite solutions of matrix equation X+A~*X~(-2)A=I, Linear Algebra Appl., 372(2003), 295-304.
[2] ANDERSON W N, MORLEY T D AND TRAPP G E. Positive solutions to X=A-BX~(-1)B~*, Linear Algebra Appl., 134(1990) 53-62.
[3] BOGGS P T AND TOLLE J W. Sequential quadratic programming, Acta Numerica, Cambridge University Press, Cambridge, UK, 1995, pp.1-51.
[4] BARLOW J L AND SLAPNICAR I. Optimal perturbation bounds for the Hermitian eigenvalue problem, Linear Algebra Appl., 309(2000), 19-43.
[5] BARRLUND A. Perturbation bounds on the polar decomposition, BIT, 31(1990), 101-113.
[6] BARRLUND A. Perturbation bounds for the LDL~H and LU decompositions, BIT, 31(1991),.358-363.
[7] BENZI M AND GOLUB G H, LIESEN J. Numerical solution of saddle point problems. Acta Numerica., 14(2005), 1-137.
[8] BHATIA R. Perturbation bounds for matrix eigenvalues, Longman, Indian, 1987.
[9] BHATIA R. Matrix Analysis, Springer Press, New York, 1997.
[10] CHEN X-S , LI W AND SUN W. Some new perturbation bounds for the generalized polar decomposition, BIT, 44(2004), 237-244.
[11] 陈小山，黎稳．酉不变范数极分解的扰动界，计算数学，2(2005)，121-128．
[12] CHEN C H, AND SUN J G. Perturbation bounds for the polar factors, J. Comput. Math., 7(1989), 397-401.
[13] CHATELIN F AND GRATTON S . On the condition numbers associated with the polar factorization of a matrix, Numer. Linear Algebra Appl., 7(2000), 337-354.
[14] DAWS C AND KAHAN W . The rotation of eigenvectors by a perturbation. Ⅲ, SIAM J. Numer. Anal., 7(1970), 1-46.
[15] DOPICO F M, MORO J AND MOLERA J M. Weyl-type relative perturbation bounds for eigensysterns of Hermitian matrices, Linear Algebra Appl., 309(2000), 3-18.
[16] DOPICO F M AND MORO J. Perturbation theory for simultaneous bases of singular subspaces, BIT, 42(2002), 84-109.
[17] DOLLAR H S AND WATHEN A J. Incomplete factorization constraint preconditioners for saddle point matrices. Technical report, NA-04/01, Oxford University Computing Laboratory, Numerical Analysis Group, 2004.
[18] ELMAN H C AND WATHEN D J A J. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90(2002), 665-688.
[19] ELSNER L AND FRIEDLAND S. Singular value, doubly stochastic matrices, and applications, Linear Algebar Appl., 220(1995), 161-169.
[20] ENGWERDA J C. On the existence of a positive definite solution of the matrix equation X+ A~TX~(-1)A=I, Linear Algebra Appl., 194(1993) 91-108.
[21] ENGWERDA J C, RAN A C M AND RIJKEBOER A L. Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A~*X~(-1)A=I, Linear Algebra Appl., 186(1993) 255-275.
[22] 戈卢布 G H，范洛恩 C F．矩阵计算，科学出版社，北京，2001．(袁亚湘等译)
[23] GULLIKSSON M, JINX Q AND WEI Y M. Perturbation bounds for constrained and weighted least squares problems, Linear Algebra Appl., 349(2002), 221-232.
[24] HANSON R J AND NORRIS M J. Analysis of measurements based on the singular value decomposition, SIAM J. Sci. Statist. Comput., 2(1981), 363-373.
[25] HASANOV V I, IVANOV I G AND UHLIG F. Improved perturbation estimates for the matrix equations X±A~*X~(-1)A=Q, Linear Algebra Appl., 379(2004), 113-135.
[26] HASANOV V I AND IVANOV I G. On two perturbation estimates of the extreme solutions to the equations X±A~*X~(-1) A=Q, Linear Algebra Appl., 413(2006), 81-92.
[27] HIGHAM N J. Computing the polar decomposition-with applications, SIAM J. Sci. Statist. Comput., 7(1986), 1160-1174.
[28] HORN R A AND JOHNSON C R. Matrix Analysis, Cambrdge University Press, New York, 1985.
[29] HORN R A AND JOHNSON C R. Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
[30] IPSEN I C F. An overview of relative sin(?) theorems for invariant subspaces of complex matrices, J. Comput. Appl. Math., 123(2000), 131-153.
[31] IPSEN I C F. Absolute and relative perturbation bounds for invariant subspaces of matrices, Linear Algebra Appl., 309(2000), 45-56.
[32] IVANOV I G, HASANOV V I AND MINCHEV B V. On matrix equations X±A~*X~(-2)A=I, Linear Algebra Appl., 326(2001), 27-44.
[33] IVANOV I G . Perturbation analysis for solutions of X±A~*X~(-n)A=Q, Linear Algebra Appl., 395(2005), 313-331.
[34] 贾仲孝，张萍．计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法，计算数学，3(2003)，293-304．
[35] 贾仲孝，牛大田．计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法 计算数学，2(2004)，13-24．
[36] KAHAN W. Numerical Linear Algebra, Canadian Mathematical Bulletin, 9(1966), 757-801.
[37] KATO T. A Sshort Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1982.
[38] KELLER C, GOULD N I M AND WATHEN A J. Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21(2000), 1300-1317.
[39] KENNEY C AND LAUB A J. Polar decomposition and matrix sign function condition estimates, SIAM J. Sci. Statist. Comput., 12(1991), 488-504.
[40] LI W AND SUN W-W. Perturbation bounds for unitary and subunitary polar factors, SIAM J. Matrix Anal. Appl., 23(2002), 1183-1193.
[41] LI W AND SUN W-W. New perturbation bounds for unitary polar factors, SIAM J. Matrix Anal. Appl., 25(2003), 362-372.
[42] LI W. Some new perturbation bounds for subunitary polar factors, Acta Math. Sinica (Engl. Sir.), 21(2005), 1515-1520.
[43] LI W AND SUN W-W, Some remarks on the perturbation of the polar decomposition for rectangular matrices, Numer. Lin. Alg. Appl., 13(2006), 327-338
[44] LI R-C. On eigenvalues of a Rayleigh quotient matrix, Linear Algebra Appl., 169(1992), 249-255.
[45] LI R-C. A perturbation bound for the generalized polar decomposition, BIT, 33(1993), 304-308.
[46] LI R C. New perturbation bounds for the unitary polar factor, SIAM J Matrix Anal. Appl., 16(1995), 327-332.
[47] LI R-C. Relative perturbation bounds for the unitary polar factor, BIT, 37(1997), 67-75.
[48] LI R-C. Relative perturbation theory:I.Eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl., 19(1998):956-982.
[49] LI R-C. Relative perturbation theory: (Ⅱ) Eigenspace and singular subspace variations, SIAM J. Matrix Anal. Appl., 20(1999), 471-492.
[50] LI R-C. On perturbations of matrix pencils with real spectra, a revist, Math. Comput., 72(2002), 715-728.
[51] LI R-C. Accuracy of computed eigenvectors via optimizing a Rayleigh quotient, BIT, 44(2004), 585-593.
[52] LIU X-G. On Rayleigh quotient theory for the eigenproblem and the singular value problem, J. Comput. Math., Supplementary issue (1992), 216-224.
[53] 刘新国，许雅各．关于Rayleign商矩阵，计算数学，2(1990)，208-213．
[54] 刘新国．不变子空间上特征值的摄动，青岛海洋大学学报．19(1989)，91-95．
[55] LIU X-G AND GAO H. On the positive definite solutions of the matrix equations X~s±A~TX~(-t)A=I_n, Linear Algebra Appl., 368(2003) 83-97.
[56] 刘新国，王卫国．关于结构KKT方程组的扰动分析，计算数学，2(2004)，179-188．
[57] 刘新国，王学峰．一类KKT系统的结构敏度分析，计算数学，4(2004)，427-436．
[58] LONDRE T AND RHEE N H. A note on relative perturbation bounds, SIAM J. Matrix Anal. Appl., 21(1999), 357-361.
[59] MATHIAS R. Perturbation bounds for the polar decomposition, SIAM J Matrix Anal. Appl., 14(1993), 588-597.
[60] Mathias R AND Veselic K. A relative perturbation bound for positive definite matrices, Linear Algebra Appl., 270(1998), 315-321.
[61] RAN A C M AND REURINGS M C B. On the nonlinear matrix equation X+A~*F(X)A=Q: solutions and perturbation theory, Linear Algebra Appl., 346(2002) 15-26.
[62] SCHONEMANN P H. A generalized solution of the orthogonal Procrustes problem, Psychometrika, 31(1966), 1-10.
[63] Stewart G. W., On the continuity of the generalized inverse, SIAM J. Appl. Math., 17(1969), 33-45.
[64] STEWART G W AND Sun J G. Matrix Perturbation Theory, Academic Press, Boston, 1990.
[65] STEWART G W. Two simple residual bounds for the eigenvalues of a Hermitian matrix, SIAM J. Matrix Anal. Appl., 12(1991), 205-208.
[66] 孙继广．一类矩阵对的广义特征值的扰动界限，计算数学，4(1982)，23-29．
[67] 孙继广，陈春晖．广义极分解，计算数学，11(1989)，262-273．
[68] SUN J-G. Eigenvalues of Rayleigh quotient matrices, Numer. Math., 59(1991), 603-614.
[69] SUN J-G. Perturbation bounds for the cholesky and QR factorizations, BIT, 31(1991), 341-352.
[70] SUN J-G. Backward perturbation analysis of certain characteristic subspaces, Numer. Math., 65(1993), 357-382.
[71] SUN J-G. Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math., 73(1996), 235-263.
[72] SUN J-G. Optimal backward perturbation bounds for linear systems and linear least squares problems, UMINF 96.15, ISSN-0348-0542, Deparment of Computing Science, Umea University, 1996.
[73] SUN J-G. Structured backward errors for KKT systems, Linear Algebra Appl., 288(1999), 75-88.
[74] 孙继广．矩阵扰动分析，科学出版社，北京，2001．
[75] SUN J-G AND XU S F. Perturbation analysis of the maximal solution of the matrix equation X+A~*X~(-1)A=P.(?), Linear Algebra Appl., 362(2003) 211-228.
[76] SUN J-G. Perturbation analysis of the matrix equation X=Q+A~H((?)-C)~(-1)A, Linear Algebra Appl., 362(2003), 211-228.
[77] SUN J-G. A note on backward errors for structured linear systems, Numer. Linear Algebra Appl., 12(2005), 585-603.
[78] STURLER E D AND EIESEN J. Block-diagonal and constraint preconditioners for nonsysmmetric indefinite linear systems, Part Ⅰ: theory, SIAM J. Sci. Comput., 26(2005), 1598-1619.
[79] Xu S-F. Perturbation analysis of the maximal solution of the matrix equation X+A~*X~(-1)A=P, Linear Algebra Appl., 336(2001), 61-70.
[80] ZHAN X. On the matrix equation X+A~*X~(-1)A=I, Linear Algebra Appl., 247(1996), 337-345.
[81] 威尔金森J H．代数特征值问题，科学出版社，北京，2001．(石钟慈，邓健新译)
[82] WEDIN P A. Perturbation bounds in connection with singular value decomposition, BIT, 12(1972), 99-111.
[83] ZHANG Y-H. On Hermitian positive definite solutions of matrix equation X+A~*X~(-2)A=I, Linear Algebra Appl., 372(2003), 295-304.