The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian ...The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.展开更多
We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put...We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.展开更多
In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system ...In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.展开更多
Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive de...Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.展开更多
A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more strin...A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.展开更多
For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matr...For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.展开更多
A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the...A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.展开更多
基金The National Natural Science Foundation of China(No.11371089)the China Postdoctoral Science Foundation(No.2016M601688)
文摘The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
基金This work is supported by NSF of Shanxi province,20011041.
文摘We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.
文摘In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.
文摘Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.
文摘A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.
文摘For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.
基金Research supported by The China NNSF 0utstanding Young Scientist Foundation （No.10525102）, The National Natural Science Foundation （No.10471146）, and The National Basic Research Program （No.2005CB321702）, P.R. China.
文摘A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.
