共查询到20条相似文献,搜索用时 78 毫秒
1.
In this paper, the nonlinear matrix equation X + A∗XqA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods. 相似文献
2.
Jian-hui Long Xi-yan Hu Lei Zhang 《Bulletin of the Brazilian Mathematical Society》2008,39(3):371-386
In this paper, we study the matrix equation X + A*X
−1
A + B*X
−1
B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two
iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the
effectiveness of the algorithms.
This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry
of Education of China 20060532014. 相似文献
3.
Jing Cai 《Applied mathematics and computation》2010,217(1):117-4466
Nonlinear matrix equation Xs + A∗X−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods. 相似文献
4.
Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation Xs±ATX-tA=In, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective. 相似文献
5.
Perturbation analysis of the matrix equation 总被引:1,自引:0,他引:1
Consider the nonlinear matrix equation X-A*X-pA=Q with 0<p1. This paper shows that there exists a unique positive definite solution to the equation. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. We also obtain explicit expressions of the condition number for the unique positive definite solution. The theoretical results are illustrated by numerical examples. 相似文献
6.
Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation Xq-A∗F(X)A=Q(q>1) always has a unique positive definite solution. An iterative method is proposed to compute the unique positive definite solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique positive definite solution is derived in the end. 相似文献
7.
8.
Vejdi Ismailov Hasanov 《Linear and Multilinear Algebra》2018,66(9):1783-1798
9.
In this paper,Hermitian positive definite solutions of the nonlinear matrix equation X + A*X-qA = Q (q ≥ 1) are studied.Some new necessary and sufficient conditions for the existence of solutions are obtained.Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions,and the convergence analysis is also given.The theoretical results are illustrated by numerical examples. 相似文献
10.
In this paper, Hermitian positive definite solutions of the nonlinear matrix equation X + A^*X^-qA = Q (q≥1) are studied. Some new necessary and sufficient conditions for the existence of solutions are obtained. Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions, and the convergence analysis is also given. The theoretical results are illustrated by numerical examples. 相似文献
11.
12.
In this paper, the nonlinear matrix equation Xs+A*X-tA=Q is investigated. Necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are derived. An effective iterative method to obtain the special solution XL (We proved that if there is a maximal Hermitian positive definite solution, then it must be XL) is established. Moreover, some new perturbation estimates for XL are obtained. Several numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates. 相似文献
13.
14.
In the paper, the split quaternion matrix equation AXAη*=B is considered, where the operator Aη* is the η-conjugate transpose of A, where η∈{i,j,k}. We propose some new real representations, which well exploited the special structures of the original matrices. By using this method, we obtain the necessary and sufficient conditions for AXAη*=B to have X=±Xη* solutions and derive the general expressions of solutions when it is consistent. In addition, we also derive the general expressions of the least squares X=±Xη* solutions to it in case that this matrix equation is not consistent. 相似文献
15.
Jacob Engwerda 《Linear and Multilinear Algebra》2013,61(6):689-700
In this note we consider the question under which conditions all entries of the matrix I???(I?+?X)?1 are nonnegative in case matrix X is a real positive definite matrix. Sufficient conditions are presented as well as some necessary conditions. One sufficient condition is that matrix X ?1 is an inverse M-matrix. A class of matrices for which the inequality holds is presented. 相似文献
16.
Igor E. Kaporin 《Numerical Linear Algebra with Applications》1998,5(6):483-509
A new matrix decomposition of the form A = UTU + UTR + RTU is proposed and investigated, where U is an upper triangular matrix (an approximation to the exact Cholesky factor U0), and R is a strictly upper triangular error matrix (with small elements and the fill-in limited by that of U0). For an arbitrary symmetric positive matrix A such a decomposition always exists and can be efficiently constructed; however it is not unique, and is determined by the choice of an involved truncation rule. An analysis of both spectral and K-condition numbers is given for the preconditioned matrix M = U−T AU−1 and a comparison is made with the RIC preconditioning proposed by Ajiz and Jennings. A concept of approximation order of an incomplete factorization is introduced and it is shown that RIC is the first order method, whereas the proposed method is of second order. The idea underlying the proposed method is also applicable to the analysis of CGNE-type methods for general non-singular matrices and approximate LU factorizations of non-symmetric positive definite matrices. Practical use of the preconditioning techniques developed is discussed and illustrated by an extensive set of numerical examples. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
17.
In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^*X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples. 相似文献
18.
矩阵方程aX2+bX+cE=O的正定解和实对称解 总被引:3,自引:1,他引:2
给出了矩阵方程aX2+bX+cE=O,a,b,c∈R,a≠0有正定解,实对称解的充分必要条件及解的一般形式. 相似文献
19.
20.
The matrix least squares (LS) problem minx ||AXB^T--T||F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ||A1X1B1^T + A2X2B2^T - T||F can also be regarded as the constrained LS problem minx=diag(x1,x2) ||AXB^T -T||F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ||A1X1B1^T+A2X2B2^T -T||F is equivalent to min x=diag(x1 ,x2) ||AXB^T - T||F whose solutions are included in the solution set of unconstrained problem minx ||AXB^T - T||F. So the general solutions of min x1,x2 ||A1X1B^T + A2X2B2^T -T||F are reconstructed by selecting the parameter matrix in that of minx ||AXB^T - T||F. 相似文献