共查询到20条相似文献,搜索用时 15 毫秒
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This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms. 相似文献
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In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A*X?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
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Zhen-yun Peng Salah M. El-Sayed Xiang-lin Zhang 《Journal of Computational and Applied Mathematics》2007,200(2):520-527
In this paper, the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation X+A*X-A=Q with the case 0<1 and the minimal positive definite solution of the same matrix equation with the case 1 are proposed. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Numerical examples to illustrate the behavior of the considered algorithms are also given. 相似文献
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Snehasish Bose Sk Monowar Hossein Kallol Paul 《Journal of Fixed Point Theory and Applications》2016,18(3):627-643
Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation \({X = Q \pm \sum^{m}_{i=1}{A_{i}}^*F(X)A_{i}}\), where Q is a positive definite matrix, A i ’s are arbitrary n × n matrices and F is a monotone map from the set of positive definite matrices to itself. We show that the presented condition is weaker than that presented by Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443]. In order to do so, we establish some fixed point theorems for mappings satisfying (\({\psi, \phi}\))-weak contractivity conditions in partially ordered G-metric spaces, which generalize some existing results related to (\({\psi, \phi}\))-weak contractions in partially ordered metric spaces as well as in G-metric spaces for a given function f. We conclude, by presenting an example, that our fixed point theorem cannot be obtained from any existing fixed point theorem using the process of Jleli and Samet [Fixed Point Theory Appl. 2012 (2012), Article ID 210]. 相似文献
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Jerzy K. Baksalary 《Linear and Multilinear Algebra》2013,61(1-4):133-139
The general nonegative definite solution to the matrix equation AXA* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered. 相似文献
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Jerzy K. Baksalary 《Linear and Multilinear Algebra》1984,16(1):133-139
The general nonegative definite solution to the matrix equation AXA* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered. 相似文献
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Chuan-Long Wang Guo-Yan Meng Xue-Rong Yong 《Advances in Computational Mathematics》2013,38(4):859-872
In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax?=?b. The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient. 相似文献
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A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations. 相似文献
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Convergence of block two-stage iterative methods for symmetric positive definite systems 总被引:2,自引:0,他引:2
Zhi-Hao Cao 《Numerische Mathematik》2001,90(1):47-63
Summary. We study the convergence of two-stage iterative methods for solving symmetric positive definite (spd) systems. The main tool
we used to derive the iterative methods and to analyze their convergence is the diagonally compensated reduction (cf. [1]).
Received December 11, 1997 / Revised version received March 25, 1999 / Published online May 30, 2001 相似文献
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研究了双线性系统中的一类广义Lyapunov矩阵方程的正定解.基于混合单调算子不动点定理,给出新的存在正定解的充分条件,构造了求其正定解的不动点迭代方法,并给出了迭代误差估计公式.数值实验表明新方法是可行的. 相似文献
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Vjeran Hari 《Numerical Algorithms》2018,79(1):221-249
The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ A AΔ A and Δ B BΔ B are small, for some nonsingular diagonal matrices Δ A and Δ B . 相似文献
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Daniele Bertaccini Gene H. Golub Stefano Serra Capizzano Cristina Tablino Possio 《Numerische Mathematik》2005,99(3):441-484
Summary. We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {An(a,p)}n discretizing the elliptic (convection-diffusion) problem with being a plurirectangle of Rd with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {Pn(a)}n, Pn(a):= Dn1/2(a)An(1,0) Dn1/2(a) where Dn(a) is the suitably scaled main diagonal of An(a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {Pn(a)}n turns out to be a superlinear preconditioning sequence for {An(a,0)}n where An(a,0) represents a good approximation of Re(An(a,p)) namely the real part of An(a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {Pn-1(a)Re(An(a,p))}n {Pn-1(a)An(a,0)}n: therefore the solution of a linear system with coefficient matrix An(a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {An(1,0)}n.Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.Mathematics Subject Classification (1991): 65F10, 65N22, 15A18, 15A12, 47B65 相似文献
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Sergey I. Solov’ëv 《Linear algebra and its applications》2006,415(1):210-229
This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem. 相似文献
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Joel Friedman 《Linear algebra and its applications》1998,280(2-3):199-216
Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ1. We consider the power method, i.e. that of choosing a vector v0 and setting vk = Akv0; then the Rayleigh quotients Rk = (Avk, vk)/(vk, vk) usually converge to λ1 as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close Rk is to λ1. They are both based on a bound on λ1 − Rk involving the difference of two consecutive Rayleigh quotients and a quantity ωk. While we do not know how to directly calculate ωk, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ1 − Rk which is proportional to (λ2/λ1)2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ). 相似文献
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In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems. 相似文献