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The structure-preserving doubling algorithm and convergence analysis for a nonlinear matrix equation

AUTOMATICA 2020年 113卷

作者： Zhang, Juan Li, Shifeng Xiangtan Univ Dept Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China

In this paper, applying special properties of doubling transformation, a structure-preserving doubling algorithm is developed for computing the positive definite solutions for a nonlinear matrix equation. Further, by mathematical induction, we establish the convergence theory of the structure-preserving doubling algorithm. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived algorithm. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Nonlinear matrix equations doubling algorithm Convergence theory

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doubling algorithm FOR THE DISCRETIZED BETHE-SALPETER EIGENVALUE PROBLEM

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MATHEMATICS OF COMPUTATION 2019年第319期88卷 2325-2350页

作者： Guo, Zhen-Chen Chu, Eric King-Wah Lin, Wen-Wei Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Monash Univ Sch Math 9 Rainforest Walk Clayton Vic 3800 Australia Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for H is an element of C-2n(x2n) with a Hamiltonian-like structure. After an appropriate transformation of H to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and the structure-preserving nature of the algorithm.

关键词： Bethe-Salpeter eigenvalue problem Cayley transform doubling algorithm

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Fast doubling algorithm for the Solution of the Riccati Equation Using Cyclic Reduction Method 2

Fast Doubling Algorithm for the Solution of the Riccati Equa...

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2nd International Conference on Mathematics and Computers in Science and Engineering (MACISE)

作者： Assimakis, Nicholas Adam, Maria Natl & Kapodistrian Univ Athens Gen Dept Psachna Evias Greece Univ Thessaly Dept Comp Sci & Biomed Informat Lamia Greece

ISBN: (纸本)9781728166957

A new iterative doubling algorithm for the solution of the discrete time Riccati equation is proposed. The algorithm is based on the Cyclic Reduction Method (CRM). The proposed doubling algorithm does not require non-singularity of the transition matrix and is faster than the classical doubling algorithm. The method can be applied to infinite measurement noise case, where the Riccati equation takes the form of the Lyapunov equation. In this case, the classical doubling algorithm is faster.

关键词： Riccati equation Lyapunov equation doubling algorithm Cyclic Reduction Method

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Highly accurate doubling algorithm for quadratic matrix equation from quasi-birth-and-death process

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LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 583卷 1-45页

作者： Chen, Cairong Li, Ren-Cang Ma, Changfeng Beihang Univ Sch Math & Syst Sci Beijing 100191 Peoples R China Univ Texas Arlington Dept Math POB 19408 Arlington TX 76019 USA Fujian Normal Univ Coll Math & Informat Fuzhou 350007 Fujian Peoples R China

A highly accurate doubling algorithm to solve the most fundamental quadratic matrix equation in the quasi-birthand-death (QBD) process is developed. It follows from the general framework of the doubling algorithm for the first standard form (SF1) but can be implemented to compute the minimal nonnegative solution with high entrywise relative accuracy for all entries, large or tiny. The algorithm is globally and quadratically convergent, except for QBD equations in the critical case where convergence is linear with the linear rate 1/2. Numerical examples are presented to demonstrate and confirm our claims. The development here parallels the recent work of Xue and Li (2017) [3] on the M-matrix algebraic Riccati equation. (C) 2019 Elsevier Inc. All rights reserved.

关键词： doubling algorithm Quasi-birth-and-death process Nonnegative solution Entrywise relative accuracy SF1

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doubling algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1327-1343页

作者： Bo Tang Yunqing Huang Ning Dong School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringKey Laboratory of Intelligent Computing and Information Processing of Ministry of EducationXiangtan UniversityXiangtan 411105HunanChina School of Science Hunan City UniversityYiyang 413000HunanChina School of Science Hunan University of TechnologyZhuzhou 412000HunanChina

We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with *** is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm(SDA)with the shift-and-shrink transformation or the generalized Cayley *** this paper,we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special ***,the doubling algorithm based on the proposed generalized transformation is presented and shown to be ***,the convergence result and the comparison theorem on convergent rate are *** numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with M-matrix.

关键词： Shift-and-shrink transformation generalized Cayley transformation doubling algorithm nonsymmetric algebraic Riccati equation

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The Structure-preserving doubling Numerical algorithm of the Continuous Coupled Algebraic Riccati Equation

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INTERNATIONAL JOURNAL OF CONTROL AUTOMATION AND SYSTEMS 2020年第7期18卷 1641-1650页

作者： Zhang, Juan Li, Shifeng Xiangtan Univ Dept Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China

In the paper, we apply a structure-preserving doubling algorithm to solve the continuous coupled algebraic Riccati equation (CCARE). Using the existence and uniqueness of the CCARE, we show that the iteration solution of the CCARE are positive semi-definite, symmetric, and unique. Further, we discuss the convergent analysis of the structure-preserving doubling algorithm. Moreover, we present two modified structure-preserving doubling algorithms. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived numerical algorithms.

关键词： Continuous coupled algebraic Riccati equation convergence analysis doubling algorithm the symmetric positive semi-definite solution

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Large-scale discrete-time algebraic Riccati equations-doubling algorithm and error analysis

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 277卷 115-126页

作者： Chu, Eric King-wah Weng, Peter Chang-Yi Monash Univ Sch Math Sci Clayton Vic 3800 Australia Acad Sinica Inst Stat Sci Taipei 115 Taiwan

We consider the numerical solution of large-scale discrete-time algebraic Riccati equations. The doubling algorithm is adapted, with the iterates for A not computed explicitly but recursively. The resulting algorithm is efficient, with computational complexity and memory requirement proportional to the size of the problem, and essentially converges quadratically. An error analysis, on the truncation of iterates, and some numerical results are presented. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Discrete-time algebraic Riccati equation doubling algorithm Large-scale problem

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A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2015年第3期22卷 393-409页

作者： Lu, Linzhang Yuan, Fei Li, Ren-Cang Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Guizhou Normal Univ Sch Math & Comp Sci Guiyang 550001 Peoples R China Univ Texas Arlington Dept Math Arlington TX 76019 USA

Recently, Guo and Lin [SIAM J. Matrix Anal. Appl., 31 (2010), 2784-2801] proposed an efficient numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) ((2)A(T)+Q + A)z = 0 arising from the vibration analysis of high speed trains, where A, Q is an element of C-nxn have special structures: both Q and A are, among others, m x m block matrices with each block being k x k (thus, n = mk), and moreover, Q is block tridiagonal, and A has only one nonzero block in the (1,m)th block position. The key intermediate step of the method is the computation of the so-called stabilizing solution to the n x n nonlinear matrix equation X + A(T)X(-1)A = Q via the doubling algorithm. The aim of this article is to propose an improvement to this key step through solving a new nonlinear matrix equation having the same form but of only k x k in size. This new and much smaller matrix equation can also be solved by the doubling algorithm. For the same accuracy, it takes the same number of doubling iterations to solve both the larger and the new smaller matrix equations, but each doubling iterative step on the larger equation takes about 4.8 as many flops than the step on the smaller equation. Replacing Guo's and Lin's key intermediate step by our modified one leads to an alternative method for the PQEP. This alternative method is faster, but the improvement in speed is not as dramatic as just for solving the respective nonlinear matrix equations and levels off as m increases. Numerical examples are presented to show the effectiveness of the new method. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： palindromic quadratic eigenvalue problem nonlinear matrix equation solvent approach doubling algorithm vibration analysis of high speed train

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A new class of complex nonsymmetric algebraic Riccati equations with its ω-comparison matrix being an irreducible singular M-matrix

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2021年第1期98卷 75-105页

作者： Dong, Liqiang Li, Jicheng Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Northwest A&F Univ Coll Sci Yangling Shaanxi Peoples R China

In this paper, we propose and discuss a new class of complex nonsymmetric algebraic Riccati equations (NAREs) whose four coefficient matrices form a matrix with its omega-comparison matrix being an irreducible singular M-matrix. We also prove that the extremal solutions of the NAREs exist uniquely in the noncritical case and exist in the critical case. Some good properties of the solutions are also shown. Besides, some classical numerical methods, including the Schur methods, Newton's method, the fixed-point iterative methods and the doubling algorithms, are also applied to solve the NAREs, and the convergence analysis of these methods are given in details. For the doubling algorithms, we also give out the concrete parameter selection strategies. The numerical results show that our methods are efficient for solving the NAREs.

关键词： Complex nonsymmetric algebraic Riccati equation extremal solution classical numerical method doubling algorithm parameter selection strategy

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Solving large-scale nonsymmetric algebraic Riccati equations from two-dimensional transport models by doubling

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 391卷

作者： Weng, Peter Chang-Yi Dongguan Univ Technol Guangdong Taiwan Coll Ind Sci & Technol Dongguan 523808 Guangdong Peoples R China

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX - XD - AX + B = 0, with M equivalent to [D, -C;-B, A] is an element of R(n1+n2)x(n1+n2) being a nonsingular M-matrix, and A, D being sparse-like, with the products A(-1)u, A(-T)u, D(-1)v and D(-T)v computable in O(n(1)) or O(n(2)) complexity, for some vectors u and v. In the nonsymmetric algebraic Riccati equation arose from a two-dimensional transport model, B, C are low-ranked corrections of some invertible diagonal matrices. The structure-preserving doubling algorithm by Guo, Lin and Xu (2006) is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the sparse plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration (with n = max{n(1), n(2)}) and converges essentially quadratically, as illustrated by the numerical examples. (c) 2021 Elsevier B.V. All rights reserved.

关键词： doubling algorithm Krylov subspace M-matrix Nonsymmetric algebraic Riccati equation Transport theory

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