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Optimal Parameters for doubling algorithms

数学研究 2017年第4期50卷 339-357页

作者： Tsung-Ming Huang Ren-Cang Li Wen-Wei Lin Linzhang Lu Department of Mathematics National Taiwan Normal University Taipei 116Taiwan Department of Mathematics University of Texas at Arlington P.O.Box 19408Arlington TX 76019 USA Department of Applied Mathematics National Chiao Tung UniversityHsinchu 300 Taiwan School of Mathematics and Computer Science Guizhou Normal University School of Mathematical Science Xiamen University Xiamen 361005 P.R.China

Inusing the structure-preservingalgorithm (SDA) [Linear Algebra Appl.,2005,vol.396,pp.55-80] to solve a continuous-time algebraic Riccati equation,a parameterdependent linear fractional transformation z → (z-γ)/(z + γ) is first performed in order to bring all the eigenvalues of the associated Hamiltonian matrix in the open left half-plane into the open unit *** closer the eigenvalues are brought to the origin by the transformation via judiciously selected parameter γ,,the faster the convergence of the doubling iteration will be later *** the first goal of this paper,we consider several common regions that contain the eigenvalues of interest and derive the best γ,so that the images of the regions under the transform are closest to the *** our second goal,we investigate the same problem arising in solving an M-matrix algebraic Riccati equation by the alternating-directional doubling algorithm (ADDA) [SLAM *** ***.,2012,vol.33,pp.170-194] which uses the product of two linear fractional transformations (z1,z2) → [(z2-γ2) / (z2 + γ1)] [(z1-γ1) / (Z1 + γ2)] that involves two *** examples are presented to demonstrate the efficiency of our parameter selection strategies.

关键词： doubling algorithm SDA ADDA optimal parameter algebraic Riccati equation

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A novel extended precise integration method based on Fourier series expansion for periodic Riccati differential equations

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OPTIMAL CONTROL APPLICATIONS & METHODS 2017年第6期38卷 896-907页

作者： Tan, Shu-Jun Zhou, Wen-Ya Peng, Hai-Jun Wu, Zhi-Gang Dalian Univ Technol Sch Aeronaut & Astronaut Dalian Peoples R China Dalian Univ Technol Dept Engn Mech Dalian 116024 Peoples R China Dalian Univ Technol State Key Lab Struct Anal Ind Equipment Dalian Peoples R China

A new, reliable algorithm for nonnegative, stabilizing solutions for the periodic Riccati differential equation (PRDE) is proposed based on Fourier series expansion and the precise integration method (PIM). Taking full advantages of periodicity, we expand coefficient matrices of the underlying linear time-varying periodic Hamiltonian system associated with the PRDE in Fourier series, and a novel extended PIM for the transition matrix of linear time-varying periodic systems is developed by combining the doubling algorithm with the increment-storage technique. This method needs to compute the matrix exponential and its related integrals only once for all evenly divided subintervals, which greatly improves the computational efficiency. Further, by introducing the Riccati transformation, a fast recursive formula for the PRDE is derived based on the block form of the transition matrix computed by the extended PIM. Finally, two numerical examples are presented to verify the numerical accuracy and efficiency of the proposed algorithm with compared results.

关键词： doubling algorithm Fourier series expansion periodic Riccati differential equation periodic system precise integration method

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Performance enhancement of doubling algorithms for a class of complex nonsymmetric algebraic Riccati equations

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IMA JOURNAL OF NUMERICAL ANALYSIS 2015年第1期35卷 270-288页

作者： Guo, Chun-Hua Liu, Changli Xue, Jungong Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Sichuan Univ Sch Math Sci Chengdu 610065 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China

A new class of complex nonsymmetric algebraic Riccati equations has been studied by Liu & Xue (2012. Complex nonsymmetric algebraic Riccati equations arising in Markov modulated fluid flows. SIAM J. Matrix Anal. Appl., 33, 569-596), which is related to the M-matrix algebraic Riccati equations. doubling algorithms, with properly chosen parameters, are used there for equations in this new class. It is pointed out that the number of iterations for the doubling algorithms may be relatively large in some situations. In this paper, we show that the performance of the doubling algorithms can often be improved significantly if a proper preprocessing procedure is used on the given Riccati equation. There are some difficult cases for which the preprocessing procedure does not help much by itself. We then propose new strategies for choosing parameters for doubling algorithms after using the preprocessing procedure. Numerical experiments show that our preprocessing procedure and the new parameter strategies are very effective.

关键词： algebraic Riccati equation M-matrix doubling algorithm

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A FAST algorithm FOR FAST TRAIN PALINDROMIC QUADRATIC EIGENVALUE PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第6期38卷 A3410-A3429页

作者： Lu, Linzhang Wang, Teng Kuo, Yueh-Cheng Li, Ren-Cang Lin, Wen-Wei Guizhou Normal Univ Sch Math & Comp Sci Guiyang Peoples R China Xiamen Univ Sch Math Sci Xiamen Peoples R China Nanchang Univ Sch Sci Dept Math & Numer Simulat Nanchang Jiangxi Peoples R China Nanchang Univ Sch Sci High Performance Comp Lab Nanchang Jiangxi Peoples R China Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Univ Texas Arlington Dept Math Arlington TX 76019 USA Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

In the vibration analysis of high speed trains arises such a palindromic quadratic eigenvalue problem (PQEP) (lambda(2) A(T)+ lambda Q + A)z = 0, where A;Q is an element of C-nxn have special structures: both Q and A are m x m block matrices with each block being k x k (thus n = m x k), and Q is complex symmetric and tridiagonal block-Toeplitz, and A has only one nonzero block in the (1, m)th block position which is the same as the subdiagonal block of Q. This PQEP has eigenvalues 0 and infinity each of multiplicity (m - 1) k just by examining A, but it is its remaining 2k eigenvalues, usually nonzero and finite but with an extreme wide range in magnitude, that are of interest. The problem is notoriously difficult numerically. Earlier methods that seek to deflate eigenvalues 0 and infinity first often produce eigenvalues that are too inaccurate to be useful due to the large errors introduced in the deflation process. The solvent approach proposed by Guo and Lin in 2010 changed the situation because it can deliver sufficiently accurate eigenvalues. In this paper, we propose a fast algorithm along the line of the solvent approach. The theoretical foundation of our algorithm is the connection we establish here between this fast train PQEP and a k x k PQEP defined by the subblocks of A and Q without any computational work. This connection lends itself to a fast algorithm: solve the k x k PQEP and then use its eigenpairs to recover the eigenpairs for the original fast train PQEP. The so-called alpha-structured backward error analysis that preserves all possible structures in the fast train PQEP to the extreme is studied. Finally numerical examples are presented to show the effectiveness of the new fast algorithm.

关键词： palindromic quadratic eigenvalue problem PQEP fast train nonlinear matrix equation solvent approach doubling algorithm

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On algebraic Riccati equations associated with regular singular M-matrices

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LINEAR ALGEBRA AND ITS APPLICATIONS 2016年 493卷 108-119页

作者： Guo, Chun-Hua Lu, Di Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada

We further study algebraic Riccati equations associated with regular singular M-matrices. An M-matrix M is said to be regular if Mv >= 0 for some v > 0, so every irreducible singular M-matrix is a regular singular M-matrix. We prove a property about the product of the minimal nonnegative solution of such an algebraic Riccati equation and the minimal nonnegative solution of its dual equation. In the critical case, we show that the alternating-directional doubling algorithm (which includes the structure-preserving double algorithm as a special case) has linear convergence with rate 1/2. The results enhance our understanding of the behaviour of doubling algorithms for finding the minimal nonnegative solutions. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Algebraic Riccati equation Singular M-matrix Minimal nonnegative solution doubling algorithm Convergence

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On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 288卷 18-32页

作者： Dong, Ning Yu, Bo Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Peoples R China Hunan Univ Technol Sch Sci Zhuzhou 412000 Peoples R China

For the large-scale nonlinear matrix equations with low rank structure, the well-developed doubling algorithm in low rank form (DA-LR) is known as an efficient method to compute the stabilizing solution. By further analyzing the global efficiency index constructed in this paper, we propose a tripling algorithm in low rank form (TA-LR) from two points of view, the cyclic reduction and the symplectic structure preservation. The new presented algorithm shares the same pre-processing complexity with that of DA-LR, but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset. Under the solvability condition, the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order. Numerical experiments including some from nano research show that the TA-LR is highly efficient to compute the stabilizing solution of large-scale nonlinear matrix equations with low rank structure. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Large-scale nonlinear matrix equations doubling algorithm Tripling algorithm Cyclic reduction Low rank Nano research

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A CONVERGENCE RESULT FOR MATRIX RICCATI DIFFERENTIAL EQUATIONS ASSOCIATED WITH M-MATRICES

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TAIWANESE JOURNAL OF MATHEMATICS 2015年第1期19卷 77-89页

作者： Guo, Chun-Hua Yu, Bo Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Hunan Univ Technol Sch Sci Zhuzhou 412000 Hunan Peoples R China

The initial value problem for a matrix Riccati differential equation associated with an M-matrix is known to have a global solution X(t) on [0,infinity) when X(0) takes values from a suitable set of nonnegative matrices. It is also known, except for the critical case, that as t goes to infinity X(t) converges to the minimal nonnegative solution of the corresponding algebraic Riccati equation. In this paper we present a new approach for proving the convergence, which is based on the doubling procedure and is also valid for the critical case. The approach also provides a way for solving the initial value problem and a new doubling algorithm for computing the minimal nonnegative solution of the algebraic Riccati equation.

关键词： Riccati differential equation M-matrix Global solution Convergence doubling algorithm Algebraic Riccati equation Minimal nonnegative solution

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SOLVING A STRUCTURED QUADRATIC EIGENVALUE PROBLEM BY A STRUCTURE-PRESERVING doubling algorithm

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2010年第5期31卷 2784-2801页

作者： Guo, Chun-Hua Lin, Wen-Wei Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

In studying the vibration of fast trains, we encounter a palindromic quadratic eigen-value problem (QEP) (lambda(2)A(T)+lambda Q+A)z = 0, where A, Q is an element of C(nxn) and Q(T) = Q. Moreover, the matrix Q is block tridiagonal and block Toeplitz, and the matrix A has only one nonzero block in the upper-right corner. So most of the eigenvalues of the QEP are zero or infinity. In a linearization approach, one typically starts with deflating these known eigenvalues for the sake of efficiency. However, this initial deflation process involves the inverses of two potentially ill-conditioned matrices. As a result, large error might be introduced into the data for the reduced problem. In this paper we propose using the solvent approach directly on the original QEP, without any deflation process. We apply a structure-preserving doubling algorithm to compute the stabilizing solution of the matrix equation X + A(T)X(-1)A = Q, whose existence is guaranteed by a result on the Wiener-Hopf factorization of rational matrix functions associated with semi-infinite block Toeplitz matrices and a generalization of Bendixson's theorem to bounded linear operators on Hilbert spaces. The doubling algorithm is shown to be well defined and quadratically convergent. The complexity of the doubling algorithm is drastically reduced by using the Sherman-Morrison-Woodbury formula and the special structures of the problem. Once the stabilizing solution is obtained, all nonzero finite eigenvalues of the QEP can be found efficiently and with the automatic reciprocal relationship, while the known eigenvalues at zero or infinity remain intact.

关键词： palindromic quadratic eigenvalue problem nonlinear matrix equation stabilizing solution structure-preserving doubling algorithm

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SOLVING LARGE-SCALE NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS BY doubling

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2013年第3期34卷 1129-1147页

作者： Li, Tiexiang Chu, Eric King-Wah Kuo, Yueh-Cheng Lin, Wen-Wei Southeast Univ Dept Math Nanjing 211189 Jiangsu Peoples R China Monash Univ Sch Math Sci Melbourne Vic 3800 Australia Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX - XD - AX + B = 0, with M = [D, -C;-B, A] is an element of R(n perpendicular to+n2)x(n perpendicular to +n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A(-1)u, A(-T)u, D(-1)v, and D(-T)v computable in O(n) complexity (with n = max{n(1), n(2)}), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392-412] 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 and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

关键词： doubling algorithm M-matrix nonsymmetric algebraic Riccati equation numerically low-ranked solution

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Solving large-scale continuous-time algebraic Riccati equations by doubling

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年第1期237卷 373-383页

作者： Li, Tiexiang Chu, Eric King-wah Lin, Wen-Wei Weng, Peter Chang-Yi Southeast Univ Dept Math Nanjing 211189 Jiangsu Peoples R China Monash Univ Sch Math Sci Clayton Vic 3800 Australia Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for A not explicitly computed but in the recursive form A(k) = A(k-1)(2) - (DkSk-1)-S-(1)[D-k((2))](T), with D-k((1)) and D-k((2)) being low-ranked and S-k(-1) being small in dimension. For the continuous-time case, the algebraic Riccati equation will be first treated with the Cayley transform before doubling is applied. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) computational complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 5.2, Example 3, of dimension n = 20 209 with 204 million variables in the solution X, was solved using MATLAB on a MacBook Pro within 45 s to a machine accuracy of O(10(-16)). (C) 2012 Elsevier B.V. All rights reserved.

关键词： Continuous-time algebraic Riccati equation doubling algorithm Krylov subspace Large-scale problem

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