Browsing by Author "Kuo, Yueh-Cheng"

Now showing 1 - 12 of 12

A FAST ALGORITHM FOR FAST TRAIN PALINDROMIC QUADRATIC EIGENVALUE PROBLEMS
(2016-01-01)
Lu, Linzhang
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Wang, Teng
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Kuo, Yueh-Cheng
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Li, Ren-Cang
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Lin, Wen-Wei
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.
A hyperplane-constrained continuation method for near singularity in coupled nonlinear Schrodinger equations
(2010-05-01)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Shieh, Shih-Feng
;
Wang, Weichung
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應用數學系
The continuation method is a useful numerical tool for solving differential equations to obtain multiform solutions and allow bifurcation analysis. However, when a standard continuation method is used to solve a type of time-independent m-coupled nonlinear Schrodinger (NLS) equations that can be used to model nonlinear optics, nearly singular systems arise in the computations of prediction and correction search directions and detections of bifurcations. To overcome the stability and efficiency problems that exist in standard continuation methods, we propose a new hyperplane-constrained continuation method by adding additional constraints to prevent the singularities while tracking the solution curves. Aimed at the 3-coupled DNLS equations, we conduct theoretical analysis to the solutions and bifurcations on the primal stalk solution curve. The proposed algorithms have been implemented successfully to demonstrate numerical solution profiles, energies, and bifurcation diagrams in various settings. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrodinger equations
(2009-11-01)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Shieh, Shih-Feng
;
Wang, Weichung
;
應用數學系
We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrodinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly. (C) 2009 Elsevier Inc. All rights reserved.
A structure preserving flow for computing Hamiltonian matrix exponential
(2019-11-01)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Shieh, Shih-Feng
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應用數學系
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Department of Applied Mathematics
This article focuses on computing Hamiltonian matrix exponential. Given a Hamiltonian matrix H, it is well-known that the matrix exponential e(H) is a symplectic matrix and its eigenvalues form reciprocal (lambda, 1/(lambda) over bar). It is important to take care of the symplectic structure for computing e(H). Based on the structure-preserving flow proposed by Kuo et al. (SIAM J Matrix Anal Appl 37:976-1001, 2016), we develop a numerical method for computing the symplectic matrix pair (M, L) which represents e(H).
Complex symmetric stabilizing solution of the matrix equation X plus A(T)X(-1)A = Q
(2011-09-15)
Guo, Chun-Hua
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Kuo, Yueh-Cheng
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Lin, Wen-Wei
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數學建模與科學計算所(含中心)
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Graduate Program of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics
We study the matrix equation X + A(T)X(-1)A = Q, where A is a complex square matrix and Q is complex symmetric. Special cases of this equation appear in Green's function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The stabilizing solution is the solution of practical interest. In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption that is satisfied for the two special applications. Moreover, our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm. (C) 2011 Elsevier Inc. All rights reserved.
Continuation methods for computing Z-/H-eigenpairs of nonnegative tensors
(2018-10-01)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Liu, Ching-Sung
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應用數學系
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Department of Applied Mathematics
In this paper, a homotopy continuation method for the computation of nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair. Additionally, using degree analysis, we show that the number of positive Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented. (C) 2018 Elsevier B.V. All rights reserved.
Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research
(2012-11-01)
Guo, Chun-Hua
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Kuo, Yueh-Cheng
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Lin, Wen-Wei
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應用數學系
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Department of Applied Mathematics
The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form X + (C* + i eta D*)X-1(C + i eta D) = R+i eta P. where R and P are Hermitian, P + lambda D* + lambda D-1 is positive definite for all lambda on the unit circle, and eta -> 0(+). For each fixed eta > 0, we show that the required solution is the unique stabilizing solution X-eta. Then X-center dot = lim(eta -> 0+) X-eta is a particular weakly stabilizing solution of the matrix equation X + C*X-1C = R. In nano applications, the matrices R and C are dependent on a parameter, which is the system energy E. In practice one is mainly interested in those values of g for which the equation X + C*X-1C = R has no stabilizing solutions or, equivalently, the quadratic matrix polynomial P(lambda) = lambda C-2* - lambda R + C has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute X-eta efficiently even for very small values eta, thus providing good approximations to X-*. We also explain how the solution X-* can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of P(lambda) should be included in the computation. In some applications the matrices C, D, R, P have very special sparsity structures. We show how these special structures can be exploited to drastically reduce the complexity of the doubling algorithm for computing X-eta. (C) 2012 Elsevier B.V. All rights reserved.
ON A NONLINEAR MATRIX EQUATION ARISING IN NANO RESEARCH
(2012)
Guo, Chun-Hua
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Kuo, Yueh-Cheng
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Lin, Wen-Wei
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數學建模與科學計算所(含中心)
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Graduate Program of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics
The matrix equation X + A (vertical bar) X(-1)A - Q arises in Green's function calculations in nano research, where A is a real square matrix and Q is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution X-*, which is the limit of the unique stabilizing solution X-eta of the perturbed equation X + A(inverted perpendicular) X(-1)A = Q + i eta I, as eta -> 0(+). It has been shown that a doubling algorithm can be used to compute X-eta efficiently even for very small values of eta, thus providing good approximations to X-*. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing X-eta for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant and of their generalizations. We also show that the imaginary part X-I of the matrix X-* is positive semidefinite and we determine the rank of X-I in terms of the number of unimodular eigenvalues of the quadratic pencil lambda(2)A(inverted perpendicular) - lambda Q + A. Finally we present a new structure-preserving algorithm that is applied directly on the equation X + A(inverted perpendicular) X(-1)A = Q. In doing so, we work with real arithmetic most of the time.
Solving large-scale nonlinear matrix equations by doubling
(2013-08-15)
Weng, Peter Chang-Yi
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Chu, Eric King-Wah
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Kuo, Yueh-Cheng
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Lin, Wen-Wei
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應用數學系
We consider the solution of the large-scale nonlinear matrix equation X + BX-1 A - Q = 0, with A, B, Q, X is an element of C-nxn, and in some applications B = A(star) (star = T or H). The matrix Q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system Qv = r in O(n) computational complexity. Furthermore, B and A are respectively of ranks ra, rb << n. The type 2 structure-preserving doubling algorithm by Lin and Xu (2006) [241 is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the lowrank updates of various iterates. Two resulting large-scale doubling algorithms have an O((r(a) + r(b))(3)) computational complexity per iteration, after some pre-processing of data in O(n) computational complexity and memory requirement, and converge quadratically. These are illustrated by the numerical examples. (C) 2012 Elsevier Inc. All rights reserved.
SOLVING LARGE-SCALE NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS BY DOUBLING
(2013)
Li, Tiexiang
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Chu, Eric King-Wah
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Kuo, Yueh-Cheng
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Lin, Wen-Wei
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應用數學系
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.
STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS
(2016-01-01)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Shieh, Shih-Feng
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應用數學系
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Department of Applied Mathematics
We construct a nonlinear differential equation of matrix pairs (M(t); L(t)) that are invariant (structure-preserving property) in the class of symplectic matrix pairs {(M, L) = ((sic)S-2, (sic)S-1)| X = [X-i j](1
The asymptotic analysis of the structure-preserving doubling algorithms
(2017-10-15)
Kuo, Yueh-Cheng
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Lin, Wen-Wei
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Shieh, Shih-Feng
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應用數學系
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Department of Applied Mathematics
This paper is the second part of [15]. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similar transformation introduced by [18] to reduce H to a Hamiltonian Jordan canonical form J. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e(Jt). The convergence of the SDA as well as its rate can thus result from the study of the structure preserving flows. A complete asymptotic dynamics of the SDA is investigated, including the linear and quadratic convergence studied in the literature [3,12,13]. (C) 2017 Elsevier Inc. All rights reserved.