Browsing by Author "Guo, Chun-Hua"

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A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor
(2019-02-01)
Guo, Chun-Hua
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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
We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.
A POSITIVITY PRESERVING INVERSE ITERATION FOR FINDING THE PERRON PAIR OF AN IRREDUCIBLE NONNEGATIVE THIRD ORDER TENSOR
(2016-01-01)
Liu, Ching-Sung
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Guo, Chun-Hua
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Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter theta(k) in the kth iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that theta(k) - 1 near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be 1 directly) that for each iteration of the Ng-Qi-Zhou algorithm, which is linearly convergent for essentially positive tensors.
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)
;
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.
CONVERGENCE ANALYSIS OF THE DOUBLING ALGORITHM FOR SEVERAL NONLINEAR MATRIX EQUATIONS IN THE CRITICAL CASE
(2009)
Chiang, Chun-Yueh
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Chu, Eric King-Wah
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Guo, Chun-Hua
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Huang, Tsung-Ming
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Lin, Wen-Wei
In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate 1/2. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.
Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory
(2010-01-01)
Guo, Chun-Hua
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Lin, Wen-Wei
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應用數學系
;
Department of Applied Mathematics
We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory. (C) 2009 Elsevier Inc. All rights reserved.
Newton-Noda iteration for finding the Perron pair of a weakly irreducible nonnegative tensor
(2017-09-01)
Liu, Ching-Sung
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Guo, Chun-Hua
;
Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
We present a Newton-Noda iteration (NNI) for computing the Perron pair of a weakly irreducible nonnegative mth-order tensor A, by combining the idea of Newton's method with the idea of the Noda iteration. The method requires the selection of a positive parameter theta(k) in the kth iteration, and produces a scalar sequence approximating the spectral radius of A and a positive vector sequence approximating the Perron vector. We propose a halving procedure to determine the parameters theta(k), starting with theta(k) for each k, such that the scalar sequence is monotonically decreasing. Convergence of this sequence to the spectral radius of A (and convergence of the vector sequence to the Perron vector) is guaranteed for any initial positive unit vector, as long as the sequence {theta(k)} so chosen is bounded below by a positive constant. In this case, we always have theta(k) = 1 near convergence and the convergence is quadratic. Very often, the halving procedure will return theta(k)(= 1 i.e., no halving is actually used) for each k. If the tensor is semisymmetric, m >= 4, and theta(k) = 1, then the computational work in the kth iteration of NNI is roughly the same as that for one iteration of the Ng-Qi-Zhou algorithm, which is linearly convergent for the smaller class of weakly primitive tensors.
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
;
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
;
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
數學建模與科學計算所(含中心)
;
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 A STRUCTURED QUADRATIC EIGENVALUE PROBLEM BY A STRUCTURE-PRESERVING DOUBLING ALGORITHM
(2010)
Guo, Chun-Hua
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Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
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.
THE MATRIX EQUATION X + A(T)X(-1) A = Q AND ITS APPLICATION IN NANO RESEARCH
(2010)
Guo, Chun-Hua
;
Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
The matrix equation X + A(T)X(-1) A = Q has been studied extensively when A and Q are real square matrices and Q is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green's function calculations in nano research, but the matrix Q there is usually indefinite (so the matrix equation has no positive definite solutions), and one is interested in the case where the matrix equation has no positive definite solutions even when Q is positive definite. The solution of interest in this nano application is a special weakly stabilizing complex symmetric solution. In this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably.