Browsing by Author "Chu, Eric King-Wah"

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A NOTE ON GOLDEN MEANS, NONLINEAR MATRIX EQUATIONS AND STRUCTURED DOUBLING ALGORITHMS
(2010-06-01)
Chiang, Chun-Yueh
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Chu, Eric King-Wah
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Lin, Wen-Wei
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應用數學系
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Department of Applied Mathematics
Several beautiful formulae for the solutions of some nonlinear matrix equations were proposed by Yongdo Lim in 2007, in terms of the matrix golden means. Numerically, these formulae will not be applicable when some matrices involved are ill-conditioned. In this note, we propose to partially fill in this gap of applicability with the structured doubling algorithm, under some favourable conditions. We also discuss how some pre-processing or scaling procedures can be applied to the matrix equations, to improve their condition. More generally, we also explore the possibility of computing the matrix golden mean using structured doubling algorithms. Some numerical examples will be presented for illustrative purposes.
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.
DOUBLING ALGORITHM FOR THE DISCRETIZED BETHE-SALPETER EIGENVALUE PROBLEM
(2019-09-01)
Guo, Zhen-Chen
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Chu, Eric King-Wah
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Lin, Wen-Wei
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應用數學系
;
Department of Applied Mathematics
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.
On the star-Sylvester equation AX +/- X-star B-star = C
(2012-05-01)
Chiang, Chun-Yueh
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Chu, Eric King-Wah
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Lin, Wen-Wei
;
應用數學系
;
Department of Applied Mathematics
We consider the solution of the star-Sylvester equations AX +/- (XB star)-B-star = C, for star = T, H and A, B, is an element of C-nxn, and the related linear matrix equations AXB(star) +/- X-star = C, AXB(star) +/- (CXD star)-D-star = E and AX +/- X(star)A(star) = C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m = n but the rectangular cases will be considered. (C) 2012 Elsevier Inc. All rights reserved.
Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model
(2011-10-01)
Li, Tiexiang
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Chu, Eric King-Wah
;
Juang, Jong
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Lin, Wen-Wei
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應用數學系
For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B(-) -XF(-) -F(+) X + XB(+)X = 0, where F(+/-) = (I -F) D(+/-) and B(+/-) = BD(+/-) with positive diagonal matrices D(+/-) and possibly low-ranked matrices F and B. We prove the existence of the minimal positive solution X* under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newton's method and the doubling algorithm. We shall also study several special cases. For example when B and F are low ranked then X* = Gamma circle(Sigma(r)(i=1)U(i)V(i)(T)) with low-ranked U(i) and V(i) that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.
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.
The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs
(2011-02-15)
Chu, Eric King-Wah
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Fan, Hung-Yuan
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Jia, Zhongxiao
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Li, Tiexiang
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Lin, Wen-Wei
We extend the Rayleigh-Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh-Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors. (C) 2010 Elsevier B.V. All rights reserved.