Browsing by Author "Weng, Peter Chang-Yi"

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Large-scale Stein and Lyapunov equations, Smith method, and applications
(2013-08-01)
Li, Tiexiang
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Weng, Peter Chang-Yi
;
Chu, Eric King-wah
;
Lin, Wen-Wei
;
應用數學系
We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with A(k) = A(2k) not explicitly computed but in the recursive form A(k) = A(k-1)(2), and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.
Solving large-scale continuous-time algebraic Riccati equations by doubling
(2013-01-01)
Li, Tiexiang
;
Chu, Eric King-wah
;
Lin, Wen-Wei
;
Weng, Peter Chang-Yi
;
應用數學系
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.
Solving large-scale nonlinear matrix equations by doubling
(2013-08-15)
Weng, Peter Chang-Yi
;
Chu, Eric King-Wah
;
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
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.