Title: Solving large-scale nonlinear matrix equations by doubling
Authors: Weng, Peter Chang-Yi
Chu, Eric King-Wah
Kuo, Yueh-Cheng
Lin, Wen-Wei
Department of Applied Mathematics
Keywords: Doubling algorithm;Green's function;Krylov subspace;Leaky surface wave;Nano research;Nonlinear matrix equation;Surface acoustic wave
Issue Date: 15-Aug-2013
Abstract: 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.
URI: http://dx.doi.org/10.1016/j.laa.2012.08.008
ISSN: 0024-3795
DOI: 10.1016/j.laa.2012.08.008
Volume: 439
Issue: 4
Begin Page: 914
End Page: 932
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