標題: ON A NONLINEAR MATRIX EQUATION ARISING IN NANO RESEARCH
作者: 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
關鍵字: nonlinear matrix equation;complex symmetric solution;weakly stabilizing solution;fixed-point iteration;structure-preserving algorithm;Green's function
公開日期: 2012
摘要: 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.
URI: http://hdl.handle.net/11536/15856
http://dx.doi.org/10.1137/100814706
ISSN: 0895-4798
DOI: 10.1137/100814706
期刊: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume: 33
Issue: 1
起始頁: 235
結束頁: 262
顯示於類別:期刊論文


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