NDLI: On the Complexity of Some Hierarchical Structured Matrix Algorithms

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Xia, Jianlin
Copyright Year	2012
Abstract	In recent years, hierarchical structured matrices have been widely used in fast solutions of integral equations, PDEs, structured matrix (such as Toeplitz) problems, companion eigenproblems, etc. In this paper, we systematically study the complexity of some hierarchical structured matrix algorithms, in terms of hierarchically semiseparable (HSS) matrices. Several important aspects are considered. We perform detailed complexity analysis for some typical HSS algorithms, with the aid of certain graph techniques. This analysis helps us provide some significant improvements to classical HSS methods. One improvement is to propose more efficient HSS construction and solution algorithms. Another improvement is a recompression procedure which reorthonormalizes some HSS generators and converts noncompact HSS forms to compact ones. A precise theoretical justification of the compactness is also given. The third improvement is to relax the rank requirement in HSS operations. Unlike many classical HSS methods where the appropriate off-diagonal (numerical) ranks are often required to be bounded, we allow the ranks to increase. Certain general rank patterns are proposed, so that similar performance can be achieved with the maximum rank unbounded. These improvements significantly enhance both the efficiency and the applicability of HSS methods. Numerical examples from some applications are included to support the analysis.
Starting Page	388
Ending Page	410
Page Count	23
File Format	PDF
ISSN	08954798
DOI	10.1137/110827788
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	2
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-05-17
Access Restriction	Subscribed
Subject Keyword	HSS construction and ULV factorization Other matrix algorithms relaxation of rank requirement hierarchically semiseparable (HSS) matrix Direct methods for linear systems and matrix inversion recompression rank structure
Content Type	Text
Resource Type	Article
Subject	Analysis

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