IMPROVED BALANCED INCOMPLETE FACTORIZATION

• Published in: SIAM journal on matrix analysis and applications Vol. 31; no. 5; pp. 2431 - 2452
• Main Authors: BRU, Rafael, MARIN, José, MAS, José, TUMA, Miroslav
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Algebra; Algebraic geometry; Algorithms; Balancing; Decomposition; Exact sciences and technology; Factorization; Inverse; Inverse problems; Linear and multilinear algebra, matrix theory; Linear systems; Mathematical models; Mathematics; Matrix; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Optimization algorithms; Robustness; Sciences and techniques of general use; Studies; Sherman-Morrison formula; Decomposition method; Estimator robustness; Algorithmics; Iterative method; nonsymmetric matrices; Decomposition; preconditioned iterative methods; Implementation; 65F50; Inverse matrix; 65F10; 15A23; Primary; Inverse; 65H10; approximate inverses; Incomplete; Non symmetrical matrix; Secondary; incomplete decompositions; 15A06; Algorithm; Mass; Numerical analysis; Linear system; Linear algebra; Influence; Preconditioning; 65N22; sparse matrices
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/090747804

In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marin, J. Mas, and M. Tuma, SIAM J. Sci. Comput., 30 (2008), pp. 2302-2318]. The improvements are based on a deeper understanding of the inverse Sherman-Morrison (ISM) decomposition, and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors numerically influence each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems. [PUBLICATION ABSTRACT]