Balanced Incomplete Factorization

• Authors: Rafael Bru, José MarÍn, José Mas;M. TŮMA
• Venue: SIAM Journal on Scientific Computing
• Year: 2008

In this paper we present a new incomplete factorization of asquare matrix into triangular factors in which we get standard $LU$or $LDL^T$ factors (direct factors) and their inverses (inversefactors) at the same time. Algorithmically, we derive this methodfrom the approach based on the Sherman-Morrison formula [R. Bru, J.Cerdán, J. Marín, and J. Mas, SIAM J. Sci.Comput., 25 (2003), pp. 701-715]. In contrast to the robustincomplete decomposition (RIF) algorithm [M. Benzi and M.Tůma, Numer. Linear Algebra Appl., 10 (2003), pp.385-400] the direct and inverse factors here directly influenceeach other throughout the computation. Consequently, the algorithmto compute the approximate factors may mutually balance dropping inthe factors and control their conditioning in this way. For thesymmetric positive definite case, we derive the theory and presentan algorithm for computing the incomplete $LDL^T$ factorization,and we discuss experimental results. We call this new approximate$LDL^T$ factorization the balanced incomplete factorization (BIF).Our experimental results confirm that this factorization is veryrobust and may be useful in solving difficult ill conditionedproblems by preconditioned iterative methods. Moreover, theinternal coupling of the computation of direct and inverse factorsresults in much shorter setup times (times to compute approximatedecomposition) than RIF, a method of a similar and very high levelof robustness. We also derive and present the theory for thegeneral nonsymmetric case, but do not discuss itsimplementation.