Balanced Incomplete Factorization

In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard $LU$ or $LDL^T$ factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from 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 robust incomplete decomposition (RIF) algorithm [M. Benzi and M. Tůma, Numer. Linear Algebra Appl., 10 (2003), pp. 385–400] the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an 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 very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of the computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness. We also derive and present the theory for the general nonsymmetric case, but do not discuss its implementation.