Balanced Incomplete Factorization

Abstract

Abstract. In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU/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 [16]. In contrast to the RIF algorithm [9], 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. Although we describe the theory behind the factorization for general nonsymmetric matrices, in implementation and experiments we restrict for clarity and conciseness only to the case when the system matrix is symmetric and positive definite. In this case, we call the new approximate LDL T factorization 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 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.

Preconditioners based on the ISM factorization

“…The ISM factorization is strongly related with the LU factorization as was proved in [9]. Theorem 2.…”

“…So it is called Balanced Incomplete Factorization (BIF). This balance gives raise to robust preconditioners as numerical experiments presented in [9] confirm. In addition they also show that the BIF preconditioner has a very good behavior from the point of view of the selection of the dropping parameter.…”

“…In [9] the relation between the ISM and LDU factorizations, that was previously stated for pivots, was extended in a fruitful way as stated in Theorem 2. s . This situation is specially interesting when A is a symmetric positive definite matrix.…”

An overview of approximate inverse methods

“…This article is a computational comparison of some preconditioners alternatives based on approximations of inverse [3,4,7,8,9,10,13,14]. We have implemented sequential codes, in Matlab and/or C and tested these codes by solving some academic problems.…”

Greville’s method for preconditioning least squares problems

“…In [3], R. Bru, J. Martín, J. Mas and M. Tůma proposed Balanced Incomplete Factorization, which was based on inverse Sherman-Morrison formula. Suppose that the general nonsymmetric matrix A can be written as…”